- Email: [email protected]
Modelling of sheet metal testing
COMPUTATIONAL MATERIALS SCIENCE ELSEVIER
Computational Materials Science 7 (1996) I73- 180
Modelling of sheet metal testing C.O. Gusek Institute of Ferrous Metallurgy,
*,
W. Bleck, W. Dahl
Aachen llniversiry of Technology, IntzestraJe I, D-52072
Aachen, Germany
Abstract Flow stress, work hardening rate and anisotropy characterize plastic behavior of sheet metals during pressforming operations. This paper concentrates on modelling forming limits and limiting drawing ratios by using a new flow curve description. The flow curve description is based on the dislocation theory and allows the separate analysis of yield stress and work hardening rate effects. The theoretical investigations are compared with experiments. Keywords:
Deep drawing; Forming limits; Limiting drawing ratio; Kocks model; Flow curve
1. Introduction Recently developed cold-rolled steels differ in the work hardening behaviour from conventional deepdrawing qualities. The traditional description of flow curves by Hollomon’s well known exponential law and the strain hardening exponent n fails in describing the work hardening of multi-phase steels, of interstitial-free steels and of metastable austenitic steels. New equations for the flow curve description have been proposed, new models have been developed in materials science for a better understanding of work hardening behavior. It is the main objective of this paper to introduce a more sophisticated description of work hardening behavior and to apply this new approach to the modelling of forming limit curves and limiting drawing ratios.
* Corresponding author. Tel.: + 49-241-805807; fax: + 49-2418888253; e-mail: [email protected].
2. Materials 2.1. Ferritic steels
The rising demand of the automotive industry for high strength and ductile steel sheets have led to the development of different ferritic steels. The steels investigated cover a wide range of chemical compositions, microstructures and mechanical properties. The chemical compositions are given in Table 1.
2.2. Metastable austenitic steels Four melts with different chemical compositions were processed. The stability of austenite in Cr-Nisteels can mainly be influenced by the Ni-content. The work hardening rate depends on the strain induced phase transformation which increases as the Ni-content decreases [ 11. The chemical composition is given in Table 2.
0927-0256/%/s 15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved. Plf SO927-0256(96)00077-8
C.O. Gusek et ul./ Con~pututionul Materiols Science 7 (1996) 173-180
174 Table 2 Chemical composition
of metastable
austenitic
by a phenomenological model of Kocks [8,9]. This model is based on the dislocation theory and describes work hardening rate as a function of dislocation density. The stress od caused by dislocation density can be written as:
steels (mass%)
Swel
Ni (%)
Cr (%o)
c (o/o)
N (%)
s (%)
A0 Al A2 A3
1 I .57 5.69 6.53 7.60
16.40 16.62 16.87 16.16
0.05 I 0.049 0.06 I 0.024
0.005 0.005 0.005 0.005
0.010 0.016 0.015 0.015
a, = aGbfi
3. Modelling of forming limits and limiting drawing ratios
3.1. Flow curve description
dp G = k,fi
The
most customary flow curve description is Hollomon’s law, which can be written as: u= kq”
(1)
with n: exponent of Hollomon’s law, and k: coefficient of Hollomon’s law. Plow curves derived from tensile tests are usually approximated with Hollomon’s law up to uniform elongation, stresses for higher amounts of strain are extrapolated. Previously it was shown that especially at higher strains this approximation is not accurate [2]. Hollomon’s flow curve description is purely empirical and not based on materials science [2,3]. In general plastic deformation of metals can be described following 5 stages [4,5]. In polycrystalline metals only stages III, IV and V have to be regarded due to multiple slip. Plastic deformation never reaches stage V in pressforming operations, thus only stages III and IV are considered in this paper. Stage III is characterized by decreasing work hardening rate while stage IV is defined by a nearly constant work hardening behavior [6,7]. The work hardening rate for stage III deformation can be described Table I Chemical composition
(2)
with G: shear modulus, (Y: geometrical factor, b: Burgersvector, and p: dislocation density. The evolution of the dislocation density during plastic deformation is given by: - k,p
(3)
k, defines dislocation storage and k, dynamic annihilation. k, can be written as:
&
(4)
bP
with (5) with p: geometrical factor, and L: mean free path for dislocation motion. k, is directly related to the mean free path of dislocations and depends on microstructure, similar to precipitations or grain size. k, relates to the temperature and the stacking fault energy. A combination of E&J.(2) and (3) gives the work hardening rate: k, da aGb -
= fj=
-k,
_
2
dv
m.$,.
2
In accordance with experimental results the work hardening rate decreases linearly with the stress a,
of ferritic steels (mass%)
Steel
Steel No.
c (%I
Si (%)
Mn (%)
s (%I
P(a)
Al (o/o)
N (o/o)
Nb (%)
Ti (%)
ZStE 220 BH ZStE 180 BH Z ZStE 220 P Z ZStE 260 P ZStE3OOP IF IF HS St 1403 St 250i
FI F2 F3 F4 F5 F6 F7 F8 F9
0.02 0.0 I 0.07 0.055 0.055 0.004 0.003 0.04 0.04
0.007 0.0 I 0.06 0.06 0.45 0.04 0.04 0.01 0.0 I
0.23 0.19 0.3 0.55 0.3 0.2 0.35 0.2 0.2
0.006 0.015 0.02 0.012 0.015 0.012 0.01 0.01 0.01
0.01 0.04 0.035 0.08 0.08 0.015 0.05 0.01 0.01
0.05 0.06 0.05 0.05 0.04 0.03 0.03 0.04
0.0045 0.0055 0.005 0.007 0.007 0.003 0.003 0.004
-
-
0.04
0.004
-
-
0.035
0.06 0.02
0.02
C.O. Gusek et al. / Computational
caused by dislocation density in stage III of plastic deformation. Fig. 1 presents the work hardening behavior of steels up to stage IV of the plastic deformation. In stage III of the plastic deformation the work hardening rate decreases linearly with stress ad caused by the dislocation density as predicted by the Kocks model. The starting value of the work hardening rate 13, depends on the microstructure k, and the slope on dynamic annihilation k,. Considering the saturation stress q and the integration of Eq. (6) in relation to CTand 50, regarding boundary conditions gives the Vote equation: ‘+=a,+((~~-a,)exp
cp
( 1 -k
Materials Science 7 (1996) I73- I80
175
Stage IV
200 -Oo 0
(7)
0
I 0.2
I 0.4
I 0.6
I 0.8
I
I
1.2
V
with a,: saturation stress, ~~0:yield stress, and k,: dynamic annihilation = fl k, ). The work hardening rate remains constant for higher deformation (stage IV) as shown in experimental and theoretical studies and is defined by Blv [2,3]. The flow curve in stage IV of the plastic deformation can be described by a linear equation: u= a + e,,(p.
(8) The flow curve description in [2,3] is given in Fig. 2. Stage III is characterized by Vote and stage IV by a linear equation. The yield stress is a, and the average work hardening rate can be described by V- (q - uJ/k, (stage III) and 8,, (stage IV).
Equivalent strain cp Fig. 2. Vote and linear equation characterizing stages III and IV of plastic deformation.
3.2. Modelling of forming limits The Swift and Hill models for calculating forming limits were modified with the new flow curve description [2,3,10,1 I]. The influence of the yield stress and of the work hardening rate on forming limits was obtained by changing crO, V and 8,,. Thereby flow curves with a constant yield stress and varying work hardening rates or flow curves with a constant work hardening rate and varying yield stresses were used (Figs. 3 and 4). The corresponding influence of yield stress on FLC curves is shown in Fig. 5 and of work hardening rate in Fig. 6. The modified models predict increasing forming limits with an increasing work hardening rate or a decreasing yield stress, which is in accordance with the experiments. 3.3. Model@
limiting drawing ratios
3.3.1. Calculation of the damage conditions ’ a,=czGbk,/k, s!,
,O”’ 660 8;10 ,I0 Equivalent stress caused by dislocation density CT,,[MPa]
Fig. 1. Work hardening in stages III and IV of plastic deformation.
Damage occurs in the cup wall under plane strain conditions with (p2= 0 and is assumed to start when the principal stress in the cup wall reaches the principal stress necessary for localized necking (necking stress), which can be easily derived from the modified Hill model [ 121.
C.O.
176
Gusek et cd./
Compututiomd
Materids
Science
7 (1996)
173-180
0.6
/
100 0 0
/ 02
I
04
Eqmvalcnt Fig. 3. Theoretical
I
06
strain
I
0.8
cp
Limit strain
flow curves with varying yield stresses.
3.3.2. Calculation of the principal stress in the wall
For the new model based on Hill’s anisotropic plasticity theory and the new flow curve description the complex equations could only be solved numerically. Therefore the whole deep drawing operation is divided into 400 steps. For each step the actual diameter of the circular blank in the flange, punch force, punch displacement, and the principal stresses, -strains, equivalent stresses and strains acting in the elements of the cup wall are calculated.
1000 ,,
tp2
Fig. 5. Influence of yield stress on forming limits.
The required force for plastic deformation in the flange is calculated by integration considering each element in the flange. Therefore Hill’s anisotropic plasticity theory and the flow curve in [2,3] are used. The bending force at the die ring and the forces resulting from friction in the flange and at the die ring are calculated by well known formulas [ 131. For each element shifting from the flange into the cup wall equivalent stresses and strains resulting from deformation in the flange as well as bending, 0.6
0.5
0.4
0.3
0.2
01 125 0 0
, 02
! 04
Equivalent Fig. 4. Theoretical
I 06 stram
I 0.8
0 -0.3
up
flow curves with varying work hardening
-0.2
-0.1
0
Limit strain rates.
Fig. 6. Influence of work hardening
0.1
0.2
0.3
cpz rates on forming limits.
C.O. Gusek et uI./Computationnl
Materials Science 7 (1996) 173-180
length and thickness of elements are evaluated and stored for following calculations. The punch force needed for the deep drawing operation is defined by addition of all individual forces mentioned above and the punch displacement is obtained from addition of all element lengths in the wall. The equivalent stresses and strains in the elements of the cup wall for plane strain conditions resulting from the applied punch force are evaluated. If equivalent strains and stresses caused by the punch force exceed strains and stresses from prior bending and deformation in the flange, thinning occurs in the cup wall. The actual thickness and principal stresses are evaluated and compared with Hill’s principal stress needed for localized necking. Damage is assumed to occur when the principal stress in the cup wall reaches Hill’s principal necking stress. Principal stresses in the cup wall and necking stresses for materials of changing work hardening rates or yield stresses are shown in Figs. 7 and 8. As long as no thinning occurs in the wall the principal stress in the wall increases linearly with blank radius. After local thinning has been started, stress rapidly increases up to the necking stress. The intersection of principal stress and necking stress defines the maximum drawable blank radius, which can be deduced from the abscissa. As shown in Fig. 7 the necking stress remains constant and the principal stress increases according to the yield stress. Intersection takes place for smaller
Table 3 Mechanical
177
5
Fig. 7. Influence of yield stress on the drawable blank radius.
blank radii with increasing yield stresses and the limiting drawing ratio decreases. As shown in Fig. 8, necking stress and principal stress in the cup wall increase with the work hardening rate. However, necking stress builds up at a faster rate, which results in the shift of intersection. For that reason drawable blank radii and limiting drawing ratios increase with the work hardening rate.
4. Experimental procedure Testing was performed on a mechanical tensile testing machine with constant cross bar velocity. The
properties of investigated steels
Steel
Steel No.
Rp0.2 (MPa)
R, (MPa)
A, (%)
A,, (%)
r90
iI
ZStE220 BH
Fl F2 F3 F4 F5 F6 F7 F8 F9 A0 Al A2 A3
256 215 278 266 310 120 196 177 250 289 322 298 257
359 336 343 404 455 311 367 307 372 606 960 726 620
19.1 21.7 23 20.2 19.2 25 22.6 24.3 24.5 35.9 17.8 42.3 49.5
32.9 40.3 36.1 32.4 32.6 42.7 38.3 43.9 37.9 41.1 19.1 42.8 50.1
1.28 1.79 1.39 1.92 1.65 2.34 2.23 2.05 1.02 0.91 0.53 0.92 0.91
I.0 0.8 0.8 0.9 1.0 1.0 0.8 0.8 0.85 0.8 0.8 0.8 0.8
ZStE 180 BH Z ZStE 220 P z ZStE 260P ZStE 300 P IF IF HS St 1403 ST 250 i Cr 16 Ni 12 Cr 16 Ni 6 Cr 16Ni7 Cr 16 Ni 8
C.O. Gusek et al./Compututionul
178
45
47
49
51
Motcrials Science 7 f 19961 173-180
53
Blankradius [mm]
Limit strain cpz
Fig. 8. Influence of the work hardening rate on the drawable blank radius.
Fig. IO. Calculated and measured forming limits (steel A2).
mechanical properties of the investigated steels are listed in Table 3. Flow curves for high deformation of ferritic steels were extrapolated with the procedure suggested in [2,3]. Flow curves of metastable austenitic steels were measured with a video-optic tensile test method [14]. Cylindrical tensile specimens, which were identically heat treated as the sheets, were used. Forming limits were determined with the proce-
-0.3
-0.1
0.1
dure suggested by the International Deep Drawing Research Group. A hemispherical punch of 100 mm diameter and sheets of varying width were used. Punch velocity was 20 mm/min, the punch was stopped when the first crack appeared. Grids in the direct neighborhood of the crack were measured with a microscope. To exclude surface and friction influences drawing foil and oil were used.
0.3
Austemtic steel (number)
Limit strain cpz Fig. Fig. 9. Calculated and measured forming limits (steel Fl).
I I.
steels.
Calculated and experimental
FLD,
values for austenitic
C.O. Gusek et uI./ComputationulMuteriuls
Science 7 (1996) 173-180
179
Limiting drawing ratios were determined for all steels with an accuracy of 0.1 mm by changing blank diameter. For the ferritic steels a punch diameter of 50 mm and blank holder force of 20 kN and for metastable austenitic steels a punch diameter of 50 mm and blank holder force of 70 kN was used. In addition, drawing foil and oil were utilized. 90
I
I
FL
F9
F-3
b
i8
F6
F7
5. Results and discussion Fig. 13. Calculated and measured drawable blank diameters for fenitic steels.
The new flow curve description and anisotropy parameter r of the tensile test were used as input data for calculating forming limits and limiting drawing ratios. Calculated and experimental forming limit curves for a ferritic and an austenitic steel are shown in Figs. 9 and 10. Forming limits calculated with the modified model better match experimental data than forming limits determined with the original model. FLD,-values (limit strain ‘p, for plane strain conditions with qo2= 0) for all investigated steels are shown in Fig. 11 for austenitic and in Fig. 12 for ferritic steels. Calculated and experimental limiting drawing ra-
I
0.6,
_c_
experimental data modified model original model
-S-----m---
o~ll--_A I
2
3
4
5
6
7
Ferritic steel (number) Fig. 12. Calculated and experimental FLD, values for fenitic steels.
tios for all ferritic steels showed a very good agreement with experimental data as shown in Fig. 13.
6. Conclusions A new flow curve description was introduced, which allows evaluation of materials behavior up to high deformation based on materials science. It can be derived from the simple tensile test for ferritic steels. Models from Swift and Hill for calculating forming limits were modified with the new flow curve description. A theoretical investigation showed that forming limits increase with an increasing work hardening rate, but decrease with increasing yield stress. It could be shown that the agreement of the calculated data and the experiment was better for the modified than for the original model. A new model for calculating limiting drawing ratios based on Hill’s theory and the new flow curve description were introduced. A theoretical investigation showed that limiting drawing ratios increase with an increasing work hardening rate, but decrease with increasing yield stress. Experiments showed that limiting drawing ratios could be predicted quantitatively.
References [I] T. Angel, J. Iron Steel Inst. 177 (1954) 165. [21 X.F. Fang and W. Dahl, Z. Metallkd. 86( 1) (1995)41. [3] X.F. Fang, VDI Fortschrittsberichte, Reihe 5, Nr. 289 (1990).
180
C.O.
Gusek et
ul./Computukmal
[4] M. Zehetbauer and W. Seumer, Acta Metall. Mater. 41(Z) (1993) 577. [5] M. Zehetbauer, Acta Metall. Mater. 41(2) (1993) 589. [6] J.G. Sevilliano, V. Houtte and E. Aemoudt, in: Progress in Materials Science, eds. J.W. Christian, P. Haasen and T.B. Massalski, Vol. 25 (Pergamon Press, Oxford, 1982) p. 69. [7] J.G. Sevillano and E. Aemoudt, Mater. Sci. Eng. 86 (1987) 31. 181 U.F. Kc&s. Trans. A.S.M. 98(l) (1976) 76. [9] J. Es&in, Stoffgesetze der Plastischen Verfotmung und Instrbilititen des Plastischen FlieOens. Habitilationsschrift, TU Hamburg Harburg, 7 Mai (1986).
Muterids
Science 7 (1996) 173-180
[lOI R. Hill, J. Mech. Phys. Solids 1 (1952) 19. [l I] H. Swift, J. Mech. Phys. Solids 1 (1952) 1. [12] C.O. Gusek, W. Bleck and W. Dahl, Effects of Work Hardening Rate on Formability of Recently Developed DDQSteels, IDDRG Congr., IO.-14. 06. %, Miskolc, Hungary, to be published. 1131 K. Lange, Lehrbuch der Umformtechnik, Band 3 (Springer Verlag, Berlin, 1990). [I41 C.O. Gusek, J. Hagedom, F.S. Doust. W. Dahl and W. Bleck, Z.F. Materialpriifung 37(9) (1995) 328.
Computational Materials Science 7 (1996) I73- 180
Modelling of sheet metal testing C.O. Gusek Institute of Ferrous Metallurgy,
*,
W. Bleck, W. Dahl
Aachen llniversiry of Technology, IntzestraJe I, D-52072
Aachen, Germany
Abstract Flow stress, work hardening rate and anisotropy characterize plastic behavior of sheet metals during pressforming operations. This paper concentrates on modelling forming limits and limiting drawing ratios by using a new flow curve description. The flow curve description is based on the dislocation theory and allows the separate analysis of yield stress and work hardening rate effects. The theoretical investigations are compared with experiments. Keywords:
Deep drawing; Forming limits; Limiting drawing ratio; Kocks model; Flow curve
1. Introduction Recently developed cold-rolled steels differ in the work hardening behaviour from conventional deepdrawing qualities. The traditional description of flow curves by Hollomon’s well known exponential law and the strain hardening exponent n fails in describing the work hardening of multi-phase steels, of interstitial-free steels and of metastable austenitic steels. New equations for the flow curve description have been proposed, new models have been developed in materials science for a better understanding of work hardening behavior. It is the main objective of this paper to introduce a more sophisticated description of work hardening behavior and to apply this new approach to the modelling of forming limit curves and limiting drawing ratios.
* Corresponding author. Tel.: + 49-241-805807; fax: + 49-2418888253; e-mail: [email protected].
2. Materials 2.1. Ferritic steels
The rising demand of the automotive industry for high strength and ductile steel sheets have led to the development of different ferritic steels. The steels investigated cover a wide range of chemical compositions, microstructures and mechanical properties. The chemical compositions are given in Table 1.
2.2. Metastable austenitic steels Four melts with different chemical compositions were processed. The stability of austenite in Cr-Nisteels can mainly be influenced by the Ni-content. The work hardening rate depends on the strain induced phase transformation which increases as the Ni-content decreases [ 11. The chemical composition is given in Table 2.
0927-0256/%/s 15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved. Plf SO927-0256(96)00077-8
C.O. Gusek et ul./ Con~pututionul Materiols Science 7 (1996) 173-180
174 Table 2 Chemical composition
of metastable
austenitic
by a phenomenological model of Kocks [8,9]. This model is based on the dislocation theory and describes work hardening rate as a function of dislocation density. The stress od caused by dislocation density can be written as:
steels (mass%)
Swel
Ni (%)
Cr (%o)
c (o/o)
N (%)
s (%)
A0 Al A2 A3
1 I .57 5.69 6.53 7.60
16.40 16.62 16.87 16.16
0.05 I 0.049 0.06 I 0.024
0.005 0.005 0.005 0.005
0.010 0.016 0.015 0.015
a, = aGbfi
3. Modelling of forming limits and limiting drawing ratios
3.1. Flow curve description
dp G = k,fi
The
most customary flow curve description is Hollomon’s law, which can be written as: u= kq”
(1)
with n: exponent of Hollomon’s law, and k: coefficient of Hollomon’s law. Plow curves derived from tensile tests are usually approximated with Hollomon’s law up to uniform elongation, stresses for higher amounts of strain are extrapolated. Previously it was shown that especially at higher strains this approximation is not accurate [2]. Hollomon’s flow curve description is purely empirical and not based on materials science [2,3]. In general plastic deformation of metals can be described following 5 stages [4,5]. In polycrystalline metals only stages III, IV and V have to be regarded due to multiple slip. Plastic deformation never reaches stage V in pressforming operations, thus only stages III and IV are considered in this paper. Stage III is characterized by decreasing work hardening rate while stage IV is defined by a nearly constant work hardening behavior [6,7]. The work hardening rate for stage III deformation can be described Table I Chemical composition
(2)
with G: shear modulus, (Y: geometrical factor, b: Burgersvector, and p: dislocation density. The evolution of the dislocation density during plastic deformation is given by: - k,p
(3)
k, defines dislocation storage and k, dynamic annihilation. k, can be written as:
&
(4)
bP
with (5) with p: geometrical factor, and L: mean free path for dislocation motion. k, is directly related to the mean free path of dislocations and depends on microstructure, similar to precipitations or grain size. k, relates to the temperature and the stacking fault energy. A combination of E&J.(2) and (3) gives the work hardening rate: k, da aGb -
= fj=
-k,
_
2
dv
m.$,.
2
In accordance with experimental results the work hardening rate decreases linearly with the stress a,
of ferritic steels (mass%)
Steel
Steel No.
c (%I
Si (%)
Mn (%)
s (%I
P(a)
Al (o/o)
N (o/o)
Nb (%)
Ti (%)
ZStE 220 BH ZStE 180 BH Z ZStE 220 P Z ZStE 260 P ZStE3OOP IF IF HS St 1403 St 250i
FI F2 F3 F4 F5 F6 F7 F8 F9
0.02 0.0 I 0.07 0.055 0.055 0.004 0.003 0.04 0.04
0.007 0.0 I 0.06 0.06 0.45 0.04 0.04 0.01 0.0 I
0.23 0.19 0.3 0.55 0.3 0.2 0.35 0.2 0.2
0.006 0.015 0.02 0.012 0.015 0.012 0.01 0.01 0.01
0.01 0.04 0.035 0.08 0.08 0.015 0.05 0.01 0.01
0.05 0.06 0.05 0.05 0.04 0.03 0.03 0.04
0.0045 0.0055 0.005 0.007 0.007 0.003 0.003 0.004
-
-
0.04
0.004
-
-
0.035
0.06 0.02
0.02
C.O. Gusek et al. / Computational
caused by dislocation density in stage III of plastic deformation. Fig. 1 presents the work hardening behavior of steels up to stage IV of the plastic deformation. In stage III of the plastic deformation the work hardening rate decreases linearly with stress ad caused by the dislocation density as predicted by the Kocks model. The starting value of the work hardening rate 13, depends on the microstructure k, and the slope on dynamic annihilation k,. Considering the saturation stress q and the integration of Eq. (6) in relation to CTand 50, regarding boundary conditions gives the Vote equation: ‘+=a,+((~~-a,)exp
cp
( 1 -k
Materials Science 7 (1996) I73- I80
175
Stage IV
200 -Oo 0
(7)
0
I 0.2
I 0.4
I 0.6
I 0.8
I
I
1.2
V
with a,: saturation stress, ~~0:yield stress, and k,: dynamic annihilation = fl k, ). The work hardening rate remains constant for higher deformation (stage IV) as shown in experimental and theoretical studies and is defined by Blv [2,3]. The flow curve in stage IV of the plastic deformation can be described by a linear equation: u= a + e,,(p.
(8) The flow curve description in [2,3] is given in Fig. 2. Stage III is characterized by Vote and stage IV by a linear equation. The yield stress is a, and the average work hardening rate can be described by V- (q - uJ/k, (stage III) and 8,, (stage IV).
Equivalent strain cp Fig. 2. Vote and linear equation characterizing stages III and IV of plastic deformation.
3.2. Modelling of forming limits The Swift and Hill models for calculating forming limits were modified with the new flow curve description [2,3,10,1 I]. The influence of the yield stress and of the work hardening rate on forming limits was obtained by changing crO, V and 8,,. Thereby flow curves with a constant yield stress and varying work hardening rates or flow curves with a constant work hardening rate and varying yield stresses were used (Figs. 3 and 4). The corresponding influence of yield stress on FLC curves is shown in Fig. 5 and of work hardening rate in Fig. 6. The modified models predict increasing forming limits with an increasing work hardening rate or a decreasing yield stress, which is in accordance with the experiments. 3.3. Model@
limiting drawing ratios
3.3.1. Calculation of the damage conditions ’ a,=czGbk,/k, s!,
,O”’ 660 8;10 ,I0 Equivalent stress caused by dislocation density CT,,[MPa]
Fig. 1. Work hardening in stages III and IV of plastic deformation.
Damage occurs in the cup wall under plane strain conditions with (p2= 0 and is assumed to start when the principal stress in the cup wall reaches the principal stress necessary for localized necking (necking stress), which can be easily derived from the modified Hill model [ 121.
C.O.
176
Gusek et cd./
Compututiomd
Materids
Science
7 (1996)
173-180
0.6
/
100 0 0
/ 02
I
04
Eqmvalcnt Fig. 3. Theoretical
I
06
strain
I
0.8
cp
Limit strain
flow curves with varying yield stresses.
3.3.2. Calculation of the principal stress in the wall
For the new model based on Hill’s anisotropic plasticity theory and the new flow curve description the complex equations could only be solved numerically. Therefore the whole deep drawing operation is divided into 400 steps. For each step the actual diameter of the circular blank in the flange, punch force, punch displacement, and the principal stresses, -strains, equivalent stresses and strains acting in the elements of the cup wall are calculated.
1000 ,,
tp2
Fig. 5. Influence of yield stress on forming limits.
The required force for plastic deformation in the flange is calculated by integration considering each element in the flange. Therefore Hill’s anisotropic plasticity theory and the flow curve in [2,3] are used. The bending force at the die ring and the forces resulting from friction in the flange and at the die ring are calculated by well known formulas [ 131. For each element shifting from the flange into the cup wall equivalent stresses and strains resulting from deformation in the flange as well as bending, 0.6
0.5
0.4
0.3
0.2
01 125 0 0
, 02
! 04
Equivalent Fig. 4. Theoretical
I 06 stram
I 0.8
0 -0.3
up
flow curves with varying work hardening
-0.2
-0.1
0
Limit strain rates.
Fig. 6. Influence of work hardening
0.1
0.2
0.3
cpz rates on forming limits.
C.O. Gusek et uI./Computationnl
Materials Science 7 (1996) 173-180
length and thickness of elements are evaluated and stored for following calculations. The punch force needed for the deep drawing operation is defined by addition of all individual forces mentioned above and the punch displacement is obtained from addition of all element lengths in the wall. The equivalent stresses and strains in the elements of the cup wall for plane strain conditions resulting from the applied punch force are evaluated. If equivalent strains and stresses caused by the punch force exceed strains and stresses from prior bending and deformation in the flange, thinning occurs in the cup wall. The actual thickness and principal stresses are evaluated and compared with Hill’s principal stress needed for localized necking. Damage is assumed to occur when the principal stress in the cup wall reaches Hill’s principal necking stress. Principal stresses in the cup wall and necking stresses for materials of changing work hardening rates or yield stresses are shown in Figs. 7 and 8. As long as no thinning occurs in the wall the principal stress in the wall increases linearly with blank radius. After local thinning has been started, stress rapidly increases up to the necking stress. The intersection of principal stress and necking stress defines the maximum drawable blank radius, which can be deduced from the abscissa. As shown in Fig. 7 the necking stress remains constant and the principal stress increases according to the yield stress. Intersection takes place for smaller
Table 3 Mechanical
177
5
Fig. 7. Influence of yield stress on the drawable blank radius.
blank radii with increasing yield stresses and the limiting drawing ratio decreases. As shown in Fig. 8, necking stress and principal stress in the cup wall increase with the work hardening rate. However, necking stress builds up at a faster rate, which results in the shift of intersection. For that reason drawable blank radii and limiting drawing ratios increase with the work hardening rate.
4. Experimental procedure Testing was performed on a mechanical tensile testing machine with constant cross bar velocity. The
properties of investigated steels
Steel
Steel No.
Rp0.2 (MPa)
R, (MPa)
A, (%)
A,, (%)
r90
iI
ZStE220 BH
Fl F2 F3 F4 F5 F6 F7 F8 F9 A0 Al A2 A3
256 215 278 266 310 120 196 177 250 289 322 298 257
359 336 343 404 455 311 367 307 372 606 960 726 620
19.1 21.7 23 20.2 19.2 25 22.6 24.3 24.5 35.9 17.8 42.3 49.5
32.9 40.3 36.1 32.4 32.6 42.7 38.3 43.9 37.9 41.1 19.1 42.8 50.1
1.28 1.79 1.39 1.92 1.65 2.34 2.23 2.05 1.02 0.91 0.53 0.92 0.91
I.0 0.8 0.8 0.9 1.0 1.0 0.8 0.8 0.85 0.8 0.8 0.8 0.8
ZStE 180 BH Z ZStE 220 P z ZStE 260P ZStE 300 P IF IF HS St 1403 ST 250 i Cr 16 Ni 12 Cr 16 Ni 6 Cr 16Ni7 Cr 16 Ni 8
C.O. Gusek et al./Compututionul
178
45
47
49
51
Motcrials Science 7 f 19961 173-180
53
Blankradius [mm]
Limit strain cpz
Fig. 8. Influence of the work hardening rate on the drawable blank radius.
Fig. IO. Calculated and measured forming limits (steel A2).
mechanical properties of the investigated steels are listed in Table 3. Flow curves for high deformation of ferritic steels were extrapolated with the procedure suggested in [2,3]. Flow curves of metastable austenitic steels were measured with a video-optic tensile test method [14]. Cylindrical tensile specimens, which were identically heat treated as the sheets, were used. Forming limits were determined with the proce-
-0.3
-0.1
0.1
dure suggested by the International Deep Drawing Research Group. A hemispherical punch of 100 mm diameter and sheets of varying width were used. Punch velocity was 20 mm/min, the punch was stopped when the first crack appeared. Grids in the direct neighborhood of the crack were measured with a microscope. To exclude surface and friction influences drawing foil and oil were used.
0.3
Austemtic steel (number)
Limit strain cpz Fig. Fig. 9. Calculated and measured forming limits (steel Fl).
I I.
steels.
Calculated and experimental
FLD,
values for austenitic
C.O. Gusek et uI./ComputationulMuteriuls
Science 7 (1996) 173-180
179
Limiting drawing ratios were determined for all steels with an accuracy of 0.1 mm by changing blank diameter. For the ferritic steels a punch diameter of 50 mm and blank holder force of 20 kN and for metastable austenitic steels a punch diameter of 50 mm and blank holder force of 70 kN was used. In addition, drawing foil and oil were utilized. 90
I
I
FL
F9
F-3
b
i8
F6
F7
5. Results and discussion Fig. 13. Calculated and measured drawable blank diameters for fenitic steels.
The new flow curve description and anisotropy parameter r of the tensile test were used as input data for calculating forming limits and limiting drawing ratios. Calculated and experimental forming limit curves for a ferritic and an austenitic steel are shown in Figs. 9 and 10. Forming limits calculated with the modified model better match experimental data than forming limits determined with the original model. FLD,-values (limit strain ‘p, for plane strain conditions with qo2= 0) for all investigated steels are shown in Fig. 11 for austenitic and in Fig. 12 for ferritic steels. Calculated and experimental limiting drawing ra-
I
0.6,
_c_
experimental data modified model original model
-S-----m---
o~ll--_A I
2
3
4
5
6
7
Ferritic steel (number) Fig. 12. Calculated and experimental FLD, values for fenitic steels.
tios for all ferritic steels showed a very good agreement with experimental data as shown in Fig. 13.
6. Conclusions A new flow curve description was introduced, which allows evaluation of materials behavior up to high deformation based on materials science. It can be derived from the simple tensile test for ferritic steels. Models from Swift and Hill for calculating forming limits were modified with the new flow curve description. A theoretical investigation showed that forming limits increase with an increasing work hardening rate, but decrease with increasing yield stress. It could be shown that the agreement of the calculated data and the experiment was better for the modified than for the original model. A new model for calculating limiting drawing ratios based on Hill’s theory and the new flow curve description were introduced. A theoretical investigation showed that limiting drawing ratios increase with an increasing work hardening rate, but decrease with increasing yield stress. Experiments showed that limiting drawing ratios could be predicted quantitatively.
References [I] T. Angel, J. Iron Steel Inst. 177 (1954) 165. [21 X.F. Fang and W. Dahl, Z. Metallkd. 86( 1) (1995)41. [3] X.F. Fang, VDI Fortschrittsberichte, Reihe 5, Nr. 289 (1990).
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Gusek et
ul./Computukmal
[4] M. Zehetbauer and W. Seumer, Acta Metall. Mater. 41(Z) (1993) 577. [5] M. Zehetbauer, Acta Metall. Mater. 41(2) (1993) 589. [6] J.G. Sevilliano, V. Houtte and E. Aemoudt, in: Progress in Materials Science, eds. J.W. Christian, P. Haasen and T.B. Massalski, Vol. 25 (Pergamon Press, Oxford, 1982) p. 69. [7] J.G. Sevillano and E. Aemoudt, Mater. Sci. Eng. 86 (1987) 31. 181 U.F. Kc&s. Trans. A.S.M. 98(l) (1976) 76. [9] J. Es&in, Stoffgesetze der Plastischen Verfotmung und Instrbilititen des Plastischen FlieOens. Habitilationsschrift, TU Hamburg Harburg, 7 Mai (1986).
Muterids
Science 7 (1996) 173-180
[lOI R. Hill, J. Mech. Phys. Solids 1 (1952) 19. [l I] H. Swift, J. Mech. Phys. Solids 1 (1952) 1. [12] C.O. Gusek, W. Bleck and W. Dahl, Effects of Work Hardening Rate on Formability of Recently Developed DDQSteels, IDDRG Congr., IO.-14. 06. %, Miskolc, Hungary, to be published. 1131 K. Lange, Lehrbuch der Umformtechnik, Band 3 (Springer Verlag, Berlin, 1990). [I41 C.O. Gusek, J. Hagedom, F.S. Doust. W. Dahl and W. Bleck, Z.F. Materialpriifung 37(9) (1995) 328.