An analysis of the steady-state wire drawing of strain-hardening materials

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Journal of ELSEVIER Journal of Materials Processing Technology 47 (1995} 201-229 Materials Processing Technology An analysis of the steady-state wire drawing of strain-hardening materials U.S Dixit, P.M Dixit* Department o/" Mechanical En~ineerinyL Italian Institute q[" Teclmologo', Kanpur 208 016 India (Received May 24, 1993) Industrial Summary A comprehensive investigation of the steady-state wire drawing process has been done to study the effects of various process variables on important drawing parameters and deformation, the process variables considered being the reduction ratio, the die semi-angle, the coefficient of friction and the back tension, whilst the drawing parameters studied are the diepressure, the drawing stress and the separation force The deformation is represented by contours of equivalent strain and of equivalent strain-rate The quantitative effects of strain hardening on the drawing parameters and qualitative effects on the deformation are studied also A comparison of the drawing parameters is made for three materials (copper, aluminium and steel) Notation A f F2 tr, t: [K] K l li lo I1 I1 area of the d o m a i n average coefficient of friction global right-hand side vector drawing force unit vectors along the r and z directions global coefficient matrix factor in the strain-hardening relationship taper length of the die length of the inlet zone length of the exit zone multiplication factor for the calculation of inlet and exit zone length exponent in the strain-hardening relationship unit normal vector *Corresponding author 0924-0136/95/$09.50 ~O 1995 Elsevier Science S.A All rights reserved SSDI - ( ) - Z 202 U.S Dixit, P.M Dixit / Journal o f Materials Processing Technology 47 (1995) 201-229 P P P Pf Pb r r% r, O, z R1 RE S S Sij , S g t t ti tr, tz tn ts U1 U2 Vr~ Yz Fn Fs Wp, Wr, C( 6~j {A} V r.F ? /:'ij, g ~;kk i}j, ~' P (Tij~ ~7o ~Tf O'xb O'xf hydrostatic pressure power drawing force in Eq (49) power due to plastic work friction power power due to back tension reduction ratio (fractional) percentage reduction cylindrical coordinates initial radius of the wire final radius of the wire coordinate along the taper length of the die separation force deviatoric part of the stress tensor equivalent deviatoric stress time traction vector components of the traction vector components of the traction vector along the r & z directions die- pressure interfacial shear stress inlet velocity of the wire drawing speed velocity vector components of the velocity vector along the r & z directions velocity normal to interface velocity tangential to interface W z weight functions die semi-angle Kronecker's delta global vector of primary variables vector differential operator, del parts of the domain boundary equivalent (plastic) strain equivalent strain-rate strain-rate tensor trace of the strain-rate tensor deviatoric strain-rate tensor density stress tensor yield stress of material at zero plastic strain flow stress of material (Fig 3) back tension average drawing stress U.S Dixit P.M Dixit / Journal of Materials Processblg Technology 47 (1995) 201 229 203 Gy It Ho () d() variable yield stress of material proportionality constant coefficient of friction in Eq (49) constant used for non-dimensionalization of/~ non-dimensional quantity differential value of( ) Introduction Since the appearance of the first paper on metal flow in wire drawing in 1886, by Smith [1], many investigators have studied the process Wistreich [2] and Majors [-3] have obtained the die-pressure experimentally using the split-die technique and by measuring the hoop-strain respectively Avitzur [-4] has described an experimental approach for determining the drawing force as a function of reduction, cone angle and friction Cook and Wistreich [5] have described a method for measuring the diepressure by the photo-elastic method The visioplasticity method introduced by Thomsen et al [6] combines the results of experiment and analysis Hoffman and Sachs [-7] proposed the slab method for the wire drawing process Siebel [8] introduced a theory of wire drawing in which he assumed that the effects of homogeneous deformation, friction, and non-useful deformation were additive, giving an equation for drawing force Avitzur [9] has proposed an extra term to account for redundant power in the drawing stress expression of Sachs [7] Shield [10] has shown that when the von Mises criterion is used for an axisymmettic problem, the general equation is of elliptic type, the slip lines becoming real only under a special condition that reduces the problem to that of a plane-strain type Avitzur [11,12] has applied the upper-bound theorem to the problem of wire drawing, dividing the wire into three zones, in each of which the velocity field was assumed to be continuous: at the interface, however, the tangential component of velocity was discontinuous Avitzur [13] has applied this technique to a strain-hardening material also, where he considered a linear strain-hardening coefficient, using the upper-bound method, he also analyzed the central-burst defect in an extruded or drawn product [14] The early applications of the finite-element method (FEM) to metal forming were based on the plastic stress-strain matrix developed from the Prandtl Reuss equations Iwata et al [15] made an elastoplastic analysis of hydrostatic extrusion using FEM, the analysis being performed for the non-steady state in both plane-strain and axisymmetric extrusion Lee et al [16] also applied the FEM to the non-steady extrusion process, calculating the residual stresses for plane-strain and axisymmetric problems after the external loads were removed From the results of non-steady state extrusion, results for the steady state were speculated Lee et al [17] undertook stress and deformation analysis of steady-state plane-strain extrusion with friction-less curved dies, using the elastic-plastic finite-element method, whilst Shah and Kobayashi [ 18] analyzed axisymmetric extrusion through friction-less conical dies by the rigid-plastic finite-element method, the technique involving the construction of 204 u.s Dixit, P.M Dixit / Journal ~?fMaterials Processing Technology 47 (1995) 201 229 flow lines from velocities and the integration of strain-rates numerically along flow lines to determine the strain distribution Zienkiewicz et al [19] have presented the flow-formulation approach in forming and extrusion, investigating two techniques, viz the pressure-velocity formulation with Lagrangian constraints and the penalty-function approach Tayal and Natarajan [20], Gunasekara et al [21], Balaji et al [22] carried out deformation analysis of the extrusion process by FEM Bianchi and Sheppard [23] compared the viscoplastic finite-element method with, slip-fine field and upper-bound solutions, showing that the FEM gives better results Some analysis of the extrusion process has been done by modifying standard packages [24,25] Wire-drawing bibliography is not so rich as that of extrusion Chen et al [26] obtained the steady-state deformation characteristics in extrusion and drawing as functions of material properties, die work interface friction, die-angle and reduction They observed that although it appears that the differences between extrusion and wire drawing are merely in geometrical quantities and hydrostatic stress components (extrusion being essentially a process of compression, whilst drawing is a process of tension), the finite-element results obtained in extrusion cannot be extrapolated to obtain results in drawing by taking into account the geometrical conditions and the concepts of pushing in extrusion and pulling in drawing Chevalier [27] studied the influence of geometrical parameters and the friction condition on the quality of the final wire using finite-element simulation, an elasto-plastic model being used in the analysis From the literature survey, it is evident that not much work has been done in the area of application of FEM to the wire drawing process considering the strainhardening effect 1.1 Modelling o[" the drawing process In the present study, only the steady-state part of the process is considered, hence an Eulerian formulation is used The process is considered to be axisymmetric, a conical die shape only being considered The material is assumed to be rigid plastic strain hardening and yielding according to the von Mises criterion The elastic effects at the entry and exit are neglected, as these are small The effects of temperature and strain rate (viscoplasticity effects) on the yield strength of the material are ignored in this work, the inclusion of these effects rendering the analysis quite complex, whilst the temperature rise in the presence of lubricants and at low speeds is quite low At high speeds, whatever increase the strain rates may produce in the yield strength, most of this increase is compensated for by a decrease in the yield strength due to the temperature rise The effect of die deformation on various design parameters and product quality has not been reported so far It is believed that the expressions for the die deformation, which will be consistent with the present finite-element formulation, can be obtained only in an iterative way, i.e., the interracial pressure found in the first iteration by assuming the die to be rigid can be used to find the geometry of the deformed die by the finite-element method, this geometry of the deformed die then being used in the second iteration, the iterations being continued until the change in the die U.S Dixit, P.M Dixit / Journal 01 Materials Processing Technology 47 (1995) 201-229 205 deformation between two successive iterations becomes negligible After the deformation, a conical die surface will deform to some other surface Since conical dies only are considered, it was decided not to include this effect in the present study The plastic behaviour of the materials is represented by a relationship between the deviatoric part of the stress tensor and the rate of deformation (i.e the symmetrical part of the velocity gradient) tensor, also called as strain-rate tensor Since the constitutive relationship used is applicable only to the plastic deformation zone, the domain should consist of the portion of wire bounded by the die and the plastic boundaries However, the plastic boundaries are not known a priori: initially therefore, the domain also includes a reasonable length of the wire in the inlet and exit regions, the plastic boundaries being determined later using a suitable criterion on the strain-rate invariant At the die wire interface, the friction is modelled by Coulomb's law, subject to the constraint that the local shear stress cannot exceed the yield shear stress Although it is true that the coefficient of friction varies along the line of contact because of its dependence on factors such as the interracial relative velocity, the die-pressure, the yield strength, etc., it was decided to use a constant coefficient of friction only, the main difficulty in using a variable coefficient of friction being the lack of a wellestablished relationship specifying this variation Continuity and momentum equations of the metal flow in the domain are converted into non-linear algebraic equations using the Galerkin finite-element technique A mixed pressure-velocity formulation is used, the resulting equations being solved by iteration using the Householder [28] method, to find the nodal velocities and pressure To update the value of yield stress in each iteration, flow lines are constructed from the velocity field and the integration of strain-rates along the flow lines is carried out using Simpson's rule to determine the strain distribution From the nodal velocities and strain distribution along the flow lines, the strain rates and stresses are found first at the Gauss points and then extrapolated to find the secondary quantities such as the die-pressure, the drawing and separation forces, the contours of strains and strain-rates and the plastic deformation zone The validity of the model is tested by comparing the results for the drawing and separation forces with available experimental results Problem formulation 2.1 Material behavior In flow formulation, the measure of deformation is the rate of strain tensor, which may be expressed mathematically as g = ~(Vv + (Vv)T) (1) In order to express the stress as a function of strain rate in a convenient form, the stress and strain-rate tensors are divided into two parts: tTij = p U + Sij , (2) 206 U.S Dixit P.M Dixit / Journal of Materials Processing Technology 47 (1995) 201-229 where p = ~ t r a is the hydrostatic part, S is the deviatoric part and u is Kronecker's delta Similarly, eij = ~kkfi~ + gu, (3) where g'is the deviatoric part of the strain-rate tensor In plastic deformation, since there is no change in volume, the hydrostatic part of stress is not related to the deformation Another consequence of volume constancy is that the hydrostatic part of the strain rate is zero and its deviatoric part is the strain-rate tensor itself For a rigid-plastic material, the deviatoric parts of the stress and strain-rate tensors are related by Sij = 2#~ u, (4) where 2/t is the proportionality constant Further, making the definition g (5) and defining the second invariant of strain rate (also known as the equivalent strain rate) as ~, = ~ , ~• • • (6) then it follows that g = 3/~' (7) For metals which yield according to von Mises criterion, g = % (8) Then,/~ for such metals is /~ = O'y/(3~), (9) where Cry is the yield strength of the metal In general, a s is a function of strain, temperature and strain rate However, as stated in Section 1, the temperature and visco-plastic effects can be neglected, and hence it is a function of strain only For a strain-hardening metal, if it is assumed that there is isotropic hardening and that the components of strain-increments bear a constant ratio to one another, then the yield stress can be expressed in terms of the equivalent plastic strain The most common approximation for metals relating equivalent plastic strain to yield stress is given by O'y Cro(1 + K g ) " , = (10) where ao is the yield stress of the metal at zero plastic strain and K and n are metal-dependent coefficients determined from experiment The equivalent plastic strain is obtained by the time integration of equivalent strain-rate as given below: t (11) u.s Dixit, P.M Dixit / Journal of Materials Processing Technology 47 (1995) 201-229 207 Since the material is assumed to be rigid-plastic, the elastic part of the strain is identically zero Therefore, henceforth the adjective plastic is dropped when referring to the equivalent strain 2.2 Governing equations and boundary conditions When the deformation is axi-symmetric, it is convenient to use cylindrical polar co-ordinates In polar co-ordinates, the governing equations for the steady, constant-volume flow can be written as (12) ~ + ~00 + ~= = 0, (13) vr p \{v r =(?v= - + ,,= ~v.'~ ~c z=]/ - \ r{10(ro',,)~ 4- ~ =0 (14) CZ f The boundary conditions for the domain of the present problem (Fig 1) are as follows: (1) Entry and exit boundaries tAB & EF): The control volume is so selected that its entry and exit boundaries are sufficiently distant from both sides of the deformation zone Since the drawing velocity U2 is specified, the z-component of the velocity at every point of EF is known° the r-component of this velocity obviously being zero at both EF and AB The z-component of the velocity at AB (U~) can then be found from the continuity equation (v~-A)AB = (v:A)Ev (15) However, AEv/AAn = (1 r), where r is the reduction ratio Hence, (Vz)AB = (1 - r)(v=)Ev (16) /> r~ [ vr=O, t z =0 /5 vo_Vr = O , t z : Vz =UI RI Vr = D E]v.°21 Vr = ]-i I _1 Vr = O , t z = -t D - "Lo Fig T h e d o m a i n a n d the b o u n d a r y c o n d i t i o n s IR /t- Z~I z (Axis of symmetry} 208 U.S Dixit, P.M Dixit / Journal of Materials Processing Technology 47 (1995) 201 229 Thus the boundary conditions on these boundaries are v~=(1-r)U2, vr=0 on AB, v~= U2, vr=0 on EF (17) (2) The top f r e e surfaces (BC & D E ) and the axis o f s y m m e t r y (AF): The following boundary condition exists on these surfaces: tz 0, vr = (18) (3) The Die wire interface (CD): The component of velocity in the direction normal to the die wire interface at any point on the interface is zero, i.e v,=0 or vr+v~tan~=0, (19) where ~ is the die semi-angle The second boundary condition is given by Coulomb's friction relationship Itsl I (20) wherefis the coefficient of friction, ts is the tangential component of the stress vector and t is the normal component of the stress vector Further, the shear stress on the interface is subject to the constraint Itsl ~< O'y/N/3 • (21) The finite-element formulation presented here does not require any pressure boundary conditions to be satisfied However, there may be a spurious pressure distribution in the solution, i.e the pressure values may be determined only up to an additive constant This latter constant can be determined from the condition that, at the inlet boundary, the pressure values should be equal to one third of the back tension (zero, if there is no back tension) Since there may be slight difference in pressure values at different nodes, the average of these values is taken 2.3 Non-dimensionalization The non-dimensionalization of various physical quantities is undertaken using the following relationships: = r/R2, Z = z/R2; = p/(ao/3); /~ ~r = Ur/U2, Uz ~- u z / U ; (22) (23) where (7 o I~o - 3(Uz/R2)" (24) U.S Dixit, P.M Dixit / Journal o/ Materials Processing Technology 47 (1995) 201 229 209 The non-dimensionalization of continuity and momentum equations are obtained by substituting Eqs (22) (24) into Eqs (12)-(14): ~rr + L00 + ~= = 0, (25) t'o \c,~+~':~)- I'o ~Vr~+V:~) \ a~ + 8e ~ + r Z =0, =0 (26) (27) Since Reynolds number (pU2R2/I~o) is found to be typically of the order of 10 v for all of the cases considered, inertial terms are ignored 2.4 Mixed.finite-element formulation Many workers have used the Galerkin weighted-residual method coupled with direct penalization for finite-element formulation of the governing equations of metal forming However, in the penalty formulation, pressure values converge only for a range of penalty number and there is no systematic method for determining this range For this reason, the mixed pressure-velocity formulation has been used here To avoid the problem of ill-conditioning, the Householder method for solving the system of simultaneous equation has been used This method does not require pivoting, and is a very effective method for the solution of an ill-conditioned set of equations [28] 2.4.1 Galerkin (weak)Jormulation Let g:, /:r, /5 be the functions that satisfy all the essential boundary conditions exactly Then v:, vr, p constitute a weak solution if the following integral equation is satisfied: (/,',r+5,00+L=)wp+\ i ( i + ?5 : Wr A (28) 7- + \ ~ + -c5 / 2rtfd~d,q = O, "j where wp, Wr and w= are the weight functions that satisfy the homogeneous versions of the boundary conditions and A represents the area of the domain Integrating the second and third parts of Eq (28) fl,2~fdFdY.+ f122r~FdFdS-f I32=FdFdS-f142~d?de=O, A A F= F~ (29) 210 U.S Dixit, P.M Dixit / Journal t~f Materials Processing Technology 47 (1995) 201 229 where I, = - w~[-~-rr + ~00 + ~ -=3, (30) /2 = p E ~ ( w ) + ~oo(W) + ~.=(w)] + 2fi[/;.,,~,,(w) + ~oo~oo(W) + k=~==(w) + 2k,=~,=(w)], 13 = t-=w=, I4 = rrw,, (31) (32) and Fr and F= are respectively those parts of the boundary where the traction components t-r and ~ are specified The terms {,~(w), ~oo(W), -~=(w) and L.,(w) are the components of the tensor ~(w) = ½(Vw + (Vw)T), (33) where w (34) = w , i , + w=i= 2.4.2 Finite-element equations The finite-element formulation of the Galerkin integral (29) and the non-dimensional version of the boundary conditions (17)-(20) are similar to that of a flow problem of a non-Newtonian incompressible fluid, the details of which are described in any standard text, viz [29] In the present work, 9-noded rectangular elements are used to descretise the domain (Fig 2), with bi-quadratic approximation for the velocity components and bi-linear approximation for the pressure Since the term ~oo~.oo(W)rof the integrand of Eq (29) contains a 1/f term, x 10 Gauss points are used in the Gauss-Legendre integration scheme for the evaluation of the elemental coefficient matrix The assembly of the elemental coefficient matrices and the right-hand side vectors into the global matrix and vector is done by transferring the elements corresponding to a local degree of freedom in each elemental matrix/vector to positions of the corresponding global degrees of freedom in the global matrix/vector The essential boundary conditions, except for those on the die-wire interface, are applied in the 55 Elements Velocity nodes Pressure nodes • • Nodes for velocity 8L pressure Nodes only for velocity ~, "U////~////~//~ e/ Fig The finite-elementmesh and a typical element ~e' U.S Dixit, P.M Dixit / Journal of Materials Processing Technology 47 (1995) 201-229 215 lO Metal : Copper b~ o o t:~ O80 Experimenf01 [2] FEM 0n01ysJs (f = 0.05) o =40% o o o o o o o _2 o 0.60 0.40 u -~ 0.20 ~c 10%"o"~ 0.% 2'.0 o 4!0 o _ _ _ - 6!0 ~ ~ I 8.0 I ,0'.0 ,20 J ,4.0 ,6.0 Die semiengle, a(deg) Fig Comparison of the relative drawing stress obtained from FEM analysis and from Wistreich's experiments Table Comparison of the separation force obtained from FEM analysis and from Wistreich's equation [2] (~ = 0.03) Reduction (%) Die semi-angle (deg) S/{R 20-0) S/(R22ao) (FEM) (Wistreich) 10 10 10 20 20 20 40 40 40 14 14 14 4.636 3.449 2.840 11.180 6.405 4.446 26.230 19.130 4.090 6.41 4,08 2,55 14.84 7.36 4.80 29.88 16,10 5.30 Error(w.r.t S by Wistreich) (%) 27.7 15.5 11.4 24.7 13.0 7.4 12.2 18.8 22.8 o b s e r v e d that the present F E M analysis gives a linear relationship between the d r a w i n g stress a n d R~/R2, as p r e d i c t e d also by the F E M analysis of Ref [26] H a v i n g c o m p a r e d the results of the present F E M analysis with e x p e r i m e n t a l l y available results, it was ascertained that the present m o d e l w o u l d be able to predict the m a g n i t u d e s of various d r a w i n g p a r a m e t e r s with r e a s o n a b l e engineering a c c u r a c y over m o s t of the range of process variables 3.2 Parametric stud), 3.2.1 Effect of reduction ratio and die semi-angle Fig shows the v a r i a t i o n of d r a w i n g stress with die semi-angle at different reductions W h e n the die-angle is very small, the length of c o n t a c t between the wire 216 U.S Dixit, P.M Dixit / Journal o[ Materials Processing Technology 47 (1995) 201 229 bo Metal : steel o o 1.2 f = 0.10 Unlubricated]~ + ÷ + Lubricated - - FEM J" r_xp , r l / to, I~,b] Analysis 1.0 ~" = * ~ ~ f =0 0.8 "X3 0.6 04 0.2 i g Z OC I I.I 1.2 13 (R~ / R ) Fig Comparison of drawing stress obtained from FEM analysis and from the experimentsof Chen et al Metal : steel 16 o3 f = 003 r=45 L2 % 0.8 1:3 c
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