and Materials Engineering Simulation of Friction in Hydrostatic Extrusion Process* Pankaj TOMAR**, Raj Kumar PANDEY*** and Yogendra NATH**** **MAE Department, GGSIPU (I.G.I.T.), Delhi, India E-mail: Pankaj_1343@rediffmail.com ***Department of Mechanical Engineering, I.I.T. Delhi, India ****Formerly professor, Department of Applied Mechanics, I.I.T. Delhi, India Abstract Accurate determination of friction at the die/billet interface in hydrostatic extrusion is a complex issue due to involvement of various operating parameters viz. billet velocity, die geometry, contact pressure, material parameter and the regime of lubrications prevailing at die/billet interface. Therefore, the objective of this paper is to investigate friction and friction stress at die/billet interface in hydrostatic extrusion process of aluminium based alloys. The friction stress at die/billet interface is numerically computed for three lubricants whose rheology is represented by Roelands viscosity model. Investigations have been carried out for friction stress variations along the work zone for a wide range of extrusion ratios (A = to 10), semi-die angles (θ = 10 0 to 0 0 ) and material parameters (G = 0.67 to 1.86). Moreover, the validation of the proposed model has been done with the published work available in the literature. Key words: Lobatto Quardature Method, Extrusion Ratio, Strain Hardening Model, Material Parameter, Roelands Viscosity Model 1. Introduction Hydrostatic extrusion is a metal forming process. It is used in the forming of hard materials which are difficult to deform. In this process, lubrication plays vital role. Existence of thin lubricating film between the die and billet ease the forming operation. Moreover, lubricating film at the interface also reduces the extrusion pressure and improves the life of die (due to reduced wear) and quality (finish/tolerance) of the products. In the hydrostatic extrusion process billet is completely surrounded by pressurized lubricant in a hydrostatic container as illustrated in Fig. 1. Hydrostatic pressure in the surrounding fluid/lubricant is controlled externally through a hydraulic ram. The deformation of billet at the interface is accomplished by the existence of hydrostatic pressure and the pressure developed at the interface due to dynamics. *Received 17 Sep., 01 (No. 1-0375) [DOI: 10.199/jmmp.7.35] Copyright 013 by JSME After the early studies of Robertson [1] and Bridgman [], the roles of friction in extrusion process have been investigated by Avitzur [3-4] using energy approach for small and large semi-die angles. Moreover, experimental studies for determining the coefficient of friction in the hydrostatic extrusion process have been reported by Pugh [5] & Erans and Avitzur [6]. In a recently published review paper, Wang et al. [7] have nicely summarized the recent developments on the studies of the friction in extrusion processes. 35
Fig. 1 Schematic diagram of hydrostatic extrusion process The objective of the present work is to study friction stress at the die/billet interface in hydrostatic extrusion process at various operating parameters; extrusion ratio (A = to 10), semi-die angle (θ = 10 0 to 0 0 ), material parameters (G = 0.67 To 1.86), and three different lubricant oils. Coupled solutions of governing equations (Reynolds equation, energy equation, rheological relation, and geometrical equations) have been achieved using an accurate technique presented by Elrod and Brewe [8]. The values of minimum film thickness, pressure, and temperature determined in inlet zone are used as boundary condition for the mathematical modeling of contact zone. In investigation of frictional stress in the contact zone, the combined effects of strain hardening of the billet material & viscous shearing of lubricant and temperature rise due to plastic deformation of materials have been considered.. Formulation of model The general Lobatto quardature technique developed by Elrod and Brewe [8], and further used by Ghosh and pandey [9], Pandey and Ghosh [10], Singh et al. [11] and Tomar et al. [1] in modeling of different types of their respective problems, has been adopted herein. Coupled solutions of partial differential governing equations (Reynolds and energy equations) have been achieved with appropriated boundary conditions. The billet material passes through severe plastic deformation; therefore a strain hardening material model proposed by Ludwik [13] is incorporated in the mathematical formulation of contact zone. Coordinate system can be referred in Fig. for the clarity in understanding the expressions appearing in the following subsections..1 Generalized Reynolds equation The following Roelands viscosity relation is adopted for representing the rheology of the lubricating oil in the extrusion process: ' -9 { 10 } z η= η exp[(ln η + 9.67) 1 + (1 + 5.1 X p) γ ( T T ] (1) 0 0 0 In the proposed model, the viscosity has been expressed in terms of fluidity ( ξ =1/ η) for the convenience in mathematical formulation. The expression for the fluidity variation across the lubricating film thickness is expressed by Legendre polynomial of order as follows: ( ) 3 ξ ζ = ξ k Pk ( ζ ) () k = 0 36
Using the eq. (), the expression for velocity variation across the lubricating film is simplified and written as follows: ζ ζ u = A ξ d ζ + B ξζ d ζ (3) -1-1 1 1 Where - h A = U / = b B ξζdζ ξdζ and B p -1-1 In order to develop generalized Reynolds equation, the expression for lineal mass flux is taken as below: + h/ m = - h/ ρ u dy (4) Divergence of mass flux leads to the generalized Reynolds equation, which is written as:.( m / ρ ) = 0 ξ ( ξ / ξ ) (5) 3 Or ( h p) = 6 U h ( U h) p b 1 0 b Where, ξ ( p = ξ0 + 0.4 ξ ξ1 /3ξ 0). Energy equation The energy equation used in the present analysis neglects the influence of dilatational viscosity. It incorporates heat convection along the lubricant film and heat conduction across the lubricant film. Thus, energy equation is as follows: ( ) ρucp T/ x= / y k T/ y + η ( u/ y) (6) The zeroth and first moments of energy equation (6) have been taken across the film thickness as shown in eq. 7 and eq. 8, respectively. This has been done in order to develop four equations (including boundary condition expressions) for computations of four unknown temperatures at Lobatto locations. h T T T 1 h u dζ = χ + φdζ x y y ρc + h/ + h/ h/ + p h/ (7) h T h T T 1 h u ζdζ = χ + 4 φζdζ x y y ρc 4 + h/ + h/ h/ + p h/ The subscripts + and - are used to denote the upper (billet) and lower (die) surfaces respectively. 3. Computational procedure 3.1 Inlet zone analysis The solution of the proposed governing equations starts for the inlet zone as indicated in the Fig.. The lubricant film thickness h in the inlet zone is expressed by the following relation. h= h + ( x - x ) tan( θ ) (9) 1 1 (8) 37
y Axis Fig. Computational domain in hydrostatic extrusion process The convergence criterions for pressure and temperature in the finite difference solution of the governing equations are as follows: For pressure: For temperature: -3 ( pi) ( pi) ( p i) < - / 10 N-1 N N -3 ( Ti) ( Ti) ( T i) < - / 10 N-1 N N Where N is the iteration number and i denotes the nodal position in the domain. The converged solution for the pressure and temperature are achieved when the convergence criteria for both pressure and temperature are satisfied simultaneously. 3. Work zone analysis In the work zone (refer Fig. ), the film thickness is assumed to be linear and is represented as: h = h ( x / x ) 1 1 Billet velocity at any location x is determined using the conservation of mass principle. Expression for billet velocity is taken as: U = U b ( x1 / x) (11) (10) The lubricant velocity u across the lubricating film at a section x is written as: U 1 dp u = y y yh h η dx ( ) (1) The shear stress for the Newtonian rheology of the lubricant is expressed as: ηu h dp τ = h dx Ludwik strain hardening model [13] is considered in the analysis of work zone (Table 1 for strain hardening constants A 3 and A 4 ) which is written as: (13) 38
A4 σ y = σ 0 + A3ε (14) Where ε is the true strain of billet at any location x ε = ε1 ln x1 where 1 β ε1 = cot β Sin β ε ε x x = 1 ln x1 (15) Due to plastic deformation, the steady state billet temperature is taken as: T T σ ln x y 1 b = 0 + ρ bc x The material parameter (G) is defined as a product of pressure coefficient of viscosity with yield stress of material; G = ασ. (17) y (16) 4. Results and discussions In order to develop the confidence in the proposed mathematical model, the results achieved by the model is compared with the work of Wilson and Walowit [14]. A good correlation between the film thickness results can be seen in Fig.3. This develops a reasonable confidence in the proposed model. Figure 4 presents the pressure distribution in the work zone for the input data listed in the Table 1. It can be seen in this figure that the pressure at the interface reduces in the direction of extrusion. This happens due to loss of pressure in the lubricating film towards the exits side of work zone due to increase in the flow velocity of the material. Variations of friction stress with the lubricating oils are shown in Fig.5. It decreases along the work zone and increases by increase in the viscosity of lubricants. Moreover, the friction stress along the dimensionless work zone is given in Fig. 6 for various values of semi-die angles (θ=10 0 to 0 0 ). With increase in semi-die angle, friction stress increases due to formation of thick lubricating film toward the inlet zone, which consumed power in its shearing. The variation of friction stress along the dimensionless work zone is also shown in Fig. 7 for various values of material parameters (G = 0.67 to 1.86). Friction stress increases by increase in the values of the material parameters. Increase in the values of material parameter takes place due to increase in yield stress of the billet material and increase in the pressure coefficient of viscosity. Variation of the average co-efficient of friction with extrusion ratio (A = to 10) is given in Fig. 8 for three values of semi-die angle (θ = 10 0 to 0 0 ). Friction coefficient decreases by increase in extrusion ratio. It is believed that this happens due to better lubrication at the contact due to pressurized reach of lubricant in the interface at the high extrusion pressure. 39
Table 1: Input Data (1) Material properties 3 Density of billet (ρ b ), Kg/m 713 Initial diameter (D i ), cm 8 Specific heat (C p ), J/Kg-k 870 Thermal conductivity (k b ), W/m-K 37 Billet velocity (U b ), mm/s 40 Yield strength (σ y ), MPa Strain hardening constant (A 3 ) Strain hardening constant (A 4 ) 138 6.1 0.355 () Lubricants properties Lubricant-1 Initial viscosity (η 0 ), N.s/m 0.1 Pressure co-efficient of viscosity (α), Pa -1 7.5x10-9 Temperature co-efficient of viscosity, K -1 0.04 Lubricant- Initial viscosity (η 0 ), N.s/m 0. Pressure co-efficient of viscosity (α), Pa -1 1.5x10-8 Temperature co-efficient of viscosity, K -1 0.03 Lubricant-3 Initial viscosity(η 0 ), N.s/m 0.3 Pressure co-efficient of viscosity (α), Pa -1 1.75x10-8 Temperature co-efficient of viscosity, K -1 0.05 Fig.3 Validation of computational results of minimum film thickness of lubricant in inlet zone (h 1 ) With Wilson and Walowit [14] Formula for different billet velocity (σ y =138.5 MPa, β=15 0, η 0 =0.3, α=1.5 10-8, A=10) 40
Fig. 4 The variation of contact pressure along the dimensionless work zone with various material parameters [Lub-1, Extrusion ratio (A) =6, Semi-die angle (θ) = 15 0 ] Fig. 5 The variation of friction stress along the dimensionless work zone with lubricants [Extrusion ratio (A) = 6, Semi-die angle (θ) = 15 0, Material parameter (G) = 1.19] 41
Fig. 6 The variation of friction stress along the dimensionless work zone with semi die angles [Lub-1, Extrusion ratio (A) = 6, Material parameter (G) = 1.19] Fig. 7 The variation of friction stress along the dimensionless work zone with various material parameters (Lub-1, Extrusion ratio (A) = 6, Semi-die angle (θ) =15 degree) 4
Fig. 8 The variation of friction stress along the dimensionless work zone with various semi die angles ( Lub-1, Material parameter (G) =1.19) 5. Conclusions Following conclusions have been drawn from the mathematical model proposed herein: Friction stress at the contact surface of die is a strong function of lubricating conditions. It increases with increase in viscosity of the lubricant. With increase in semi-die angle, interfacial frictional stress increases significantly. In work zone, friction stress decreases. However, it increases by increase in material parameter. The average co-efficient of friction at die/billet interface reduces with increase in extrusion ratio. References [1] Robertson, J., Method of an apparatus for forming metal articles (1894), British Patent No, 19356, US Patent No, 54504. [] Bridgman, P. W., Studies in large plastic flow and fracture (195), McGraw-Hill, New York. [3] Avitzur, B., Analysis of wire drawing and extrusion through conical dies of small cone angle, ASME Journal of Engineering for Industries, Vol. 85 (1963), pp. 89-95. [4] Avitzur, B., Analysis of wire drawing and extrusion through conical dies of large cone angles, ASME Journal of Engineering for Industries, Vol. 86 (1964), pp. 305-311. [5] Pugh, H. L. D., Redundant work and friction in the hydrostatic extrusion of pure aluminum and aluminum alloy, Journal of mechanical engineering science, Vol. 6 (1964), pp. 36-370. [6] Erans, W. and Avitzur, B., Measurement of friction in drawing, extrusion and rolling, ASME Journal of Lubrication Engineering, Vol. 90(1968), pp. 7-80. [7] Wang, L., Zhou, J., Duszczyk, J. and Katgerman, L., Friction in aluminium extrusion-part 1: A 43
review of friction testing techniques for aluminium extrusion. Tribology International, Vol. 56 (01), pp. 89-98. [8] Elrod, H. G. and Brewe, D. E., Thermohydrodynamic analysis for laminar lubricating films. NASA technical memorandum No. 88845(1986). [9] Ghosh, M. K. and Pandey, R. K., Thermal elastohydrodynamic lubrication of heavily loaded line contacts- an efficient inlet zone analysis. ASME Journal of Tribology, Vol. 10 (1998), pp. 119-15. [10] Pandey, R. K. and Ghosh, M. K., A Thermal analysis of traction in elastohydrodynamic Rolling/Sliding Line Contacts. Wear, Vol. 16 (1998), pp. 106-114. [11] Singh, P., Pandey, R. K. and Nath, Y., An efficient thermal analysis for the prediction of minimum film thickness in inlet zone at high speed lubricated cold strip rolling, Journal of Materials Processing Technology, Vol. 00 (008), pp. 38-49. [1] Tomar, P, Singh, P, Pandey, R. K. and Nath, Y., Thermohydrodynamic analysis of inlet zone for minimum film thickness in hydrostatic cold extrusion process, Proceedings of the 4 th Internatioal Conference of Tribology in Manufacturing Processes (ICTMP-010), Nice-France, 13-15 June, Vol. (010), pp. 477-485. [13] Ludwik, P., Elemente der technologischen Mechanik, 1909 (Berlin) [14] Wilson, W. R. D. and Walowit, J. A., An isothermal hydrodynamic lubrication theory for hydrostatic extrusion and drawing process with conical dies, ASME Journal of Lubrication Technology, Vol. 9 (1971), pp. 69-74. 44