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Metal-forming process analysis by the visioplasticity method: Application to stationary and non-stationary processes
Journal of Materials Processing Technology, 31 (1992) 199-208
199
Elsevier
Metal-forming process analysis by the visioplasticity method: Application to stationary and non-stationary processes L. Chevalier and N. Dahan Laboratoire de Mdcanique et Technologie, E.N.S. Cachan,C.N.R.S., Universitd Paris 6, 61, Avenue du Prdsident Wilson, 94235 Cachan Cedex, France (Accepted in revised form August 31, 1991 )
Industrial Summary In this paper the authors describe an application of the experimental method called "visioplasticity": the latter allows the determination of strain and strain-rate information at every point within a deformed piece. This information can be compared with finite-element simulation in order to validate a model. Using a rigid-plastic model, this experimental method allows the calculation of the stress components. In the present work, this latter step is developed for stationary problems, where a single image sums up the whole process. To solve non-stationaryproblems, the authors propose the employment of a simplified method by using the minimum energy condition. Finally, it is demonstrated how the stress components can give the frictional conditions between the workpiece and the tool in metal-forming processes and help to develop process models.
1. Introduction
Numerical simulation of metal-forming processes requires a model of the tool-workpiece contact. The Coulomb friction coefficient, for example, is not easy to measure experimentally and friction models are difficult to validate. However, accurate information on this point can be obtained by using the visioplasticity method [ 1-3 ]. The visioplasticity method can be used for either plane-strain or axi-symmetric analysis, the present analysis being limited to the second class of problems. The first step of the method is an experimental simulation of the process. The workpiece is cut into two parts along a radial plane on which a rectangular grid is then marked. The two parts are put back together and the forming process is carried out. The deformed grid is digitised and the displacement components measured. In a stationary problem, a single image of the specimen at time z sums up the whole process. For a non-stationary problem, visioplastic images at every 0924-0136/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
200
time z are needed to determine D by finite differences between two times r + Az and z, after which the history of o~can be integrated. This is a quite heavy and expensive method because an experimental simulation must be carried out for each step z. The minimum energy theorem developed by Dammame [4] gives an upper-bound approximation for ~ by calculating the eigenvalues of the Lagrangian dilatation tensor C. By using this approximation, only the first and last steps are needed, together with a step near the end of the process to &. The authors have tested this method on billet extrusion, on compression and, more recently, on an industrial process to form conical heads. 2. S t r a i n a n d s t r a i n - r a t e t e n s o r s
The kinematics concept of large displacement analysis is now given (see Fig. 1). At the initial instant, a particle M is in position Mo. The same particle moves to Mt at time t, after the transformation. The displacement U(Mo,t) can be measured directly on the deformed grid. Let F be the deformation gradient F = 1 +grad U
(1)
The local initial configuration So following particle M, becomes St by the F linear transformation
S t = F So
(2)
For an axi-symmetric process, the components of the F tensor are given by Ort
F=
0
Ort
r~
ro
Oz~ ~r 0
0
(3)
Oz~ 0
Oz0
(er,eO,ez)
A strain measure can be defined by using the Lagrangian C tensor or Eulerian ----4 er
A
F
Cr
_A
Fig. 1. Description of large displacements.
201
B tensor representation [5 ]. A Lagrangian formulation will be used and strain distributions are calculated by using Green-Lagrange: E=½(C-1)
where
C=FTF
(4)
Green-Lagrange strain measure is given by
E=
1
I
2L\Oro] \OroJ
_.1
0
0
!(Ort 05 Ozt Oz,~
=L\ro)
2\Oro Oz---oo÷ Or---~oOZo]
0
!(Oft Or, Ozt Oz~
,
~\aro OZo aro
1 J
iF/Ort'2
OZoJ
0
1
(5)
[ Ozt'~2 ]
2[k-~Zo) +k-~Zo) - 1
The linear transformation F can be calculated by replacing the space derivatives by finite differences (see Fig. 2). Any strain measure (Green-Lagrange E; Logarithmic zt; Almansi A, etc. ) can be obtained from F.
Ort rt(ro +ao;Zo)-rt(ro-ao;Zo) 0ro ~
2ao
(6)
Ozt ,, zt(ro;Zo +bo) -zt(ro;Zo -bo) 0Zo
2bo
where ao and bo are the initial grid dimensions. For the time derivative, the same particle must be followed, which can be done by working on two consecutive configurations t and t' = t + At.
~ AFter,
j~ AEt~t,
At
(7)
At 7(to+ Oo, Zo)
~(ro+ Oo,Zo).
I*0+ O 0 Zo I
IF
/ So "-" /
Mo
ro zo- bo
= ~ro
~ M 7 ( to, zo- bo} ( ro, zo- bo)_ _ ~
/~f~~ ~
z°+ bo r0 - o 0 ;t o
Fig. 2. Finite differences approximation.
-r-( to- ao,Zo) T ( ro-Oo,Zo)
"f'( r°' z°+ b°) (r°' z°+ b°)
202
The strain rate becomes
D = F T ~ F ~ F w-hEt_+t, - F At
sothat
JD..~DAt,~FTAEt~t,F
(8)
3. Deviatoric stress and hydrostatic p r e s s u r e
It is assumed that the material is rigid-plastic, and that the von Mises yield criterion associated with the Ludvick hardening law is used.
f=x/-~ (aD:aD)--aeq--R with R=ga n
(9)
A standard thermodynamic study gives the evolution laws, where _~ is the cumulative plastic strain and R the associated force. D is the strain-rate tensor and o"D the deviator of Cauchy stress
D=~
and
&=-4
-
8f
OR
is obtained by the consistency condition _a=x/~ ( D : D )
and
D=~ (a_/R)aD
(10)
dr= O.Thus (11)
The stress deviator is given by ~
~,
IR.DIIIDII
(12)
The complete Cauchy stress can be obtained by calculating the hydrostatic pressure in the workpiece. This is done by integrating the equilibrium equation between a point belonging to the free surface and the current point
(13)
div (aD) -- grad (p) = 0
where p = - 1 / 3 tr (a) is the opposite of the hydrostatic stress all. Equation (13) can be summed between a free surface of the workpiece to obtain the hydrostatic pressure at all points (see Fig. 3):
O
Fig. 3. Boundary conditions.
A
203
P(M)=p(O)÷IOM diV(OD)dM
(14)
4. Visioplastic analysis of a stationary problem: Extrusion In a stationary problem a single image of the specimen at time z sums up the whole process. The strain rate can be obtained by replacing time derivatives by derivatives along a trajectory. For example:
~, E(ro,Zo +bo) -E(ro,Zo) M(ro,zo)M(ro,Zo +bo) M ( ro,zo )M ( ro,Zo + bo ) At
(15)
where the first term on the right-hand side of eqn. (15) is ~ and the second term is V(M). The latter is constant for a stationary problem. A smoother evolution for E is obtained by using an average value calculated between two points placed after and before M(ro,Zo)
l[ (E(ro,zo +bo)-E(ro,Zo) )
=2L
(E(ro,Zo)-E(ro,zo-bo) ) 1
J
I : 0.00 2:0.07 3:0.15
(a)
(c)
4:0.24
5:0,30
I: -0.20 2:-0.15 3;-0.10 4: - 0.05 5: 0.00
(b)
(d)
(16)
I:-0.13 2: 0.00 3:0.15 4: 0.33 5:0.50
I:-0.40 2:-0.30 3:-0.20 4:-0.10 5: 0.00
Fig.4. Green-Lagrangestraindistribution: (a) dr; (b) ~z; (c) ¢~;and (d) ¢P~.(stationaryproblem)
204
(a)
(C)
I:0 2:-30 3:-50 4:-75 5:-100
(b)
4:-60 5:-80
1:10 2:0 3:-20
4:-40
(d)
5:-60
Fig. 5. Cauchy stress distribution: (a) arr; (b)
1:0 2:-20 3:-40
1:-35 2:-25 3:-15 4:-5
1
5:5
6:20
aoe;
(c) azz;and (d) ar~.
with fixed ro. The cumulative plastic strain _~ can be computed easily, so that R and o"D can be known. By integrating the equilibrium equation along a flow line the pressure can be obtained, so that all of the stress tensor is known. The following results are given for a 30-CD-4 tempered steel. This billet has been extruded: its initial diameter was 35 mm and the extrusion conditions were 30 ° semi-cone die angle and a 30% reduction rate. Figure 4 shows the Green-Lagrange strain distribution and Fig. 5 shows the Cauchy stress components.
5. Visioplastic analysis of non-stationary problem For a non-stationary problem, visioplastic images at every time 3 are needed to determine D by finite differences between two times 3+ A3 and 3, after which the history of_~ can be integrated. This is a quite tedious and expensive method because an experimental simulation must be carried out for each time step 3. Figure 6 shows the different billet-compression steps needed to obtain accurate information on the time evolution of all the variables of the problem. The minimum energy theorem developed by D a m m a m e [4 ] gives an upper bound approximation for _~, by calculating the eigenvalues of the Lagrangian dilatation tensor C.
205 2o 1.
.$S ....... ti .'~o. ~L S.
Fig. 6. Billet compressionsteps (20, 30, 35, 40, 45, 50, 55, 65, 70, 75, 80, 90 and 100).
I
I
Fig. 7. Conicalhead formingprocess.
~=
~1 i=3 ~ log (Ci) 2
(16)
i=1
By using this approximation, only the first and last steps are needed in order to be able to calculate _a. With a step near to the end of the process & can be computed. The authors have tested the above method on an industrial-process to form conical heads. A 20-MB-5 steel specimen is used, being first laminated and then subjected to backward extrusion, after which the conical head is obtained by a forming process (see Fig. 7). The photographs of Fig. 8 show the initial and final grids printed on the workpiece. Figure 9 shows the G r e e n -
Fig. 8. Initial (on left) and final grids on a 20-MB-5 conical head.
2: 3: 4: 5: 6:
(a)
(b)
Fig. 9. Green-Lagrange strain distribution: (a) &; (b) &; (c) &; and (d) tYz.(non-stationary problem)
207
Fig. 10. Von Mises equivalent stress distribution, a~: (1) 0; (2) 20; (3) 40; (4) 60; (5) 80; and (6) 100 daN/mm 2.
f= ~
A 0,2+ /
0,35 0 30 "~,15 z
Fig. 11. Value of the friction coefficient.
208 Lagrange strain distribution, whilst the von Mises equivalent stress is shown in Fig. 10. 6. Friction coefficient
Examining the ratio of the shear stress r to the normal stress a along the workpiece boundary gives information on the Coulomb friction coefficient f. The evolution o f / g i v e n in Fig. 11 shows clearly t h a t f is not a constant coefficient depending on the two interacting materials only. 7. Conclusions
Although the visioplasticity method is an approximate approach (the billet is cut, then deformed), the strain distribution, especially, gives relevant information on the mechanical behavior of the material. In all cases the stress distribution needs three successive discretizations, so t h a t the results are not very smooth: a special effort needs to be made to obtain accurate results. The friction coefficient obtained can be used in a finite-element simulation of the same process to gain more precise information on the stress distribution. This kind of simulation has been carried out already for the extrusion process [ 6 ] and a comparison of experimental and numerical solutions has been made [ 7 ]. F u r t h e r development of the method can be undertaken, if more precise information on the boundary surface of the workpiece is required. The grids are so distorted in these areas t h a t it is difficult to gain relevant information on the displacement field. An effort should be made to analyze the contact properties using the visioplasticity method.
References
1 S. Kobayashi and A. Shabaik, Computer application to the visioplasticitymethod, J. Eng. Ind., (1967) 339-346. 2 J.P. Bernardou, Calcul des ddformations minimales dans les pi~ces axisym~triquesforgoes, 4~me CoUoque Mdcanique et MdtaUurgie de Tarbes, 1980. 3 P. le Nevez and N. Dahan, M~thodede calculsdes grandes ddformationsplastiques en endommagementdans les pi~cesextrud~es,M~m. Etud. Sci. Rev. MetaU., (1983) 557-566. 4 G. Dammame, Minimum de la deformation gSnSralis~e d'un ~lSment de mati~re, pour des chemins de d~formationpassant d'un Stat initial/l un dtat final donn~s, C. R Acad. Sci., 287 (1978). 5 N. Dahan, B. Gathouffi and M.F. Molez, Sur les mesures en grandes d~formations,Rapport interne Laboratoire de M$caniqueet Technologie,No. 37, 1983. 6 L. Chevalier, Caracteristique des materiaux tr$fil~s apres l'opdration de tr$filage, Th~se de l'Universit~ Paris 6, May, 1988. 7 L. Chevalier, Simulation of cold extrusion process: A numerical experimental comparison, Third Int. Conf. on Structural Analysis Systems, Stafford, UK, 1987, pp. 25-34.
199
Elsevier
Metal-forming process analysis by the visioplasticity method: Application to stationary and non-stationary processes L. Chevalier and N. Dahan Laboratoire de Mdcanique et Technologie, E.N.S. Cachan,C.N.R.S., Universitd Paris 6, 61, Avenue du Prdsident Wilson, 94235 Cachan Cedex, France (Accepted in revised form August 31, 1991 )
Industrial Summary In this paper the authors describe an application of the experimental method called "visioplasticity": the latter allows the determination of strain and strain-rate information at every point within a deformed piece. This information can be compared with finite-element simulation in order to validate a model. Using a rigid-plastic model, this experimental method allows the calculation of the stress components. In the present work, this latter step is developed for stationary problems, where a single image sums up the whole process. To solve non-stationaryproblems, the authors propose the employment of a simplified method by using the minimum energy condition. Finally, it is demonstrated how the stress components can give the frictional conditions between the workpiece and the tool in metal-forming processes and help to develop process models.
1. Introduction
Numerical simulation of metal-forming processes requires a model of the tool-workpiece contact. The Coulomb friction coefficient, for example, is not easy to measure experimentally and friction models are difficult to validate. However, accurate information on this point can be obtained by using the visioplasticity method [ 1-3 ]. The visioplasticity method can be used for either plane-strain or axi-symmetric analysis, the present analysis being limited to the second class of problems. The first step of the method is an experimental simulation of the process. The workpiece is cut into two parts along a radial plane on which a rectangular grid is then marked. The two parts are put back together and the forming process is carried out. The deformed grid is digitised and the displacement components measured. In a stationary problem, a single image of the specimen at time z sums up the whole process. For a non-stationary problem, visioplastic images at every 0924-0136/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
200
time z are needed to determine D by finite differences between two times r + Az and z, after which the history of o~can be integrated. This is a quite heavy and expensive method because an experimental simulation must be carried out for each step z. The minimum energy theorem developed by Dammame [4] gives an upper-bound approximation for ~ by calculating the eigenvalues of the Lagrangian dilatation tensor C. By using this approximation, only the first and last steps are needed, together with a step near the end of the process to &. The authors have tested this method on billet extrusion, on compression and, more recently, on an industrial process to form conical heads. 2. S t r a i n a n d s t r a i n - r a t e t e n s o r s
The kinematics concept of large displacement analysis is now given (see Fig. 1). At the initial instant, a particle M is in position Mo. The same particle moves to Mt at time t, after the transformation. The displacement U(Mo,t) can be measured directly on the deformed grid. Let F be the deformation gradient F = 1 +grad U
(1)
The local initial configuration So following particle M, becomes St by the F linear transformation
S t = F So
(2)
For an axi-symmetric process, the components of the F tensor are given by Ort
F=
0
Ort
r~
ro
Oz~ ~r 0
0
(3)
Oz~ 0
Oz0
(er,eO,ez)
A strain measure can be defined by using the Lagrangian C tensor or Eulerian ----4 er
A
F
Cr
_A
Fig. 1. Description of large displacements.
201
B tensor representation [5 ]. A Lagrangian formulation will be used and strain distributions are calculated by using Green-Lagrange: E=½(C-1)
where
C=FTF
(4)
Green-Lagrange strain measure is given by
E=
1
I
2L\Oro] \OroJ
_.1
0
0
!(Ort 05 Ozt Oz,~
=L\ro)
2\Oro Oz---oo÷ Or---~oOZo]
0
!(Oft Or, Ozt Oz~
,
~\aro OZo aro
1 J
iF/Ort'2
OZoJ
0
1
(5)
[ Ozt'~2 ]
2[k-~Zo) +k-~Zo) - 1
The linear transformation F can be calculated by replacing the space derivatives by finite differences (see Fig. 2). Any strain measure (Green-Lagrange E; Logarithmic zt; Almansi A, etc. ) can be obtained from F.
Ort rt(ro +ao;Zo)-rt(ro-ao;Zo) 0ro ~
2ao
(6)
Ozt ,, zt(ro;Zo +bo) -zt(ro;Zo -bo) 0Zo
2bo
where ao and bo are the initial grid dimensions. For the time derivative, the same particle must be followed, which can be done by working on two consecutive configurations t and t' = t + At.
~ AFter,
j~ AEt~t,
At
(7)
At 7(to+ Oo, Zo)
~(ro+ Oo,Zo).
I*0+ O 0 Zo I
IF
/ So "-" /
Mo
ro zo- bo
= ~ro
~ M 7 ( to, zo- bo} ( ro, zo- bo)_ _ ~
/~f~~ ~
z°+ bo r0 - o 0 ;t o
Fig. 2. Finite differences approximation.
-r-( to- ao,Zo) T ( ro-Oo,Zo)
"f'( r°' z°+ b°) (r°' z°+ b°)
202
The strain rate becomes
D = F T ~ F ~ F w-hEt_+t, - F At
sothat
JD..~DAt,~FTAEt~t,F
(8)
3. Deviatoric stress and hydrostatic p r e s s u r e
It is assumed that the material is rigid-plastic, and that the von Mises yield criterion associated with the Ludvick hardening law is used.
f=x/-~ (aD:aD)--aeq--R with R=ga n
(9)
A standard thermodynamic study gives the evolution laws, where _~ is the cumulative plastic strain and R the associated force. D is the strain-rate tensor and o"D the deviator of Cauchy stress
D=~
and
&=-4
-
8f
OR
is obtained by the consistency condition _a=x/~ ( D : D )
and
D=~ (a_/R)aD
(10)
dr= O.Thus (11)
The stress deviator is given by ~
~,
IR.DIIIDII
(12)
The complete Cauchy stress can be obtained by calculating the hydrostatic pressure in the workpiece. This is done by integrating the equilibrium equation between a point belonging to the free surface and the current point
(13)
div (aD) -- grad (p) = 0
where p = - 1 / 3 tr (a) is the opposite of the hydrostatic stress all. Equation (13) can be summed between a free surface of the workpiece to obtain the hydrostatic pressure at all points (see Fig. 3):
O
Fig. 3. Boundary conditions.
A
203
P(M)=p(O)÷IOM diV(OD)dM
(14)
4. Visioplastic analysis of a stationary problem: Extrusion In a stationary problem a single image of the specimen at time z sums up the whole process. The strain rate can be obtained by replacing time derivatives by derivatives along a trajectory. For example:
~, E(ro,Zo +bo) -E(ro,Zo) M(ro,zo)M(ro,Zo +bo) M ( ro,zo )M ( ro,Zo + bo ) At
(15)
where the first term on the right-hand side of eqn. (15) is ~ and the second term is V(M). The latter is constant for a stationary problem. A smoother evolution for E is obtained by using an average value calculated between two points placed after and before M(ro,Zo)
l[ (E(ro,zo +bo)-E(ro,Zo) )
=2L
(E(ro,Zo)-E(ro,zo-bo) ) 1
J
I : 0.00 2:0.07 3:0.15
(a)
(c)
4:0.24
5:0,30
I: -0.20 2:-0.15 3;-0.10 4: - 0.05 5: 0.00
(b)
(d)
(16)
I:-0.13 2: 0.00 3:0.15 4: 0.33 5:0.50
I:-0.40 2:-0.30 3:-0.20 4:-0.10 5: 0.00
Fig.4. Green-Lagrangestraindistribution: (a) dr; (b) ~z; (c) ¢~;and (d) ¢P~.(stationaryproblem)
204
(a)
(C)
I:0 2:-30 3:-50 4:-75 5:-100
(b)
4:-60 5:-80
1:10 2:0 3:-20
4:-40
(d)
5:-60
Fig. 5. Cauchy stress distribution: (a) arr; (b)
1:0 2:-20 3:-40
1:-35 2:-25 3:-15 4:-5
1
5:5
6:20
aoe;
(c) azz;and (d) ar~.
with fixed ro. The cumulative plastic strain _~ can be computed easily, so that R and o"D can be known. By integrating the equilibrium equation along a flow line the pressure can be obtained, so that all of the stress tensor is known. The following results are given for a 30-CD-4 tempered steel. This billet has been extruded: its initial diameter was 35 mm and the extrusion conditions were 30 ° semi-cone die angle and a 30% reduction rate. Figure 4 shows the Green-Lagrange strain distribution and Fig. 5 shows the Cauchy stress components.
5. Visioplastic analysis of non-stationary problem For a non-stationary problem, visioplastic images at every time 3 are needed to determine D by finite differences between two times 3+ A3 and 3, after which the history of_~ can be integrated. This is a quite tedious and expensive method because an experimental simulation must be carried out for each time step 3. Figure 6 shows the different billet-compression steps needed to obtain accurate information on the time evolution of all the variables of the problem. The minimum energy theorem developed by D a m m a m e [4 ] gives an upper bound approximation for _~, by calculating the eigenvalues of the Lagrangian dilatation tensor C.
205 2o 1.
.$S ....... ti .'~o. ~L S.
Fig. 6. Billet compressionsteps (20, 30, 35, 40, 45, 50, 55, 65, 70, 75, 80, 90 and 100).
I
I
Fig. 7. Conicalhead formingprocess.
~=
~1 i=3 ~ log (Ci) 2
(16)
i=1
By using this approximation, only the first and last steps are needed in order to be able to calculate _a. With a step near to the end of the process & can be computed. The authors have tested the above method on an industrial-process to form conical heads. A 20-MB-5 steel specimen is used, being first laminated and then subjected to backward extrusion, after which the conical head is obtained by a forming process (see Fig. 7). The photographs of Fig. 8 show the initial and final grids printed on the workpiece. Figure 9 shows the G r e e n -
Fig. 8. Initial (on left) and final grids on a 20-MB-5 conical head.
2: 3: 4: 5: 6:
(a)
(b)
Fig. 9. Green-Lagrange strain distribution: (a) &; (b) &; (c) &; and (d) tYz.(non-stationary problem)
207
Fig. 10. Von Mises equivalent stress distribution, a~: (1) 0; (2) 20; (3) 40; (4) 60; (5) 80; and (6) 100 daN/mm 2.
f= ~
A 0,2+ /
0,35 0 30 "~,15 z
Fig. 11. Value of the friction coefficient.
208 Lagrange strain distribution, whilst the von Mises equivalent stress is shown in Fig. 10. 6. Friction coefficient
Examining the ratio of the shear stress r to the normal stress a along the workpiece boundary gives information on the Coulomb friction coefficient f. The evolution o f / g i v e n in Fig. 11 shows clearly t h a t f is not a constant coefficient depending on the two interacting materials only. 7. Conclusions
Although the visioplasticity method is an approximate approach (the billet is cut, then deformed), the strain distribution, especially, gives relevant information on the mechanical behavior of the material. In all cases the stress distribution needs three successive discretizations, so t h a t the results are not very smooth: a special effort needs to be made to obtain accurate results. The friction coefficient obtained can be used in a finite-element simulation of the same process to gain more precise information on the stress distribution. This kind of simulation has been carried out already for the extrusion process [ 6 ] and a comparison of experimental and numerical solutions has been made [ 7 ]. F u r t h e r development of the method can be undertaken, if more precise information on the boundary surface of the workpiece is required. The grids are so distorted in these areas t h a t it is difficult to gain relevant information on the displacement field. An effort should be made to analyze the contact properties using the visioplasticity method.
References
1 S. Kobayashi and A. Shabaik, Computer application to the visioplasticitymethod, J. Eng. Ind., (1967) 339-346. 2 J.P. Bernardou, Calcul des ddformations minimales dans les pi~ces axisym~triquesforgoes, 4~me CoUoque Mdcanique et MdtaUurgie de Tarbes, 1980. 3 P. le Nevez and N. Dahan, M~thodede calculsdes grandes ddformationsplastiques en endommagementdans les pi~cesextrud~es,M~m. Etud. Sci. Rev. MetaU., (1983) 557-566. 4 G. Dammame, Minimum de la deformation gSnSralis~e d'un ~lSment de mati~re, pour des chemins de d~formationpassant d'un Stat initial/l un dtat final donn~s, C. R Acad. Sci., 287 (1978). 5 N. Dahan, B. Gathouffi and M.F. Molez, Sur les mesures en grandes d~formations,Rapport interne Laboratoire de M$caniqueet Technologie,No. 37, 1983. 6 L. Chevalier, Caracteristique des materiaux tr$fil~s apres l'opdration de tr$filage, Th~se de l'Universit~ Paris 6, May, 1988. 7 L. Chevalier, Simulation of cold extrusion process: A numerical experimental comparison, Third Int. Conf. on Structural Analysis Systems, Stafford, UK, 1987, pp. 25-34.