MATERIAL SPREADING AND COMPACTION IN POWDER-BASED SOLID FREEFORM FABRICATION METHODS: MATHEMATICAL MODELING

Transcription

1 MATERIAL SPREADING AND COMPACTION IN POWDER-BASED SOLID FREEFORM FABRICATION METHODS: MATHEMATICAL MODELING Yae Shanjani and Ehan Toyekani Depatment of Mechanical and Mechatonic Engineeing, Univeity of Wateloo, 00 Univeity Avenue Wet Wateloo, ON NL 3G, Canada Reviewed, accepted Septembe 0, 008 Abtact In thi tudy, the peading and compaction of powde by a counte-otating olle, in the olid feefom fabication (SFF) poce, i modeled and chaacteized. The effect of laye thickne, olle diamete, amount of looe powde, and it initial denity on the powde bed elative denity and olle compaction peue i invetigated. Intoduction Denity i a cucial chaacteitic of the pat fabicated by olid feefom fabication (SFF). Thi facto diectly affect the mechanical popetie and functionality of the pototyped pat. In powde-baed SFF method, uch a 3D-pinting and elective lae inteing (SLS), the denity of the fabicated pat ubtantially depend on the denity of the powde laye, which ae equentially pead and compacted by a counte-otating olling mechanim, a hown in Figue. Counte-Rotating Rolle Powde Bed Figue : Schematic of powde peading and compacting poce by a counte-otating olle To aive at the deied denity in the powde bed, the paamete aociated with the peading mechanim (e.g., linea and otational peed of the olle, olle uface popetie, and powde laye thickne) mut be caefully adjuted in uch a way that an appopiate compaction foce i applied on the powde bed by the olle. Since the popetie of powde mateial (e.g., peadability and flowability) ae inheently nonlinea, thi optimization poce i inticate. The cuent effot fo obtaining the optimum popetie of the pead powde laye ae baed on tial-and-eo technique, which mut be caied out fo any new powde mateial and chaacteitic. A an altenative appoach, mathematical modeling povide inight to the compaction phenomenon. By uing the mathematical modeling and imulation, enginee can 399

2 tune the poce paamete fo bette contol of the pototyped pat pooity. In pite of the appaent implicity of the counte-otating olling compaction, thee ae too many unknown apect of thi poce, fom an analytical pepective. Seveal invetigation have been conducted on powde olling and denification, and many theoie have been developed to detemine the mateial behavio in compaction condition. Johanon [] developed the fit comple model fo pedicting the mateial flow undegoing continuou hea defomation between two olle. The D lab method, in which the equilibium foce balance on a thin tip of mateial with a diffeential thickne i conideed, ha often been ued to tudy the olling, etuion and dawing of metal []. Katahinkii et al. [3, 4] developed a mathematical model of the denification of powde by two otating olle, epeenting the te-tain tate of the powde being olled in the denification zone. A complete eview about thi aea of eeach i available in the liteatue []. Thi wok i focued on the mathematical modeling of the powde compaction whee a counte-otating olle pae ove the powde bed duing the feeding tep of the SFF poce. Invetigation ae conducted to obtain the effect of the olling paamete (laye thickne and olle diamete) on the kinematical chaacteitic of the poce in the denification zone. The denity and olling contact peue within the compacted powde bed ae attained. Mathematical Modeling The paamete which ae ued in the following ection ae lited in Table, a the nomenclatue. Table : Nomenclatue Paamete Deciption paamete Deciption Shea te between olle and looe powde (N/m ) b Slab length (m) t e Rate of denification (/) Rolling angle (adian) Rate of lab defomation in e diection (/) in Rolling uppe bound (adian) Rate of lab defomation in y e y Slab width (m) diection (/) h Slab height (m) Rate of hape change (/) Coefficient of fiction between looe and h Powde laye thickne (m) p compacted powde Coefficient of fiction between olle and m Slab ma (kg) looe powde n Nomal vecto Pooity P Hydotatic te (N/m ) i Initial pooity P Rolle contact peue (N/m ) Denity (kg/m 3 ) P Rolle contact peue / Q Reaction foce (N), y, z Ste in, y, and z diection (N/m ) Ste in diection / R Rolle adiu (m) Yield te of bulk mateial (N/m ) Shea te between looe and t q compacted powde (N/m Ste deviato (N/m ) ) 400

3 The counte-otating olle compaction of powde i analyzed with the D lab appoach, fo which ome aumption ae taken into account. It i aumed that te and tain vay only in the olling diection and not in the width o thickne. Since the platic tain ae ubtantially highe than the elatic tain, the elatic defomation ae negligible. In addition, body foce (gavitational foce) can be ignoed in compaion with the fictional and compaction foce. In addition, it i aumed that the undeneath powde laye ae compacted enough o that the new laye of olling powde i pepaed onto a tationay plane plate. The effect of olle otational and linea motion peed ae taken into account by the coefficient of the fiction between the olle and powde. Since the mateial i pevented fom peading lateally duing olling, and the olle length i elatively lage than it diamete and powde laye thickne, the poce i teated a a plane tain poblem. By conideing a continuum mechanic fo thi poce, imila to the phenomenon of the lamina flow of a vicou liquid, a teady flow of the powde take place into the denification zone. The analyi i pefomed accoding to the platicity theoie fo compeible poou olid. Fo implification, it i aumed that the powe i pead and then compacted to a cetain thickne. A hown in Figue, in detemine the beginning angula poition of the denification zone beyond whee the compaction of the powde take place. In othe wod, epeent the amount of powde which flow into the denification zone. The lowet olling in gap h, which i equal to powde laye thickne, occu at in =0. The coefficient of fiction between the olle uface and the powde i aumed to be contant on entie olle-powde contact aea. Thi paamete depend on eveal facto, including the powde paticle ize and hape, olle uface moothne, and olle otational (ω) and linea (v) motion peed. Rolle Zw V Za R zda Za in P t Looe Powde Compacted Powde h Z z + d Sh Y X Q tq Denification Zone Figue : Foce and tee acting on the powde denification zone 40

4 To deive the govening equation on the powde unde the olle, a lab of powde with an infinitely mall thickne i conideed a hown in Figue. It i oiented pependicula to the olling diection and the applied foce. The tee acting on the lab, the oll peue P, and the hea tee t and t q, ae alo depicted in Figue. The equilibium of the foce in the olling diection, that i, the -diection, fo a unit length in the oll width, lead to the following epeion: d R P t t d in co q h () With the conideation of the equation of continuity, fo a compeible mateial, the denity i not contant and the continuity equation (ma conevation) i witten a follow [3]: h m h b 0 h () The tem and h ae the ate of the lab defomation in the diection and y, h epectively. Thu, e and e y h h Fo the plane defomation of compeible mateial, the compatibility equation ae peented in (4)-(6), which ae deived in [5]. Equation (4) and (5) epeent the load uface. (3) P 3 6 and P e 6 i i ; 3 i i (4) (5) (6) P i the fit invaiant of the te teno, when i the modified econd invaiant of it deviato [5] uch that (7) P ii and ij P ij ij P ij 3 The elation between the te teno and defomation ate teno i witten a ij eij eij and e eijij 3 (8) 40

5 that The te in the thid ai (along the olle length) i obtained in tem of 3 z y 3 A a eult the fit invaiant P i P y and y uch Accoding to Shten [5], in powde olling, when the angle of uppe bounday of the denification zone doe not eceed adian, the following appoimation i valid: P and 3 () y (9) (0) Theefoe, the tem P 4 y can be neglected fo, and then () Invaiant and e ae epeed in tem of the pincipal defomation ate a [3] e e e e e By applying (0)-(3), along with (5) e i found a P h e P h 6 (3) (4) In addition, fom (), the following can be computed: h e (5) h With (5) and the eultant epeion fo e in (4), the following elation i obtained: h h 6 P P (6) 403

6 Fom the geomety pepective, it i obeved that h h( ) h R co. Since the poce i analyzed in the teady-tate, i only a function of. Conequently, d in d h A P (7) co R B P whee A ; B By ubtituting (7) in the load uface equilibium (5), and olving it fo P, A B A B P A B A B A B (8) (9) whee P P and ae the dimenion-le te component. In addition, the odinay diffeential equation, obtained fom the foce equilibium, i (0) d P in f co p co f in d h co R (0) by conideing t P and t Q P co in. q p p Equation (7) and (0) fom a govening equation ytem of the compaction poce. By adjuting the initial value fo the in, coefficient of fiction p and, initial denity of powde initial, laye thickne (olling gap) h, and olle adiu R, and by uing a numeical algoithm of Eule developed in MATLB, the olle contact peue, te ditibution, and denity gadient of the powde along the olling diection ae obtained. Reult and Dicuion The deived govening equation ae calculated fo a wide ange of bounday condition and initial value which ae baed on the actual condition fo the pat fabication with the available SFF machine. Table lit the analyi paamete and ubjected value. 404

7 Table : Analyi paamete Paamete Value Deciption Fiction coefficient of looe powde and p 0.5 undelying compacted powde laye Fiction coefficient of olle-powde contact 0.3 aea R 0mm Rolle adiu initial 30% Powde initial denity in 0.ad Uppe-bound of olling angle The elative denity (RD) of the compacted powde laye i defined a bulk denity of compacted powde RD powde mateial denity () The vaiation of RD in elation to the change in the powde laye thickne, h, i hown in Figue 3. The plot i epeented a a function of the olling angle, which vaie fom in (the uppe-bound) to zeo, along the denification zone. The elected laye thicknee ae acceible in a typical 3D-pinting machine. Alo, the powde elative denity coeponding to the laye thicknee i plotted in Figue 4, fo diffeent in. The value of in epeent the amount of powde which i tapped into the denification zone and indiectly depend on eveal facto, uch a the flowability of powde and olle-powde coefficient of fiction. It can be een that the lage the thickne, the lowe the RD in the compacted powde. It i obviou that RD inceae with in. Fo a contant in, highe h mean le pecentage of looe powde i compacted which, in tun, eult in a lowe denity in the compacted powde. Howeve, if we aume the pecentage of the compacted powde to be contant (), in function with h a h in AcCo (- ), and RD vaie along the denification zone a illutated in R Figue

8 Relative Denity Relative Denity in in in in in h Rolling Angle (Radian) Figue 3: Effect of the powde laye thickne on the elative denity ditibution along the denification zone fo 0.ad in a in 0.05 Rad 0. Rad 0. Rad 0.5 Rad Laye Thickne (in) 0-3 Figue 4: Relative denity of compacted powde laye v. the adjuted laye thickne 406

9 Relative Denity in in h in in in Rolling Angle (Radian) Figue 5: Effect of the powde laye thickne on the elative denity ditibution along the denification zone The RD vaie due to the change that occu in the longitudinal tain, wheea the vetical tain i unchanged in the lab element. A a eult, accoding to the law of ma conevation, (), the elative denity ie. The uppe-bound of the olling angle, in, vaie in elation to the amount of powde in font of the olle. It i obviou that poviding moe powde to be conveyed into the denification zone, fo a contant h, eult in a dene compacted laye. Figue 6 how thi effect fo h =0.005in. Figue 6: Effect of the uppe-bound of the olling angle on the elative denity ditibution along the denification zone fo h 0.005in 407

10 Contact Peue / Yield Ste The data fom the epeimental invetigation of othe eeache [4] how that the ditibution of the powde denity in the denification zone depend on two facto: the abolute and elative magnitude of the longitudinal and nomal tee, and the geomety of the defomation zone. The ditibution of the olle nomal contact peue along the denification zone in the powde laye i gaphed in Figue 7. It i obviou that the maimum compact peue occu in the naowet gap between the olle and the undeneath powde laye, that i, at the olling angle of zeo. The contact peue deceae by chooing the lage powde laye thickne in in in in in h Rolling Angle (Radian) Figue 7: Effect of the powde laye thickne on the contact peue ditibution along the denification zone By keeping all the paamete unchanged, an inceae in the appaent denity of the tating powde (initial denity) eult the inceae in the elative denity of the compacted laye. Howeve, the amount of thi inceae depend on the initial denitie. The pecentage of the RD change fo diffeent initial denitie i peented in Figue 8. It i obeved that the powde with a lowe initial denity how a highe change in RD. 408

11 Relative Denity Relative Denity Change (%) h 0.005in 0.Rad in Initial Denity (%) Figue 8: Relative denity change v. initial appaent denity of powde The diamete of the olle ha alo a cucial influence on the powde compaction. Poviding the contant amount of the powde to convey into the denification zone, by popely adjuting the in, the RD of the powde laye can be tailoed uing the olle with diffeent diamete. The effect i hown in Figue mm 0 mm 5 mm 0 mm 5 mm 50 mm R Uppe Bound of Rolling Angle (Radian) Figue 9: Effect of olle diamete on the on the elative denity ditibution along the denification zone The eult ae ueful in etimating the elative denity of the pat, fabicated by the SFF method, uch a 3D-pinting and SLS that ue a counte-otating olle fo the powde peading. In addition, (7) to (0) can be olved fo the invee poblem. Theefoe, to fabicate a pat with a pecific denity, the equiite laye thickne, olle diamete, and powde condition (initial appaent denity and paticle ize) can be choen baed on eithe of the gaph o olving 409

12 the evee poblem. It i notewothy that the minimum and maimum value fo the laye thickne and olling velocity ae eticted fo any method of SFF. Concluion A one-dimenional mathematical modeling of powde compaction by a counte-otating olle wa conducted. The denity ditibution in the powde bed along the denification zone, in the olling diection, wa analyzed. It wa hown that how the laye thickne, olle geomety, and initial powde popetie affect on the compacted powde elative denity, which diectly yield to the popetie of the pototyped pat. In the futue wok, the model will be enhanced and validated though epeiment. Refeence [] J. R. Johanon, "A olling theoy fo ganula olid," ASME, Jounal of Applied Mechanic, vol. 3, e. E, pp , 965. [] J. C. Cunningham, "Epeimental tudie and modeling of the olle compaction of phamaceutical powde," Deeel Univeity, pp. 3, 006. [3] V. P. Katahinkii and M. B. Shten, "Ste-tain tate of powde being olled in the denification zone, I. Mathematical model of olling in the denification zone," Powde Metallugy and Metal Ceamic, vol. (5), pp. 7-, 983. [4] V. P. Katahinkii and M. B. Shten, "Ste-tain tate of powde being olled in the denification zone, II. Ditibution of denity, longitudinal tain, and contact te in the denification zone," Powde Metallugy and Metal Ceamic, vol. (5), pp. 9-3, 983. [5] M. B. Shten, "Plane defomation chaacteitic of compeible mateial," Powde Metallugy and Metal Ceamic, vol. 3(3), pp. 4-,