MODELLING OF COLD METAL EXTRUSION USING SPH

Transcription

1 Fifth International Conference on CFD in the Proce Indutrie CSIRO, Melourne, Autralia Decemer 2006 MODELLING OF COLD METAL EXTRUSION USING SPH Maheh PRAKASH and Paul W. CLEARY 1 CSIRO Mathematical and Information Science, Clayton, VIC 3169 ABSTRACT Extruion i often ued for producing long oject with contant cro-ection. The proce i ued with oth hot and cold metal and alo platic. Extruion modelling can e ued to reduce trial extruion numer ecaue of etter proce deign following from accurate defect prediction. Such model can alo e ued a input for die-deign activitie and to indicate way for improving nonperforming die. Traditionally, the extruion proce ha een modelled uing meh aed Finite Element Method (FEM). However, the large platic deformation occurring in the illet during extruion can often lead to heavy remehing requirement for uch method and exceive numerical diffuion. Here we demontrate the ue of the meh-le Smoothed Particle Hydrodynamic (SPH) method for imulating uch procee. It grid-free nature allow SPH to handle large deformation without the need for any re-mehing. Additionally SPH provide the aility to track hitory dependent propertie uch a platic train on a particle y particle ai making it very uitale for analyi and defect prediction. In thi paper we evaluate the variation in platic train level and deformation profile in a typical aluminium alloy with extruion parameter uch a die diameter and die angle. We alo invetigate the effect of thee parameter on the maximum extruion force required. INTRODUCTION Extruion can e defined a the production of a part of a deired hape y forcing a olid metal illet to flow through a die. Extruion i often ued in making automotive and uilding hardware part where high trength and cloe tolerance are required. The advantage of modelling extruion include: Aility to verify empirical knowledge Reduction of trial-preing Uaility in input for die deign program Indication of method to improve non-functional die. Large platic deformation in the metal during extruion can lead to heavy re-mehing requirement for FEM method (Tiernan et al. 2005) which can e inaccurate and diffuive. Due it meh-free nature uch large deformation can e eaily handled y SPH (Fernandez- Mendez et al. 2005). The particle aed nature of the method alo give it the aility to track hitory dependent propertie uch a urface oxide formation and platic train on a node y node ai even during very large deformation, Cleary et al. (2006). Here we evaluate, for a imple 3D cold extruded part: Variation in platic train level, Variation in metal deformation pattern and Variation in maximum extruion force required; with change in extruion parameter uch a die diameter and die angle. THE SPH METHOD A rief ummary of the SPH method i preented here. For more comprehenive detail one can refer to Gray et al., The interpolated value of a function A at any poition r can e expreed uing SPH moothing a: A A ( r) = m W ( r r, h) (1) ρ where m and r are the ma and denity of particle and the um i over all particle within a radiu 2h of r. Here W(r,h) i a C 2 pline aed interpolation or moothing kernel with radiu 2h, that approximate the hape of a Gauian function ut ha compact upport. The gradient of the function A i given y differentiating the interpolation equation (1) to give: A A ( r) = m W ( r r, h) (2) ρ Uing thee interpolation formulae and uitale finite difference approximation for econd order derivative, one i ale to convert paraolic partial differential equation into ordinary differential equation for the motion of the particle and the rate of change of their propertie. Continuity Equation From Monaghan (1992), our preferred form of the SPH continuity equation i: dρ a = m ( v a v ) Wa (3) dt where ρ a i the denity of particle a with velocity v a and m i the ma of particle. We denote the poition vector from particle to particle a y r = r r and let ( r h) Wa = W a, e the interpolation kernel with moothing length h evaluated for the ditance a r a a. Thi form of the continuity equation i Galilean invariant (ince the poition and velocitie appear only a difference), ha good numerical conervation propertie and i not affected y free urface or denity dicontinuitie. Momentum Equation The momentum equation ued for the elato-platic deformation of the olid i: i dv 1 σ i = + g, (4) j dt ρ where v i velocity, g denote the ody force and σ i the tre tenor which can e written a: σ = P δ + S, (5) 1

2 where P i the preure and S i the deviatoric tre. Auming Hooke law with hear modulu µ the evolution equation for the deviatoric tre S i (Gray et al., 2001): ds. 1. ik jk ik kj = 2µ + S Ω + Ω S dt ε δ ε (6) 3 where. i j 1 ε = + (7) j i 2 and i j 1 Ω = (8) j i 2 i the rotation tenor. The ucript in the aove ymol refer to the olid 2 tate. The ulk modulu i K = ρ and the Poion ratio ν i: ( 3K / µ 2 ) = 2( 3K / µ + 2) c 0 ν (9) Equation of State The SPH method ued here i quai-compreile with the preure calculated from the particle denity uing an equation of tate: 2 P c ( ρ ρ ) = (10) where P 0 i the magnitude of the preure and ρ 0 i the reference denity. The preure cale factor P 0 i given y: γ P = 100V = c (11) ρ 0 where V i the characteritic or maximum velocity. Thi enure that the denity variation i le than 1% and the material can e regarded a incompreile. Platicity Model The platicity model ued i a radial return platicity model originally propoed y Wilkin (1964). The trial tre S i the deviatoric part of the tre calculated Tr auming that the initial repone i elatic, S = αs (12) Tr where S i the final deviatoric tre at the end of a timetep and α i a proportionality contant given y: p 3µ ε α = 1 Tr (13) σ with 2 Tr eing the magnitude of the trial deviatoric σ 3 p tre and ε eing the increment in equivalent platic train: Tr n σ σ p y ε = (14) 3µ + H n where σ y i the final yield tre and H i the hardening modulu. The tre update i completed y adding the deviatoric and mean tre a given in equation (5). The platic train i incremented a: p p p ε = ε + ε (15) Hitory Dependent Propertie of the Metal Each SPH particle repreent a pecific volume of metal and carrie that information with it. Thi i a critical attriute of the Lagrangian method. Thi mean that information on the precie tate of each piece of metal can e known at all time and the hitory of each piece of metal i uilt into the particle data. Thi provide ignificant capaility to track propertie uch a: Cumulative platic train Damage (which i a volume averaged local meaure of cracking) leading to fracture prediction; Metal compoition (including tracking multiple metal or metal compoite) and trapped ga; Metallic phae and microtructure; and Surface oxide. Some or all of thee propertie can then e ued to feed ack into the flow dynamic uing uitale rheology model. EXTRUSION MODELLING USING SPH Figure 1 how a front view of the extruion geometry coniting of a 3D cylindrical illet of diameter D = 40 mm, which i cold extruded through a die exit diameter d = 20, 15 or 10 mm with a punch moving at a contant peed of 25 m/min. Die angle of θ = 120, 140 or 160 degree were ued for the imulation. The material propertie of the aluminium alloy are preented in Tale 1. An SPH particle reolution of 1.6 mm wa ued giving a total of 7,760 particle for thee three dimenional imulation. punch illet Figure 1: Schematic diagram of extruion geometry. D θ d In cold extruion modelling it i common to ue a rigidplatic model to decrie the material ehaviour (Tiernan et al., 2005). However in certain circumtance elatic effect can ecome important epecially in a cylindrical channel where elatic effect can have a ignificant impact on the reult (Lof, 2000). In thi paper the elato-platic model decried earlier i ued to repreent the alloy ehaviour. Effect of die diameter on platic train and deformation The effect of die diameter on platic train and internal illet deformation i demontrated uing three cae with a contant die angle of 120 o and varying die exit diameter of 20, 15 and 10 mm. 2

3 Bulk Modulu (GPa) 7.0 Shear Modulu (GPa) 2.70 Initial Yield Stre (MPa) 5.52 Hardening Modulu (MPa) 0.0 Denity (kg/m 3 ) Tale 1: Propertie of Al alloy ued for cold extruion. Figure 2 how the illet coloured y percentage platic train with lue eing minimum and red howing maximum train level fixed at 2.1 or 210% percent for all three cae. Figure 3 how metal deformation profile for the three exit diameter. In Figure 3 the illet i coloured in horizontal trata aed on initial material poition in order to track the internal metal deformation profile. A ection view of the 3D illet i hown in thi figure. At 20 m the illet ha not yet moved into the die exit for the d = 20 mm cae ince a large amount of metal need to e puhed through the die for the larger exit diameter. Platic train level are very low for thi cae at thi tage. For d = 15 mm, the metal ha moved into the die exit with approximately a quarter of the illet experiencing a low to moderate amount of train. Once the illet corner contact the converging wall of the die, the metal i quickly elatically loaded and egin undergoing platic deformation. For d =10 mm, a reaonale amount of metal ha een already een extruded through the die at 20 m with the metal experiencing high train level in exce of 210% a it pae through the relatively mall die orifice. At 30 m the metal ha progreed into the die for d = 20 mm cae with high train level developing in region adjacent to the die wall. For d = 15 mm, ignificant amount of metal have paed through the die orifice and have experienced train level greater than 210%. The central material at the leading urface ha a lower train. For d = 10 mm, the amount of high train i very large with almot the entire extruded metal length now experiencing train level greater than 210% including the central region. Thi indicate that for large extrude ratio (ratio of illet diameter to product diameter) train level can e ignificantly high throughout the extruded product and not jut region adjacent to the wall. Product with large extrude ratio will thu have a higher tendency to fail in the aence of a rigorou annealing proce after cold extruion (Shivpuri et al., 1999). By 40 m, ignificant amount of extruded product have een produced for all three die diameter. For d = 20 mm, the high platic train level adjacent to the wall have now ecome more pronounced. The central region (except at the leading edge) alo develop medium amount of train of around 160%. A econd low train cell tart to develop in the central core at thi time. For d = 15 mm, high train level of around 210% are now een even in the central region away from the leading edge. Here again there are pocket or cell of medium train level in the core coloured yellow although the ize of thee cell are much maller in comparion to the d = 20 mm cae. Such pocket are developed ince the two high platic train region from the two wall do not meet completely and could lead to a defect referred to a chevron crack (Saanouni K. et al., 2004). The occurrence of uch pocket can e reduced y increaing the extruion ratio (a een from the d = 10 mm cae). By 70 m, the central low train cell i clearly een for the d = 20 mm cae a the illet i almot fully extruded through the die. A the end of the illet approache the convergent ection a clear train concentration occur in the central core. For d = 15 mm, the central cell are more tretched and le pronounced. For d = 10 mm the metal pae through the die with high train level throughout the metal ma. The imulation how that even at very high extruion ratio leading to large deformation no pecial treatment i required with the SPH methodology. Effect of die angle on platic train and deformation The effect of die angle on platic train and internal illet deformation i demontrated uing three cae with a contant die diameter of 20 mm and varying die angle of 120, 140 and 160 o. Figure 4 how the illet coloured y percentage platic train with lue eing minimum and red howing maximum train level fixed at 2.1 or 210% percent for all three cae. Figure 5 how metal deformation profile a in the previou ection. At 20 m, the metal ha jut moved into the die opening for the 120 o cae with the metal not yet reaching the die for the higher die angle due to larger volume needing to e filled efore reaching the die entrance. The metal at the leading edge ha aumed the hape of the die entrance for the 120 cae. For the larger die angle the metal front i till fairly flat at thi tage. At 30 m, the metal ha moved into the die for all three cae with more metal already eing extruded for the maller die angle. The metal deformation profile for the three cae are quite imilar with the only difference eing a delay in the metal extruion with increaing die angle. For the 120 cae the metal deformation profile ha a v- haped tructure at thi tage a een from the green coloured and. At 40 m increaed train level can e een at region adjacent to the die wall for all three die angle. Note that in thee imulation frictional effect have not een taken into conideration with normal force accounting for high train level. The metal deformation profile for all three angle reveal a paraolic pattern at the leading edge at thi tage. At 50 m the effect of normal force leading to high train region ecome more prominent. Thi i een mot clearly with a die angle of 160 where the die wall at the entrance are almot orthogonal to the metal entering the die. At 74 m, the metal ha almot completely extruded through the die for all three cae. A central cell with metal at a low train level (coloured green) i een for all three cae. Except for a relatively higher train level adjacent to the ide wall for the 160 o angle cae the train level and metal deformation profile for all three cae are imilar. Thi indicate that the die angle ha little impact on the quality or propertie of the extruion. 3

4 (a) () (c) Figure 2: Platic train increae trongly with a decreae in die diameter (a) 20 mm () 15 mm (c) 10 mm. (a) () (c) Figure 3: The extent of metal deformation increae with decreae in die diameter (a) 20 mm () 15 mm (c) 10 mm. 4

5 (a) () (c) Figure 4: Platic train increae with increae in die angle (a) 120 deg () 140 deg (c) 160 deg. (a) () (c) Figure 5: Metal deformation doe not change ignificantly with die angle (a) 120 deg () 140 deg (c) 160 deg. 5

6 Effect of die diameter and angle on extruion force Figure 6 how the effect of die diameter on the extruion force required. The extruion force ramp up over the firt m to a maximum value a the illet fill the region around the die inlet. Between 20 and 60 m, the metal i extruded at a contant rate through the die giving a conitent force at around the maximum level. The force then ocillate a the rear of the illet approache the extruion region and then decline rapidly. A hown in Tale 2 the extruion force increae with an increae in the extruion ratio with the d = 10 mm cae experiencing the maximum force of 30 kn which i around 150% greater than the d = 20 mm cae. Thi reult i qualitatively conitent with reult otained from previou tudie of a imilar nature uing the finite element method (Tiernan et al., 2005). Cae Die exit diameter (mm) Die angle (degree) Maximum force (kn) Additionally the effect of change in extruion parameter uch a extruion ratio and die angle on platic train, metal deformation profile and maximum extruion force required can effectively and eaily e analyed uing SPH. Specifically we have hown that: Platic train on the metal increae harply with increaing extruion ratio, with high platic train oerved throughout the metal for large extruion ratio. Platic train increae moderately with an increae in die angle epecially in region adjacent to the wall of the die. The maximum extruion force increae harply with an increae in extruion ratio, with a variation of around 150% etween extruion ratio of 2:1 and 4:1.. The maximum extruion force how only a mall increae with die angle giving a variation of around 15% for the three die angle conidered Tale 2: Maximum extruion force with variation in die diameter and angle. Figure 7 how the effect of die angle on the extruion force. Again the extruion force ramp up to it maximum value y around 20 m for all three cae. It increae moderately with die angle, with the variation etween the three cae eing around 15% a een from the maximum force value in Tale 2. Figure 6: Extruion force increae with decreae in die diameter (a) 20 mm () 15 mm (c) 10 mm. With an increae in die angle there i evidence of trong ocillation in the extruion force with an almot periodic ehaviour. For the 160 cae the ocillation have an amplitude of approximately 2.4 kn and a frequency of around 125 Hz. Due to the aence of frictional effect we ee an increae in the normal force required with an increae in the die angle. Thi will only e true in ituation where a very good luricant i ued for the extruion proce uch a in hydrotatic cold extruion (Swiotek et al., 2006). However in many extruion cae one will need to include frictional effect and thi will alter the relationhip etween die angle and extruion force required. CONCLUSION The SPH method can e effectively ued for modelling metal extruion procee uing an elato-platic model for the metal rheology. For thee metal forming procee, SPH ha the advantage of eing ale to follow very high deformation (eyond what i poile with FEM and FV method) and to keep track of the pecific hitory of each part of the metal allowing fine cale control over the rheology model and potentially direct prediction of many type of flow and microtructure related defect. Figure 7: Extruion force increae with increae in die angle (a) 140 deg () 160 deg (c) 180 deg. The preent imulation aume perfect lurication (no friction). REFERENCES CLEARY, P.W., PRAKASH, M., HA, J., STOKES, N., and SCOTT, C., (2005), Smooth Particle Hydrodynamic; Statu and future potential'', Proc. 4th Int. Conf. on CFD in the Oil and Ga, Metall. & Proce Indutrie, Norway, Ed. S.T. Johanen, I.G. Page. 6

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