A comparison of some deformation models in axisymmetric extrusion

Journal of Materials Processing Technology, 33 (1992) 263-272

263

Elsevier

A comparison of some deformation models in axisymmetric extrusion S.B. Altan School of Civil Engineering, Technical University of lstanbul, Maslak, Istanbu180626, Turkey

N. Antar and E. Gultekin Faculty of Art and Science, Technical University of Istanbul, Maslak, Istanbu180626, Turkey (Received February 5, 1991; accepted November 30, 1991 )

Industrial Summary As well as in other plastic-forming processes, in extrusion the importance of knowledge of the deformation of the material during the process is beyond question. This knowledge has been used in all areas associated with the process, from planning of the plant through to the final quality of the product. As the competition between methods of production increases, this kind of knowledge becomes more important. In this study the authors propose a method of constructing kinematically admissible velocity fields appropriate for axisymmetric extrusion and give two applications of it resulting in two new deformation models for direct extrusion. These models are introduced into the upper-bound theorem to find the most appropriate values of the parameters contained in the models. Comparison of the results obtained in this study and Avitzur's spherical velocity field indicates clearly the direction for better deformation models for axisymmetric extrusion. With the method proposed in this study and in the light of experience gained on the nature of the problem, new deformation models, which are expected to give lower values in the upper-bound theorem, are under current investigations.

Notation

r, z

f yr, R1 R2 yo v,

cylindrical coordinates function used to denote a family of flow lines cylindrical components of the spatial velocity field radius of the container (entry radius) radius of the die (exit radius) speed of the ram (entry velocity) velocity of the extrude (exit velocity)

Correspondence to: S.B. Altan, School of Civil Engineering, Technical University of Istanbul, Maslak, Istanbu180626, Turkey.

0924-0136/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

264 e~

ao ~o AV

Ti B S1 $2

(ij)th component of strain-rate tensor flow stress in tension flow stress in shear (~o = ao/~3 for a Von Mises material) jump in velocity surface tractions plastic deformation zone a part of the boundary of B on which V is given complementary to the boundary of B on which Ti is given

1. Introduction

In plastic-forming processes where large plastic deformations occur, solutions for the deformation of the material during the process are not yet well established, except for a few cases. As well as in other processes, many advantages are provided by a sufficientknowledge of the mechanics of the deforming material in extrusion. Especially where precise forming is concerned, the necessity of information on the stress,the strain and the temperature distribution of the extruded material is clear. Over the lastfour decades, plastic-forming processes have become important and have attracted the attention of many scientists.M u c h effort has been devoted to understanding the mechanics of the material in plastic-forming processes, resulting in several methods of solution, such as the slab method, the uniform-energy method, the slip-linemethod, the visioplasticitymethod [14], the finite-differencemethod [5], the flow-lines method [6], the finite-element method [7,8],etc. In the application of these methods, the so-calledupper-bound theorem, which is a variational principle for boundary value problems in plasticity,is used frequently [1,2,9].The firststep in the application of this variational method is to propose a kinematically admissible velocity fieldcontaining some parameters, that is,a spatialvelocity fielddepending on some parameters and satisfying the incompressibility condition and the displacement boundary conditions. Since, to the best knowledge of the authors, there is no method available in the literature for constructing kinematically admissible velocity fieldsappropriate for the problem under consideration, it is clearly important to have available such a method for analyzing this kind of problem. Further, in all applications of the upper-bound theorem known to the authors, the proposed deformation models contain some discontinuitiesin the velocity field, which the authors feel intuitively is contrary to the second principle of mechanics, that is, the conservation of m o m e n t u m and to the experimental results [ 10 ]. Since the main purpose of this study is to compare some deformation models with the aid of the upper-bound theorem, the problem is formulated in its most simple form to make the point clear. The material is taken therefore as a rigidplastic, non-hardening Von Mises material and the effect of temperature and

265

friction between the billet and the container and between the extrude and the die is neglected. In the subsequent section, the method for constructing kinematically admissible velocity fields which was introduced in details in a previous work [11] is summarized. By applying this method two deformation models for axisymmetric extrusion are constructed. These two models are introduced into the upper-bound theorem for finding a parameter of the models: this is chosen as the length of the deformation zone. In the final section the results obtained in this study are compared with each other and with the results of employing Avitzur's spherical velocity field [9]. 2. Construction of kinematically admissible velocity fields

In this section, a method for constructing kinematically admissible velocity fields in axisymmetric extrusion is summarized. Assume that the flow lines in the plastic-deformation region can be expressed as a one-parameter family of curves

f(r,z)=C

(1)

Since the velocity vector at a point is tangential to the flow line passing through that point,

V= dz Vr-dr-

Of~Or Of/Oz

(2)

On the other hand, a kinematically admissible velocity field should satisfy the incompressibility condition

ovr _vr +oV =o Or

r

0z

(3)

which is written in cylindrical coordinates. By eliminating V~ or (2) and (3)

OVr Or

Ofl~rOVr (~ O ~flOr~ 0z Of/Oz Oz

Vr between

=0

(4)

or

OVz O[/OzOVz Oz Of/Or Or

1 Of/Oz , O Of/Oz~ r ~ - ~ -t OrOf/Or] Vz=O

(5)

respectively. The solution of these first order, quasi-linear differential equations, can be written as

Fl(gl,g2)=O or g~=F2(g2) where gl and g2 are the solutions of the adjoint system of equations:

(6)

266

(5f/Sr) , (1 50HOr~ Vr/dVr (of/Oz)/O_Z=t r_Oz

(7)

for (4) and

dz-1 ~(O[lOz) O /f r]O Vz ./dVz / a r = ~I.r (10flOz Sf/Or ~ OzOOflOz

(8)

for (5). It is an easy task to verify that one of the solutions of both equation is

f(r,z)=C1

(9)

where C1 is an arbitrary constant. It is also possible to show that the second solution of (8) is

10f Vz =C2 - -r 0r

(10)

where C2 is another arbitrary constant. The general solution of (5) can now be written as

O[f(f) _1 0 - r ~ r a(f)

v~=l-~rr

(11)

and from (2)

"dr= l Of F ( f ) = _ l ~ a ( f ) r 5z r ~z

(12)

where F (or G) is an arbitrary function which will be described by the initial conditions of the problem. The initial condition of the problem under consideration is the velocity field on either the entry or the exit surfaces of the deformation zone. Although the form of the spatial velocity fields given by (11) and (12) is not new, being known as the flow potentials in the literature [6], the dependence of the flow potentials on the flow lines indicated by these expressions is interesting and provides a powerful tool for investigating the deformation in axisymmetric extrusion. Although expressions (11) and (12) enable the formulation of the problem as a free boundary problem, the authors prefer to restrict theirselves in this study to analyzing the effect of the type of flow lines. 3. T w o d e f o r m a t i o n m o d e l s

In this section are introduced two deformation models for analyzing the effect of discontinuity in the velocity field. In both models the entry and exit surfaces are chosen as z=O and z=A, where A, being the length of the defor-

267

mation zone, is the parameter of both models, to be determined by the upperbound theorem. In the first model, the flow lines are chosen as straight lines (Fig. 1 (a)):

r=C(~-~z+R2)

(13)

where C is the parameter of the family, of value 0 ~
=,Container Dead -Metal ~ ' ~

f(r z) r/(R,- R~ ' =_.~__. R= A R,-R~

Billet v.

Die

tRz

//"

,,.~

'

///..

Deformation

Zone

_ r

(°'

,-

A3 r

A

I

~

r

Co ntai ner R~

(

Billet

Die

JR~ v~. Deformation ~

_ ~ - f (r, i~} = o

Zone Z

(b)

Fig. 1. Two deformation models: (a) the flow lines are straight lines; and (b) the flow lines are cubic.

268

Yr'~ -- Yo \R1 -Re ] \A]

( R1 ~2( z Vz = - Vo

~-4 R1 -R2

(14)

R2 ) -2

-A Rx - R 2

The cylindrical components of the strain-rate tensor that will be used in the upper-bound theorem for the deformation model (14) are R~ 2 1 z -3

~rr=__Vo(el~R2) (~)(_A ~ RI R2 ~R2 ) R--~2 ~ 2 ~ A ~ ( n erz : -- Vo ~,R1 - R 2 ] \Ae,]

--4

(15)

RI~R2)

eoo ~-err ezz -~--- 2err ero = 0 ezo = 0 In the second model, the family of flow lines is chosen as

3 r= cR1-R2

z2(3A-2z)4 R~R2 _R~ A 3)

(16)

where C is the parameter of the family, of value 0~< C~< 1 (Fig. 1 (b)). It can be verified also that the cylindrical components of the kinematically admissible velocity field corresponding to the flow lines (16) are

R2 2 A a 6rz(A-z) z2(3A-2z)-~ R1 -R2 Aa Vr = -- Vf RI~R (~

)2(

Vz='-Vf R1 R2 e3

R2

(17)

)-2

z2(3A-2Z)4 R(~R2A3

and the components of the strain-rate tensor are

2

(

--3

err=--Yf -~I~Re A~ 6z(A-z) z2(3A-2z)4 R,_R~A 3 drz = -- Vf

(

R2 A3) 2 ---R2

X[18rz ~, A - z , 2 - r , A - 2 z , ( z 2 ( 3 A - 2 z , ~ R~R1 _R~ ( ~ a ~ ) -4 × z2(3A-2z)'4 R1-R-~2 doo=d,, d= = - 2d,, drO=0 d~O= 0

(18)

269

The parameter A appearing in both models will be found by referring the upper-bound theorem in the following section. 4. Upper-bound theorem

As is well known, the upper-bound theorem states that the functional

J= 2-~aof x/½~.ii~ijdV+f .:o,~Vlda-f TiVida x/3

B

S1

(19)

$2

attends its minimum value for the solution of the problem under consideration, among all kinematically admissible velocity fields for a rigid-plastic von Mises material. As is emphasized in the introduction, the effects of friction in both the container and the die, temperature effects and strain hardening are neglected, since the aim of this study is to compare some deformation models. All the expressions given below are factorized by 2ao For the first model, the powers expended in different zones can be expressed analytically. The power expended in the deformation zone is

Vo/v/3.

PDI=3-~(Rl~R2A)2{[4+3(R1AR2)2]s/2-8}ln(~2)

(20)

the power expended on the entry and exit surfaces of the deformation zone being equal and given by PS1 = ~ (R1) 2 R1 A -R2

(21)

and the power expended on the surface between the plastic deformation zone and the dead metal zone being given by (R1) 2

1+

The total power expended for the first model is therefore

(23)

J~ =PD~ +2PS~ +PM1

For the second model, the power expended in the plastic deformation zone is calculated from PD2 .4

F2

=C1 ~ ~ 12[z(A-z)F2(z)]2+ [18rz2(A-z)2-r(A-2z)F2(z) ]2r d r d z z=0 0

(F2(z))4

(24)

270 where

CI=6(-~-R2Aa) 2

and

F2(z)=Rl~-53R2(z2(3A-2z)-~R,3)_R2A R~

and the power expended on the boundary surface between the dead-metal zone and the deformation zone from A

)~21-R2 A3 -~1+ 36[ (R1-R2)z(A-Z)R2 ]2 dz PM2 -- R(R1 o z2 (3A-2z)÷-A3 R1 - R 2

(25)

Of course, the power expended on the entry and exit surfaces of the deformation zone is zero in this model. The total power expended for the second deformation model is therefore J2 = PD2 + PM2

(26)

The deformation models corresponding to the value of the parameter A for which the functionals attends to minimum denote the best deformation models i.e., the closest ones to the exact solution of the problem, among the proposed family. Of course, the model for which the upper-bound theorem gives a lower value is comparatively better than the others. The minimum of the functionals given by (23) and (26) are searched for with the aid of a computer, the partial results being given in the following section. 5. Results and discussion

Since the aim of this study is to compare the effect of the flow lines the results of both model are given in one example: similar trends are observed for other extrusion conditions. For the example given in the following table, R1 = 50 ram, R2 = 25 m m and Vo= 10 m m / s are chosen. The results shown in Table 1 indicate that the second model, for which the spatial velocity field is continuous on the entry and exit surfaces of the deformation zone, is slightly better than the first model. However, the difference between the models is very small since the power gained by removing the discontinuity on the entry and exit surfaces of the deformation zone is given back to the system almost entirely by the increase of the trajectories of the material points in the deformation zone causing an increase in the powers expended in the deformation zone (PD2) and on the surface between the deformation zone and the dead-metal zone (PM2). This observation leads to the following conclusion: for better deformation models, the change of curvature of the flow lines should be concentrated near to the entry and the exit surfaces of the defor-

271 TABLE

1

Powers expended in models I and 2

A PD1 PD2 PM1 PM2 PS1 PS2 J1 J2

3.82 8.077 10.109 9.455 10.112 2.727 0.000 20.259 20.222

3.84 8.072 10.089 9.475 10.130 2.713 0.000 20.259 20.219

3.86 8.066 10.070 9.495 10.148 2.699 0.000 20.259 20,217

3.92 8.050 10.012 9.556 10.203 2.657 0.000 20.263 20.214

3.94 8.044 9.993 9.576 10.221 2.644 0.000 20.264 20.214

3.96 8.039 9.974 9.597 10.240 2.630 0.000 20.267 20.214

mation zone, i.e., flow lines should be almost straight lines in the deformation zone except near to the entry and the exit surfaces. The researches made in this direction indicate that higher-order polynomials may serve for this purpose: the results of this latter research are the subject of another paper which will be published elsewhere. On the other hand, the so-called 'Avitzur's spherical velocity field' [9] gives J = 18.980 for the same extrusion conditions. The unique difference between the first model in this study and 'Avitzur's spherical velocity field' is the boundaries of the deformation zone, which in the Avitzur model are concentric spheres perpendicular to the flow lines. This observation indicates the importance of the boundaries of the deformation zone for a better deformation model. With the formalism introduced in this study it is possible to leave these boundaries as 'unknowns' and the problem can be formulated as a semi-free-boundary problem. The results of the research made in this direction will be reported in a forthcoming paper.

References

1 R. Hill, The Mathematical Theory of Plasticity, Clarendon, Oxford, 1950. 2 W. Johnson and P.B. Mellor, Engineering Plasticity, Van Nostrand Reinhold, London, 1973. 3 T.Z. Blazynski, Metal Forming: Tool Profiles and Flow, MacMillan, London-Basingstoke, 1976. 4 E.G. Thomsen, Investigation of the application of visioplasticitymethod of analysis to metal deformation processing, Technical Report, University of California, 1967. 5 A.H. Shabaik, Finite difference method for the complete analysis of plane strain extrusion, Proc. NAMRC-III, Dearborn, Michigan, Society of Manufacturing Engineers, 1974, pp. 127147. 6 E.R. Lambert, H.S. Mehta and S. Kobayashi, A new upper bound method for analysis of some steady-state plastic deformation process, J. Eng. Ind., Trans. ASME, Series B, 91 (1969) 731-742. 7 N.L. Dung and O. Mahrenholt, Progress in the analysis of unsteady metal-forming processes using the finite element method, Proc. Conf. Numerical Methods in Industrial Forming Processes, Swansea, Pineridge Press, Swansea, 1982, pp. 187-196.

272 8

N.L. Dung, O. Mahrenholt and C. Westerling, Finite element modelling of precise forming processes, ASM Metals Congress, Philadelphia, 1983. 9 B. Avitzur, Metal Forming: Processes and Analysis, MacGraw-Hill, New York, 1968. 10 A. Shabaik and S. Kobayashi, Computer application to the visioplasticity method, J. Eng. Ind., Trans. ASME, Series B, 89 (1966) 339-346. 11 S.B. Altan, M. Gevrek and S. Onurlu, A method for deformation analysis in axisymmetric extrusion, Bull. Tech. Univ. Istanbul, 37(4) (1984) 489-504.