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

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 o

Journal of Materials

Processing

Technology

E L S E V I E R Journal of Materials Processing Technology 47 (1995} 201-229

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 de- formation, 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 die- pressure, 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)

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 n o r m a l vector

*Corresponding author

0924-0136/95/$09.50 ~O 1995 Elsevier Science S.A All rights reserved

SSDI 0 9 2 4 - 0 1 3 6 ( 9 5 ) 0 1 3 2 0 - Z

202 U.S Dixit, P M Dixit / Journal o f Materials Processing Technology 47 (1995) 2 0 1 - 2 2 9

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

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

coefficient of friction in Eq (49)

constant used for non-dimensionalization of/~

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 axisymmet- tic 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 ~?f Materials 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 pen- alty-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 strain- hardening 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 deforma- tion, 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 well- established relationship specifying this variation

Continuity and momentum equations of the metal flow in the domain are con- verted into non-linear algebraic equations using the Galerkin finite-element tech- nique 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

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 6 u is Kronecker's delta Similarly,

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

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)

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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, con- stant-volume flow can be written as

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

(16) (Vz)AB = (1 - r)(v=)Ev

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Thus the boundary conditions on these boundaries are

to the die wire interface at any point on the interface is zero, i.e

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

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):

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:

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, (29)

210 U.S Dixit, P.M Dixit / Journal t~f Materials Processing Technology 47 (1995) 201 229

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

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)r of the integrand of Eq (29) contains a 1/f term, 3 x 10 Gauss points are used

in the Gauss-Legendre integration scheme for the evaluation of the elemental coeffi- cient 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

U.S Dixit, P.M Dixit / Journal of Materials Processing Technology 47 (1995) 201 229 211

usual way The conditions on the interface (Eqs (19)-(20)) are applied by performing certain row operations on the global coefficient matrix and the global right-hand side vector, the details of which are given in the thesis of Dixit [30] Letting the global equation after application of all the boundary conditions be

where [K] is the global coefficient matrix, {F} is the global right-hand side vector and {A} is the global vector of primary unknowns (i.e velocity components and pressure)

2.4.3 Formulation for strain hardening

The equivalent strain at a point is obtained by the time integration of the equivalent strain rate along the particle path, as given by Eq (11) The first step in the determination of the equivalent strain field is the construction of particle paths, or 'flow lines' as they are called The slope of a flow line is given by

Now, along a flow line,

to obtain the equivalent strains at the Gauss points These values are then substituted into Eq (10) to update the values of cry for further iteration

U.S Dixit, P.M Dixit / Journal q[" Materials" Processing Technology 47 (1995) 201 229

The strategy followed for obtaining the converged solution with and without strain hardening is as follows First a converged solution (nodal velocities and pressures) without strain hardening is obtained For this part of the analysis, an initial guess for

is taken as 10 This solution is then used as the initial solution for the analysis with strain-hardening and the yield strength is updated by evaluating the strains from

Eq (39) using the immediately preceding solution in each successive iteration

A 55-element mesh is selected (Fig 2) Because of discontinuity at the corner points, smaller size elements are chosen near to the corner points Similarly, elements adjacent to the center line are also of smaller size The high value of ~ in the inlet and exit zones renders the coefficient matrix ill-conditioned To counter this problem, ~i is restricted to a threshold maximum, which is 100 in the present analysis

The convergence criteria used for primary variables is as follows All values of velocities of less than 10 4 and all values of pressures of less than 10-3 are considered

to be zero and not checked for convergence A percentage difference of less than 10-2 for velocities and of 0.1 for pressure, between the solutions of two successive iterations,

is considered to mark the convergence of the solution

2.5 Evaluation of the secondary quantities

Once the solution of the problem is obtained in the form of nodal velocities and pressures, the secondary quantities, viz the drawing force, the die-pressure, the separation force, the deformation zone, etc., are calculated as explained below

The drawing force can be calculated by integration of the drawing stress along the exit plastic boundary However, since it is difficult to determine the plastic boundary accurately, the drawing force is calculated from the relationship

where the total power P consists of the following three parts

(a) Internal power dissipation (Pp) The power dissipated due to plastic deforma- tion is given by

P p = 2r~ f Sij ~'ij r dr dz

A

(42)

u.s Dixit P.M Dixit / Journal ofi Materials Processing Technology 47 (1995) 201 229 213 Substitution of Eqs (4), (6) and (9) into this equation leads to

overcome the back tension O'xb is given by

R1

i ]

0

where U1 is the velocity at the inlet

The average drawing stress (~xf) is found by dividing the drawing force by the exit cross-sectional area of the wire

(ii) Die pressure (t,)

While calculating t,, first the stresses ~i~ are evaluated at 2 x 2 Gauss points and then these are extrapolated to various points on the die wire interface, after which t, is calculated from the expression

where

and fi is the unit outward normal to the interface

(iii) Separation force (S)

The separation force is the resultant of the vertical component of the stress vector

on the upper half of the die Since the stress vector is independent of 0, the integration along 0 direction leads to the product of the vertical component and the projected length 2r Thus the separation force is given by

l

, d

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~, over the domain

3 Results and discussion

The finite-element modelling of the physical problem of wire drawing developed in the previous section has been applied to a number of cases involving three metals and various non-dimensional process variables, the metals considered being aluminium, steel and copper, having material properties as given in Table 1

3.1 Validation

The results of the finite-element analysis have been compared with Wistreich's results [2] for copper (Fig 3), good agreement being observed in all the cases Wistreich used the following equation to relate the separation force S and the drawing force P:

where/~ is the coefficient of friction Table 2 compares the separation force obtained

by the F E M analysis with the separation force obtained from Wistreich's equation where the drawing force P has been obtained from the F E M analysis It is observed that there is good agreement between the two values, except for combinations of low cone-angle and low reduction and of high cone-angle and high reduction

Fig 4 compares the results of the F E M analysis with the experimental results of Chen et al [26] for steel, there being fairly good agreement between the two It is to be