Application of Optimization Technique in the Powder Compaction and Sintering Processes

Transcription

1 Applaton of Optzaton Tehnque n the Powder Copaton and Snterng Proesses Young-Sa Kwon and Suk-Hwan Chung CetaTeh, In. TIC, 96-3, Seonn-r, Yonghyeon-yon, Saheon, Kyongna, , Korea Chantal Bnet, Ru Zhang, Renata S. Engel, Nholas J. Salaon and Randall M. Geran Center for Innovatve Sntered Produts, P/M Lab, 47 Researh West The Pennsylvana State Unversty, Unversty Park, PA , USA 9ABSTRACT In general, the shrnkage of the powder opat durng the snterng proess s not unfor due to the non-unfor densty dstrbuton of the powder opat durng the opaton proess. We have developed an optzaton progra for the powder opaton and snterng proesses. The optzaton progra s oposed of fnte eleent analyss and optzaton routnes for powder opaton and snterng proess. We apply the optzaton progra to fnd the optu proess varables for nzng the densty varaton of the ylndral shape and the T-shape powder opats durng the de opaton proess. We an verfy our optzaton progra fro these applatons. Fnally, shrnkage dstorton durng the snterng proess was analyzed for the powder opat ade fro the optu de opaton proess. INTRODUCTION Powder etallurgy (P/M) s a proess that aheves the hgh ehanal strength of opated powder by the snterng proess of the fored shape. The powders are usually fro a xture of etal powders and era powders. Often there are addtons to the forng proess suh as lubrants for de opaton. The range of sntered oponents by powder etallurgy vares wdely n aterals and shapes, and the proessng tehnque s apable of ahevng net-shape forng, thus powder etallurgy has been developed rapdly. In reent years, sntered produts have been frequently adopted n the autooble ndustry for the purpose of redung ost and weght []. Sne the developent of powder neton oldng proess the applatons of ths tehnque has been nreased partularly n produng relatvely oplex shaped oponents []. However, up to now ost of forng proess s arred out by the de opaton. Sne there s no restrton on the shape hange of a powder opat durng the snterng proess, the dstorton of a opated body due to the non-unfor densty dstrbuton resultng fro the de opaton has presented barrers to the full utlzaton of the proess. Therefore, fnshng steps suh as ahnng ay be needed to aheve the desred shape after the snterng proess. It s expetable that frton between the partles and the tool surfae, and the non-unfor stress fro the tools along the tool-ontated surfae of opaton body ause ths non-unforty. Up to now, efforts to redue densty gradents have anly onentrated on redung the frton between powder and toolng by usng lubrants or by redung the frton effet along the surfae of the powder body, for exaple, through the double aton opaton. Moreover, due to the absene of effent tool desgn these efforts have 9-3

2 heavly depended on engneer s experene. However, the proble of non-unforty annot be elnated ust by the treatent of redung frton beause the desred shape opat annot be opated wth unfor pressure due to other geoetral onstrants. Therefore, the optzaton proedures of the de opaton and snterng proesses an provde the best ethod to produe net shape PM parts. Durng the de opaton proess, the optzaton tehnque an provde the optu proess ondtons to aheve the ost unfor densty dstrbuton nsde the powder opat. The applaton of the optzaton tehnque to the snterng proess wll provde the optu de shape for the de opaton by onsderng the dstorton of the sntered part. Ultately, t s very portant to obtan the optu de shape and optu proessng ondtons for the desred shape part by the fully oupled optzaton proedures. To apply the optzaton tehnque on the P/M proesses, the onsttutve odels to analyze the de opaton and snterng behavors are very portant. The analyss of the de opaton and snterng requres the approprate plast yeld funton for porous ateral and the approprate snterng odel for the dfferent densfaton ehans suh as nterpartle dffuson and dffusonal flow[3-4]. Up to now, dverse optzaton tehnques nludng a bakward trang shee[5], genet algorth[6], and dervatve based approahes[7-9], for forng proesses suh as extruson, rollng, forgng and powder forgng have been used. Aong these tehnques, the dervatve based approahes are generally superor to others onsderng the qualty of the desgn and te onsupton. In usng dervatve based approahes, the alulaton of the desgn senstvty s very portant. To alulate the desgn senstvty, the analytal ethod, the adont varable ethod, the dret dfferenta ton ethod, and the fnte dfferene ethod, are beng used. In the dret dfferentaton ethod the te ost for alulatng the desgn senstvty s nreased wth the nuber of desgn paraeters beause the alulaton of atrx equatons s related to the nuber of the desgn paraeters needed to alulate the dervatve of the fnte eleent solutons wth respet to the desgn paraeters. In the adont varable ethod the equaton tself s very hard to derve espeally n non-steady forng. However, n ths study the adont varable ethod wll be adapted for redung the te ost through the dervaton of the equatons for the adont varable ethod n non-steady forng of porous ateral. For the frst step of developng the optzaton tool n powder etallurgy only the de opaton proess s onsdered. Sne n general the de opaton proess s arred out n roo teperature, the ateral property depends on relatve densty and effetve stran of the base ateral. Therefore, the onsderaton of the dervatves related to relatve densty and effetve stran of the base ateral should be nluded n the dervaton of the adont varable ethod. The present paper reports the applaton of the optzaton tehnque on the de opaton proess. To analyze the de opaton proess, we used Sha and Oyane yeld funton []. To verfy the optzaton progra we apply the optzaton progra to fnd the optu proess varables for nzng the densty varaton of the ylndral shape and the T-shape powder opats durng the de opaton proess. Fnally, shrnkage dstorton durng the snterng proess was analyzed for the powder opat ade fro the optu de opaton proess. To analyze the snterng proess for the de opated part, we proposed the reep densfaton equatons based on Ashby s dffusonal reep densfaton odel [3-4]. FINITE ELEMENT FORMULATION 9-3

3 Consttutve odel In the de opaton proess, the deforaton behavor of the powder s based on the yeld rteron. Unlke bulk solds, the yeld rteron nludes the hydrostat pressure due to volue hange n opaton: ' AJ + BJ = () σ where, A and B are the funtons of relatve densty. σ s the effetve stress of powder ontnuu, whh an be expressed by the funton of the effetve stress of bulk ateral and relatve densty as follows. σ = β ( ρ) σ ( ε, ε, T ) () where, ε, respetvely. ε, and T are the effetve stran and effetve stran rate of base ateral and teperature, Fro the unaxal tenson or opresson test, the relaton of A and B s A = 3( B). Aordng to the defnton of A and β dverse rtera have been suggested [], where n ths study Sha and Oyane s rteron s used []. 3 A =, ( ρ) 5 ρ β = ( ρ) (3) Boundary Value Proble Consder a deforng body Ω wth the traton h presrbed on a part Γ h of the surfae Γ and the veloty u = u presrbed on a part. Let Γ be the reander of the surfae and assue that represents the tool-workpee nterfae. The boundary value proble assoated wth the urrent te n the nonsteady state plast deforaton proess an be gven as follows: Fnd a veloty feld - Mass balane: u satsfyng ρ ρ ε v = (4) - Equlbru equaton: σ, + f = (5) - Yeld rteron: - Consttutve equaton: 3 A( ρ) ( ) σ 3 ' A ρ J + J = (6) σ = + (7) ' pδ σ where 9-33

4 σ ' σ = A ε ε (8) p = σ εv 3(3 A) ε (9) ' ' ' ε = ε ε + ε v () A 3(3 A) & βρ ( ) ε = & ε () ρ - Boundary ondtons: σn = h on h u = u on u Γ () Γ (3) D δ σn = ξ un un + on Γ (4) Δt σt = μσ ng( Δut ) on Γ (5) where ξ s a penalty onstant leadng to onfor the noral veloty of the de-ontated node to the D noral tool veloty. n s a unt vetor of the noral dreton and u n s the noral tool veloty. In Eq. (4) δ represents the noral dstane of a de -ontated node and tool surfae and Δ t s the te step sze of non-steady state fnte eleent sulaton. In order to onfor the poston of a deontated node and de surfae durng fnte eleent sulaton, the proeton ethod, whh proets the poston of a node onsdered to be ontated wth the de on the de surfae after eah nreental step. However the proeton ethod hanges the te step sze n order to onfne the aount of proeton on de surfae. The hange of te step sze ay ause extraordnary te onsupton for the fnte eleent sulaton and oplate dervng the equatons for the adont varable ethod. Therefore n ths study a new ontat algorth suggested by Fourent et.[] s used, where new ontat algorth keeps the te step sze onstant. In Eq. (5) Δ u t represents the dfferene of tangental veloty between the de -ontated node and the de, t represents the dreton of Δ u t and μ represents Coulob frtonal oeffent. The funton g was suggested by Chen and Kobayash[3] to deal wth both stkng and sldng frton as follows. Δu g( Δu) = tan (6) π a where a s very sall postve value. Fnte Eleent Forulaton An ntegral stateent equvalent to the above boundary value proble ay be as follows: u satsfyng the veloty presrbed boundary ondtons, fnd Aong the veloty felds satsfes the followng varatonal equaton for arbtrary funtons ω (that vansh on Γ u ). u, whh 9-34

5 ' ' σ ω dω pωdω fωdω h ω d α α Γ Ω ξ u Γ n Ω u D n Ω α Γ δ + ω ndγ μξ un u Δt Γ hα D n δ + g( δut ) ωtdγ = Δt (7) ' ωkk where ω = ω + ω ) and ω = ω 3 δ. (,, In Equaton (7), the fnte eleent approxaton on the oordnate x and veloty u results n the followng fnte eleent equaton. R = R( p, X, ρ, ε, V ) (8) where R represents resdual of the fnte eleent equaton, equvalent to the state equaton n optzaton proble, and p s the desgn paraeter. X and V represent the oordnate and veloty of a nodal pont, respetvely, and ρ and ε represent relatve densty and effetve stran of the base ateral, respetvely. Varables affetng the fnte eleent equatons an be lassfed nto 3 ategores as defned by Chung, et al. The desgn paraeters, p, should be seleted by the proess desgners, state varables, S, are the solutons of the fnte eleent analyss, and governng paraeters, L, that are the funton of the desgn paraeters and deterne the state varables. In the ategory of the desgn paraeters n powder etallurgal proess, the upper and lower punh speed, de shape, frton, and reduton rato n de opaton proess and heatng yle and frton n snterng proess an be nluded. Sne only the veloty feld s the soluton of the fnte eleent analyss of porous ateral, the veloty feld of eah nreental step s the state varable. The last governng paraeters nlude the oordnates of nodal ponts, relatve densty, effetve stran of base ateral, et. The desgn paraeter affets both the state varables and governng paraeters, where t an dretly affet the ntal values of governng paraeters lke the ntal shape of powder body and also ndretly affet the state varables and governng paraeters by affetng the fnte eleent equaton lke punh speed, de shape, frton, et. The governng paraeters of eah nreental step are updated as follows usng the veloty feld obtaned fro the fnte eleent analyss. X X + V Δt (9) = ( ) Δ v t ρ = ρ e ε = ε + ε Δt ( ) ( ) ( ) () ε () Calulaton of the Desgn Senstvty The entre optzaton proble results n the proble of fndng desgn paraeters that nze the obetve funton. In ths study, two knds of obetve funtons, relatve densty dstrbuton and forng energy, are onsdered. In the snterng proess the body s free therefore the deforaton s not onfned. Dstorton of the powder opat durng snterng proess an our due to the non-unforty of the relatve densty dstrbuton n the powder opat. In general, the nzaton of forng energy s often onsdered to be the obet funton n etal forng proess. Through the nzaton of forng energy, produton ost an be redued and tool lfe an be nreased by redung tool wear. In addton dfferent obetve funtons an be onsdered and expressed as follows. Φ ( p) = Φ( p, X, ρ, ε, V ) () Also, fro the ondton that N state equatons should be satsfed, the followng Lagrangan funtonal 9-35

6 an be ntrodued. N = ( ( ) ) Λ ( p, λ) =Φ ( p) + λr p, X, ρ, ε, V (3) The dervatves of governng paraeters wth respet to the desgn paraeter, d ( ) dp ε are obtaned as follows fro Equatons (9)-(). dx dx = + dv Δt dp dp = dp ( ) ( ) dρ v v dρ & ε dx & ε dvk = A ρ Δt ρ Δt Δt dp dp X dp X dp ρ = ( & ε ) V k = = k = v dv Δ t dp ( ) ( ) ( ) ( ) d ε d ε & ε & ε ( & εv) dx = + ρ Δt Δt dp dp X X dp k k = = ρ k = k & ( ε ) ( ) dρ & ε dv + A Δ t + Δt Δt k k = ρ dp = X k = dp & ( ε ) k ( & ε v ) ρ dv Δt Δt Δt k l k l = ρ k= Xk l= dp & ( ε ) ( ε ) ( ) v k dv & & ε k ρ dv Δt Δ t + Δt k = ρ k= Vk dp = V dp dx dp, ρ dp d (4) (5) (6), and where dδ t dp an be negleted beause the te step sze Δ t does not hange aordng to the ontat algorth used n ths study. And A s defned as follows. A = e ( ε ) v = Usng the dervatves of the governng paraeter wth respet to the desgn paraeter n equatons (4)- (6), the desgn senstvty an be alulated as follows. where Δt ( ε ) d R dx dρ d (8) dp p p dp dp dp N ρ ε ρ Λ Φ X Σ +Σ ε = + λ +Σ + +Σ = ρ Φ Σ = + XX R (7) N N λ =+ X =+ X (9) Φ Σ = + R N N ρ ρ λ ρ =+ ρ =+ ρ (3) 9-36

7 Σ ε = N Φ N + λ ( ε ) = + ( ε ) = + ( ε ) R N ε X ε t =+ X (3) Σ = & Δ Σ (3) ( εv ) Σ = & Δ Σ (33) Σ N ε ρx ε ρ t =+ X ε ρ = N = + ( ) ε ε ρδt Σ (34) ρ X XX ρx εx ερx Σ = Σ Σ +Σ Σ (35) and the adont varable of -th nreental step λ s alulated as follows. ( & ε ) ( & ε ) R Φ ρ λ = Σ Δ t + Δt ( Σ +Σ ) ΔtΣ V V V V X v ε ρ ε (36) ANALYSIS FOR SINTERING PROCESS Creep Stran Rate Most of snterng densfaton an our by dffuson of ateral fro the ontat areas between powder partles, suh that the partles ove loser together and the pores fll up [3-4, 4-5]. However, t an happen that the gran sze of the powder s sgnfantly saller than the partle sze tself. In tradtonal P/M proess nludng de opaton and snterng, ost of used powders are larger than powder for the powder neton oldng proess. Then, we have to onsder a new deforaton ehans dffusonal flow to analyze the snterng densfaton behavor of de opated part. It s onvenent to dvde the proess of densfaton of powder opats nto two stages [3-4,4-5] followng the densfaton ehans. Stage s the ntal stage of densfaton and stage s the fnal stage of densfaton. The transent relatve densty of stage and stage s.9[4]. The general for of reep stran rate ε for porous powder opats an be wrtten [5] ε = σ + ( σ σ s ) δ (37) μ 3Κ where σ and σ are devator and hydrostat( = σ kk / 3 ) parts of the true stress σ, δ s Kroneker delta, and σ s s the snterng potental. Also, μ and Κ are the effetve shear vsosty and effetve bulk vsosty of porous powder opats. In Eq. (37), the effetve bulk vsosty Κ an be represented by funtons of relatve densty. The effetve bulk vsosty durng the ntal stage under dffusonal flow by gran boundary dffuson an be wrtten[4] where ρ, ρ, k, T, G, Ω, δ and / 3 ρ ρ ρ ktg Κ = ρ (38) 45 ΩδD b D denote the relatve densty, the ntal relatve densty, b 9-37

8 Boltzann s onstant, teperature, gran sze, ato volue, the boundary thkness and the gran boundary dffuson oeffent. In Eq. (38), the produt of the boundary thkness δ and the gran boundary dffuson oeffent D b an be wrtten δ D = δ D exp( Q / R T ) (39) b b b where δ D, Q and R are the pre-exponent for boundary dffuson, the atvaton energy for boundary b b dffuson and gas onstant. The effetve bulk vsosty durng the fnal stage under dffusonal flow by gran boundary dffuson an be wrtten [4] 3 ρ ktg Κ = (4) ρ Ωδ D b In Eqs. (39) and (4), ato volue Ω, the boundary thkness δ and the gran boundary dffuson oeffent D b are ateral onstants. The easurng of the values of ato volue Ω, the boundary thkness δ and the gran boundary dffuson oeffent D b are very dffult and aren t known for all aterals. Moreover, even the known values annot be used wthout odfyng the values [4]. In ths work, new ateral paraeters β and β f are defned for the pratal use. Thus, 45Ω δ Db Ωδ Db β( T) =, β f ( T) = (4) k k Fro Eqs. (39) and (4), the ateral paraeters β and β f an be sply wrtten β = β exp( Q / R b T), β f = β exp( Qb / R T) (4) where β and β f are ateral onstants whh an be deterned fro experental data. Fro Eqs. (37), (38), (4) and (4), the effetve bulk vsosty durng ntal stage Κ f an be wrtten / Κ and fnal stage 3 3 ρ ρ TG ρ TG Κ = ρ, f ρ Κ = β ρ. (43) β f The effetve bulk vsostes Κ and Κ f n Eq. (43) ust have sae values at the transent relatve densty.9 for ontnuty. The ateral onstant β an be expressed n ters of the ateral onstant β by posng the ontnuty ondton. Thus, β =.736 D.9 o Do / β. (44) Creep Stran Rate for Snterng When onsderng pressureless snterng, the frst ter of the rght sde n Eq. (37) an be gnored and the reep stran rate ε an be expressed only the effetve bulk vsosty Κ ter. In order to onsder the frton between the saple and the saple support plate and gravty of the saple durng snterng, the effetve shear vsosty ter n Eq. (37) s requred. In ths work, the effetve shear vsosty s sply assued n ters of the effetve bulk vsosty and the relaton an be wrtten [6] μ Κ =.6. (45) The reep stran rate ε durng the ntal stage an be obtaned by substtutng Eqs. (43) and (45) nto Eq. (37). Thus, 9-38

9 The reep stran rate nto Eq. (37). Thus, ( σ σ ) 3σ + /. s δ ( ρ ) β ε = / 3. (46) 3.6 ρ ( ρ ρ) T G ε durng the fnal stage an also be obtaned by substtutng Eqs. (43) and (45) ( σ σ ) 3 +. σ s δ ρ β f ε = 3. (47) 3.6 ρ TG Assung that a bulk ateral s nopressble, the densfaton rate ρ an be wrtten ρ = ρ ε kk. (48) The densfaton rate an be obtaned for overall relatve densty regon by substtutng Eqs. (46) and (47) nto Eq. (48). Gran Growth and Snterng Potental The gran growth rate G an be wrtten [4-5] k (49) G = G where k = k exp( Qs R T ). (5) Here, and k denote the gran growth exponent and pre-exponent for gran growth and atvaton energy for gran growth. Q s s the The snterng potental s the free drvng fore for snterng due to nterfaal energy of pores and gran boundares. Durng the ntal stage, the snterng potental an be wrtten [4] 3γ ρ ρ σ = ρ (5) s R ρ and durng the fnal stage γ ρ σ s =, r = R 6 (5) r ρ where γ s the surfae energy and r s the pore radus. /3 APPLICATION OF THE OPTIMIZATION Optzaton of De Copaton for Cylndral Shape Part To verfy the optzaton progra, the upper and lower punh speeds are optzed durng the de opaton of the ylndral and T-shaped opat to obtan the ost unfor relatve densty dstrbuton after opaton. The obetve funton for the unfor relatve densty s as follows. Φ ( ) = ρ ρ dω (53) Ω f where Ω f represents the analyss doan n fnal state and ρ, average relatve densty, s alulated below. 9-39

10 ρ = Ω f Ω f ρdω dω The su of the upper and lower punh speeds should be ontrolled onstant so that the fnal shape of powder opat s kept unhanged. Therefore, ust one desgn paraeter, the dfferene of the upper and lower punh speed, an be seleted, where the upper and lower punh speeds are alulated as follows. upper upper lower lower VD = VD + p, VD = VD p (55) upper lower where V D and V D are ntal guesses for the upper and lower punh speeds, respetvely, where the values are /se and /se. Also, P denotes the desgn varable. The ateral used n ths sulaton s 36L Stanless steel wth the relaton of stress-stran as follows [7]..678 σ= ε [MPa] (56) The ntal relatve densty s.457 and The theoretal densty and tap densty are 7.87 g/ 3 and 3.6 g/ 3. We assue the de daeter s 3.75 and the ntal relatve densty of the powder opat s.457. Also, the frton oeffent used n the fnte eleent sulaton s.8. Fgure shows the fnte eleent esh and the boundary ondtons. (54)..9 Obet funton teraton nuber Fgure. Fnte eleent esh and boundary ondtons for the ylndral opat Fgure. The varaton of the obet funton durng the optzaton teraton ondtons for the ylndral opat In usng the optzaton tehnque, fndng the searhng dreton s very portant. The searhng dreton s deterned by the onugate gradent ethod based on the desgn senstvty obtaned by the adont varable ethod. The steppng sze s deterned usng the seond order polynoal urve fttng. The ntal guessed desgn varable p n Eq. (55) s -.5 /se and the axu hange of p durng the optzaton teraton s.5 /se. Fgure shows the varaton of the value of obetve funton durng the optzaton teraton. After 6 9-4

11 teratons, we an obtan the fnal optu results. The optal value of the desgn paraeter s - /se and the upper and lower punh speeds are equally /s fro Eq. (55). Therefore, t an be shown that double aton pressng s the optal proess for ylndral shape part. The optzaton result s reasonable aordng to oon sense and we an verfy the optu progra fro these results. (a) (b) Fgure 3. Relatve densty ontour of the ylndral opat by fnte eleent Calulatons wth (a) ntal guessed values and (b) the optu results Fgures 3(a) and (b) show the relatve densty dstrbuton of the ylndral opat by fnte eleent sulaton wth the ntal guessed proess varables and wth the optu proess varables. The densty gradent s learly sall n the sulaton results wth the optu proess varables. Optzaton of De Copaton for T-Shaped Part To fnd the optal upper and lower punh dsplaeents for the ost unfor densty dstrbuton durng de opaton of the T-shape opat wth ntal heghts a and b, we apply the optzaton progra. We assue the gven powder ass s 9.8 g and the ntal relatve densty of the powder opat s.457. Fgure 4 shows the fnal shape and densons of the de opat part: the average relatve densty s.94. Fgure 5 shows the fnte eleent esh ontanng 47 four-noded axsyetr eleents and the approprate boundary ondtons. 9-4

12 5 5 Fgure 4. Fnal T-shape opat (unt : ) 4 Obet funton teraton nuber Fgure 5. Fnte eleent eshes and boundary ondtons for the T-shape opat Fgure 6. The varaton of the obet funton durng the optzaton teraton We guess the ntal guessed upper and lower punh speeds are equally 7 /se. The ntal guessed desgn varable p s seleted -. /se and the axu hange of p durng the optzaton teraton s.5 /se. Fgure 6 shows the optzaton results. After 7 teratons, we an obtan the fnal optu results. The results are that the upper punh should dsplae 5. and the lower punh should dsplae 4.4. Fgure 7(a) and (b) show the relatve densty dstrbuton of the ylndral opat by fnte eleent sulaton. Fgure 7(a) shows the sulaton results wth the ntal proess varables and Fgure 7(b) shows the sulaton results wth the optu proess varables. In the Fgure 7(a), the relatve densty regon s fro.63 to.95. But, n the Fgure 7(b), the relatve densty regon s fro.856 to.97. We an fnd that the optu proess varables fro the optzaton progra an learly prove the unfor densty dstrbuton n the de opaton proess. 9-4

13 (a) (b) Fgure 7. Relatve densty ontour of the T-shape opat by fnte eleent alulatons wth (a) ntal guessed values and (b) the optu results Intal guessed varables Optu results Effetve stress, MPa 5 Intal guessed varables Optu results Pressure, MPa Pressure, MPa Relatve densty (a) (b) Fgure 8. Stress paths of the lowest densty eleent durng de opaton of the T-shape opat wth (a) effetve stress and pressure and (b) pressure and relatve densty In Fgure 7, the lowest densty and severe densty gradent were observed n the nek of the T-shape powder opat. Ths eans that the nek wll be the weakest regon n the T-shape opat and an be generate the rak durng the de opaton and eeton proesses. Fgure 8(a) and (b) shows the stress paths of the lowest densty eleent n the T-shape opat durng the de opaton proess. Fgure 8(a) shows the relaton between effetve stress and Fgure 8(b) shows the relaton between pressure and relatve densty. The sold lne and dashed lne were obtaned fro the fnte eleent alulaton wth the ntal proess varables and the optu proess varables, respetvely. The sold lne shows that the nek surfae wasn t opated wth adequate pressure durng the de opaton. Moreover, the sold lne shows that durng the de opaton the appled pressure on the nek surfae was dereased and relatve densty was also dereased. The appled nu pressure value was negatve and ths eans the rak an be generated at the nek regon. 9-43

14 SINTERING ANALYSIS Deternaton of Materal Paraeters The dlatoeter data [8] of 36L stanless steel powder opat wth D=.8 durng snterng an be analyzed by usng the onsttutve Eqs. (4)-(5). The ntal relatve densty D =.48 was assued for the alulatons. Materal propertes n Eqs. (46)-(5) for 36L stanless steel powder an be deterned fro experental data. The ateral propertes obtaned fro the lterature [] and Kwon et al. [4] for 36L stanless steel powder and are shown n Table I. Table I. Materal propertes for 36L stanless steel Materal property Unt Value Surfae energy, γ J / Atvaton energy for boundary dffuson, Qb kj / ol 67 Gran growth exponent, Pre-exp. for gran growth, k / se.73 Atvaton energy for gran growth, kj / ol 46.3 Qs 5 The ateral paraeter β n Eq. (4) was obtaned by nzng the dfferene between theoretal alulatons and the easured varaton of dlatoeter data. Fgure 9 shows a oparson between experental data and theoretal alulatons (a) for the varaton of axal shrnkage wth te and (b) for the varaton of axal shrnkage wth teperature durng heatng and snterng at 5 C. The sold urves denote experental data and the dashed urves were alulated fro Eqs. (4)-(5) by usng the 8 6 deterned ateral paraeter β = 5.5 K / J se and assung the ntal gran sze s μ....5 Experental Data Calulaton results.5 Experental Data Calulaton results.. Shrnkage (%).5. Shrnkage (%) Te (n) (a) Teperature ( o C) (b) Fgure 9. Coparson between experental data and theoretal alulatons (a) for the varaton of axal shrnkage wth te and (b) for the varaton of axal shrnkage wth teperature Analyss The onsttutve Eqs. (4)-(5) were pleented nto the user defned subroutne CREEP of ABAQUS[] to analyze the snterng proess of T-shape powder opat by fnte eleent analyss. Fgure shows relatve densty ontour and shrnkage shape of T-shape opat by fnte eleent alulatons after 6 n at 5 C durng snterng wth the sulated ntal relatve densty dstrbuton fro Fgure 7(b). In Fgure, we an fnd that durng the snterng proess the non-unfor densty dstrbuton was antanng beause the densfaton shrnkage s very sall and wthn.5%. 9-44

15 Fgure. Relatve densty ontour and shrnkage shape of T-shape opat by fnte eleent alulatons CONCLUSION The optzaton progra was developed to optze the de opaton proess (PMsolver/Opt). We an verfy the optzaton progra n the de pressng proess for the ylndral powder opat. The applaton of the optzaton progra n the de pressng proess for the T-shape powder opat shows the optu proess varables fro the optzaton progra are very useful to reate an unfor densty opat, thus redue the possblty of rak foraton. The analyss of the snterng proess after de opaton shows that durng the snterng proess the nonunfor densty dstrbuton was antaned. So, the optzaton progra to provde the optu proess varables durng the de opaton proess s shown to be a very powerful tool to prove the desgn and developent of new P/M parts. ACKNOWLEGMENTS We thank N. Myers and R.K. Ennet n CISP for provdng the experental data. REFERENCE. R.M. Geran, Powder etallurgy Sene, Metal Powder Industres Federaton, Prneton, NJ, R.M. Geran and A. Bose, Ineton Moldng of Metals and Ceras, Metal Powder Industres Federaton, Prneton, NJ, 997, p.. 3. A.S. Helle, K.E. Easterng and M.F. Ashby, Hot-Isostat Pressng Dagras: New Developents, Ata etall., Vol. 33, No., 985, pp M.F. Ashby, Bakground Readng HIP 6., Unv. of Cabrdge, U.K., G. Zhao, E. Wrght and R.V. Grandh, Forgng Prefor Desgn wth Shape Coplexty Control 9-45

16 n Sulatng Bakward Deforaton, Int. J. Mah. Tools Manufat., Vol. 35,995, pp S. Roy, S. Ghosh and R. Shvpur, A New Approah to Optal Desgn of Mult-Stage Metal Forng Proesses wth Mro Genet Algorths, Int. J. Mah. Tools Manufat., Vol. 37, 997, pp S.M. Byon and S.M. Hwang, De Shape Optal Desgn n Betal Extruson by the Fnte Eleent Method, Trans. ASME, J. Manufat. S. Eng., Vol. 9, 997, pp L. Forent and J.L. Chenot, Optal Desgn for Non-Steady State Metal Forng Proesses - I. Shape Optzaton Method, Int. J. Nu. Meth. Eng., Vol. 39, 996, pp S.H. Chung, L. Fourent, J.L. Chenot and S.M. Hwang, Adont State Method for Shape Senstvty Analyss n Non-Steady Forng Applatons, Int. J. Nu. Meth. Eng., aepted.. S. Sha and M. Oyane, Plastty Theory for Porous Metals, Int. J. Meh. S., Vol. 8, 976, pp S.M. Doravelu, H.L. Gegel, J.S. Gunasekera and J.T. Morgan, A New Yeld Funton for Copressble P/M Materals, Int. J. Meh. S., Vol. 6, 984, pp L. Fourent, J. L. Chenot and K. Moelln, Nueral Forulatons and Algorths for Solvng Contat Probles n Metal Forng Sulaton, Int. J. Nu. Meth. Eng., Vol. 46, 999, pp C.C. Chen and S. Kobayash, Rgd Plast Fnte Eleent Analyss of Rng Copresson, Applatons of Nueral Methods to Forng Proesses, ASME Publaton, AMD, Vol. 8, Y.-S. Kwon and K. T. K, Hgh Teperature Densfaton Forng of Aluna Powder Consttutve Model and Experents, ASME J. Eng. Mat. Teh, Vol. 8, 996, pp Y.-S. Kwon, S.-H. Chung, H.-K. Ahn, S.-T. Chung, S.-J. Park, D. T.-S. Yoon, J.-Y. Cho, K.T. K, L.-J. Park and J.-H. K, Cae Analyss for Snterng Stage of Powder Ineton Moldng, Advanes n Powder Metallurgy & Partulate Materals,, p P.E. Mhugh and H. Redel, A Lqud Phase Snterng Model: Applaton to S3N4 and WC- Co, Ata etall., Vol. 45, No. 7, 997, pp Y.-S. Kwon, H.T. Lee and K.T. K, Analyss for Cold De Copaton of Stanless-Steel Powder, ASME, J. Eng. Mat. Teh., Vol. 9, 997, pp N. Myers, D.F. Heaney and R.K. Ennet, Densonal Control of 36L Stanless Steel, CISP Industry Meber Meetng,. 9. R.M. Geran, Snterng Theory and Prate, John Wley & Sons, 996, p ABAQUS, User s I and II Manual, Hbbt, Karlsson and Sorenson,. 9-46