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Damage evolution in a compressive forming process: ironing of beverage cans
Scripta Materialia 48 (2003) 1519–1524 www.actamat-journals.com
Damage evolution in a compressive forming process: ironing of beverage cans Jo€el Courbon
*,1
GEMPPM (UMR CNRS 5510), INSA de Lyon, B^ atiment Saint-Exup ery, 20 Avenue Albert Einstein, 69621 Villeurbanne Cedex, France Received 8 November 2002; received in revised form 21 February 2003; accepted 10 March 2003
Abstract Density has been used to investigate the evolution of damage with strain in wall ironing of an aluminium beverage can (AA3104). The observed increase of density and the fact that it reaches a maximum are explained semi-quantitatively by the hydrostatic stress inside the material and a damage criterion for porous media. 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Damage evolution; Void growth; Forming; Ductility; Aluminium alloy
1. Introduction Ironing is the forming process by which the walls of a beverage can are being thinned: after a cup has been formed by deep drawing, its wall is pulled by a punch through a die of lower inner diameter than the cup outer diameter. The material thins and stretches. It enables one to form a container with optimal material distribution (thinner in the wall where not as much strength is needed to resist pressure). Under the die, it is often assumed that the hydrostatic stress is compressive, so that high strains can be achieved. But the critical step in the process is the flow out of the die, since the principal stress becomes a tension along the wall axis (Figs. 1–3).
*
Tel.: +33-472-43-63-81; fax: +33-472-43-85-39. E-mail address: [email protected] (J. Courbon). 1 Formerly at Pechiney Centre de Recherches de Voreppe, CentrÕAlp, BP 27, 38341 Voreppe Cedex, France.
This stress has two drawbacks: it induces a compressive hoop stress that shrinks the can on the punch and that hinders its stripping, especially for tinplate cans [1]; it may be large enough to induce fracture (‘‘tear-off’’) on aluminium cans, that are made from highly cold-worked alloy (AA3104 H19) with low residual tensile elongation [2]. The observed fracture mode is ductile, with cusps around constituent Fe-rich particles about 5 lm in size. It is then relevant to study how damage nucleates and grows, in order to minimize it and reduce the risk of failure. The relative variation of density is a powerful experimental tool in order to obtain information at the scale of a few hundreds of cubic millimetres of material. A prior study by Baudelet and Grange [3] applied it to the can making process. For the deep drawing stages, mainly depressive, density decreased and damage conversely increased with the height in cup wall, therefore with increasing forming strain; the observed density profile in the cup wall could be modelled using the Latham and Cockroft damage
1359-6462/03/$ - see front matter 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-6462(03)00131-3
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J. Courbon / Scripta Materialia 48 (2003) 1519–1524
a/ REDRAW DIE
b/ IRONING DIE
die radius
punch velocity
yield stress (MPa)
Fig. 1. Sketch of the forming process: (a) redraw die with the die radius and (b) ironing die.
360 350 340 330 320 310 300 290 280
current data previous data Ludw ik fit 0
0.2 0.4 0.6 0.8
1
decreased during the last ironing step. The authors explained this phenomenon by shear stresses in the vicinity of the die, that became dominant as the thickness was reduced. This evidence of damage healing in a compressive process is new. Damage as considered by mechanics is irreversible, whereas its microstructural cause, voids in the material, may be reduced. Yet the decohesions or fractures induced previously are not necessarily wiped out, and more generally internal interfaces created in the material may be decorated by dissolved hydrogen (for instance). So the phenomenon needs further investigations, both per se and for the consequences on ductility or formability. The purpose of the present paper is to evaluate the role of thickness reduction on density and explain the observations by the hydrostatic stress at stake in single-pass ironing. It is a complement of the former study [3], since the same forming and measurements devices have been used.
1.2
Tresca equivalent strain
Fig. 2. Strain-hardening curve of the material in the ironing process.
Np die
Td ti
B1 Nd
B4 Tp
F punch
tf O y x
Fig. 3. A slice of the wall material between the ironing die and the punch. The initial and final wall thicknesses are ti and tf . The plastic flow can be modelled in local radial coordinates with centre O, between the radii ri ¼ ti = sin a and rf ¼ tf = sin a.
criterion [4], in which damage is proportional to the integral of the main tensile stress over the strain path. But for the first two ironing stages–– after which the wall thickness successively became 280 and 200 lm (original sheet thickness: 335 lm)––they reported that density increased. The cavities developed in deep drawing were partly closed by the positive hydrostatic pressure developed by the process. On the other hand, density
2. Experimental The same sheet material has been used as in [3], on the same forming devices. The material is AA3104 H19 containing 0.99 wt.% Mn, 0.99 wt.% Mg, 0.38 wt.% Fe, 0.18 wt.% Si, 0.14 wt.% Cu was used at the sheet thickness 335 lm. The mean value of the yield stress at 0.2% proof strain in tension is 280 MPa. Cups have been deep drawn at a diameter of 85 mm. Then they have been redrawn and ironed through a redraw die and a single ironing die. Two redraw dies with different radius have been tested: 2.5 and 3.25 mm. The forming section of the ironing die was a cone of half-angle a ¼ 8. Relative thickness reductions of 11%, 22% and 35% were achieved. For damage evaluation, a 9 mm high ring was carefully cut from the can wall starting at the height of 35 mm, viz. above the transition zone between the bottom and the thin wall. It was annealed in order to wipe out the contribution of dislocations to the density. The detailed experimental protocol is given in [5].
J. Courbon / Scripta Materialia 48 (2003) 1519–1524
For can wall elongation, longitudinal tensile specimens with 45 mm length, 10 mm width were cut starting at the height of 12 mm, so that the same area covered by density underwent the tensile test. Anyhow, the tensile stress in the can wall does not vary significantly in the zone of constant thickness [6].
3. Results 3.1. Density Both the ironing strain and the redraw die radius had an influence on density. Table 1 sums up the results. They are coherent with [3]: density actually increases during the ironing process, meaning that cavities are refilled by the matrix due to the hydrostatic compression. Yet this healing effect is not perfect, since the density remains lower than that of the cold-rolled sheet, that itself is lower than the theoretical one since it already contains some decohesion and fragmentation voids [7]. Also, the variations with the ironing reduction denote that a maximum healing is reached close to the intermediate reduction, since damage is larger for the highest reduction. There is a strong effect of the redraw die radius: the smaller radius induced more damage than the large one. This is coherent with the role of bending and unbending in the nucleation of damage reported by Baudelet and Grange [3] and also with empirical practice, that suggests to use a radius at least 10 times as large as the sheet thickness to avoid trouble in sheet forming. The larger radius
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(ratio 9.7) is close to compliance with that empirical rule, not the smaller (ratio 7.5). As a matter of fact, the effect on density has the same order of magnitude as the ironing reduction. 3.2. Tensile tests Table 2 sums up the results obtained on wall material. The ironing operation is inducing some work-hardening of the material, as already evidenced [5]. The proof stress results have been plotted on a stress–strain curve: Fig. 2, using a macroscopic equivalent of the Tresca plastic strain ep ¼ lnðti =tf Þ where ti and tf stand for the initial and final can wall thickness. As can be seen, the present results are coherent with previous ones [6] and allow one to propose a hardening law for the wall material r ¼ Kðe0 þ ep Þ
n
ð1Þ 2
with K ¼ 348 MPa, e0 ¼ 3 10 and n ¼ 0:062. The material achieves an extreme state with very limited tensile ductility. As shown in the following Table 3, the elongation at necking As and the (total) tensile elongation Ar follow a global decreasing trend with the ironing strain. Yet small but detectable effects of damage on the average values of As and Ar are detected: the lower redraw die (maximum damage) is always associated with the lowest ductility. Altogether, the experimental results have confirmed the healing effect of the ironing process on damage and also shown that there exists an optimum reduction for maximum healing. The following mechanical analysis is going to propose an explanation for this last phenomenon.
Table 1 Effect of strain and redraw die radius on the relative variation of density of can wall State
Redraw die 2.5 mm
Redraw die 3.25 mm
Redrawn cup Reduction 11% Reduction 22% Reduction 35%
)12.0 104 )10.3 104 )7.7 104 )9.1 104
Not available )7.3 104 )5.8 104 )7.5 104
The reference density is that of the unformed sheet material. Values are the average of 3 or 4 individual measurements and are given within 0.5 104 at the 95% confidence level.
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J. Courbon / Scripta Materialia 48 (2003) 1519–1524
Table 2 Tensile properties of longitudinal specimens cut from the wall of ironed cans State (%)
rp0:2 (MPa)
rmax (MPa)
Reduction 11 Reduction 35
309 3 328 2
326 2 339 2
4. Analytical evaluation of stresses Canova et al. [8] have proposed a micromechanical damage model in which the local stress is able to open or close a cavity neighbouring a constituent particle, depending on stress triaxiality (defined by the ratio of the hydrostatic stress to the flow stress). However, the applications rely on a collection of particle shapes and orientations collected from image analysis. We simplify the problem by assuming that the major role in damage evolution is that of stress triaxiality, through an integral of triaxiality with strain. Such an expression is part of the Oyane damage criterion [9] for porous media, but it is also consistent with GursonÕs formulation [10]. Let us then determine the variation of the hydrostatic stress with the ironing reduction. The basic forces on a thin slice of material are shown in Fig. 3. The material under the die is submitted to the longitudinal tension F , the normal reactions of the punch and die Np and Nd , the friction forces Tp and Td . Friction on the punch side is a driving force, since the relative velocity of material vs. punch points backwards. Indeed, the common industrial practice uses a rough or crosshatched punch and a smooth die in order to minimize the residual tension F . Avitzur [11] proposed an upper bound method to evaluate F . We resumed the mainlines and added developments about the hydrostatic stress.
Fig. 3 summarizes the geometry. In the coordinates system of the die, the material flows at uniform horizontal speed Vi and thickness ti before the die; after the die, speed and thickness have become Vf and tf . The conservation of flowrate imposes Vf tf ¼ Vi ti . Under the die, the flow on the die side and on the punch side is parallel to the side, so it points towards O as seen in Fig. 3. The plastic volume is comprised between four boundaries: • inlet B1 : r ¼ ri ¼ ti = sin a; 0 6 h 6 a; no applied force; • die side B2 : applied forces Td and Nd ; • punch side B3 : applied forces Tp and Np ; • outlet B4 : r ¼ rf ¼ tf = sin a; 0 6 h 6 a; applied force F . The velocity field proposed by Avitzur is ~ðrÞ ¼ V ðrÞ cos h~ V ur . It is kinematically admissible since it conserves the flowrate and the normal component of velocity across boundaries. Its drawback is the fact that the slowest flow occurs close to the die and the quickest close to the punch, whereas in reality it is the reverse due to the previously described friction conditions. With the upper bound method, the coefficients of friction are of the Tresca type on boundaries B2 and B3 (the friction shear stress is a fraction of the Tresca shear flow stress rY =2): md and mp , respectively. It is suggested to use an overestimate for md (dissipative power) and an underestimate for mp (driving power) so that the calculation remains an upper bound. The balance of powers includes that of the applied forces and that of the deformation process (shear discontinuities on B1 and B4 inducing redundant work, volumic plastic power inside the forming volume). From the calculation, the residual longitudinal tensile stress, normalized by the
Table 3 Effect of strain and redraw die radius on the elongation at necking As and the (total) tensile elongation Ar of can wall Die radius (mm)
2.5
Reduction (%)
As (%)
Ar (%)
As (%)
3.25 Ar (%)
11 35
1.1 0.1 1.1 0.15
2.2 0.2 1.9 0.2
1.7 0.2 1.35 0.1
2.7 0.2 2.2 0.15
Values are the average of 15 individual measurements; intervals of confidence are given at the 95% confidence level.
J. Courbon / Scripta Materialia 48 (2003) 1519–1524
current thickness reduction 0.0% 20.0% 40.0%
rxx F ¼ 2pRtf rY ri 2ð1 cos aÞ md cos a ri þ ln ¼ ln þ sin a 2 sin a rf rf mp ri ri 1 ln 2 sin a rf rf
0
ð2Þ
The second term is the effect of the redundant shear strains on surfaces B1 and B4 . The expression can also be used to compute the longitudinal stress under the die, with the slight modification that the redundant shear strain at the outlet should not be taken into account r 1 cos a m cos a r rxx i d i þ ln ¼ ln þ sin a 2 sin a rY r r r mp ri i 1 ln 2 sin a r r for rf < r < ri : ð3Þ Conversely, since rxx ryy ¼ rY (Tresca yield criterion), the triaxiality equals rm rxx 1 ¼ rY rY 2
ð4Þ
It is highly negative (hydrostatic compression) close to the inlet but it increases with the current reduction and may reach positive zones if the reduction exceeds a critical value, close to 30% with the case shown in Fig. 4. The integral of triaxiality (here equal to the normalized hydrostatic stress shown above) over
normalized stress
thickness reduction, % 0.0% 1 0.5 0
-0.5
20.0%
40.0% xx triax. yy
-1
Fig. 4. The principal stresses under the die for a large reduction, normalized by the yield stress of the material for md ¼ 0:12 and mp ¼ 0:03. The hydrostatic stress is negative (compressive) close to the outlet, but increases with the reduction and may reach positive (depressive) values for a large reduction.
damage factor, dimensionless
flow stress rY (supposed constant: no work-hardening) is
1523
-0.02 -0.04 -0.06 -0.08 -0.1
Fig. 5. Integral of stress triaxiality over strain path for different reductions in the conditions used in Fig. 4. If such a term governs damage evolution, it is compatible with the observations.
the strain path is plotted in Fig. 5, where it is shown to present a minimum value just like the inverse of the relative variation of density.
5. Discussion The healing evidenced by density in single-pass ironing can be explained by the compressive hydrostatic stress observed for light ironing reductions. At larger reductions, the effect still exists close to the inlet but is gradually balanced by the larger longitudinal tension close to the outlet. Semi-quantitatively, the effect can be depicted by the integral of triaxiality over the strain path, that presents a minimum at a finite reduction. The reason why it depicts the evolution of density is that the (positive or negative) growth of pre-existing voids is the major feature in this forming process where nucleation does not play a large role, at least in the bulk (Kenny et al. [12] have proven that surface particles on the die side are broken and buried under the surface during the ironing operations). As such, the model does not pretend more than a semi-quantitative agreement with the experimental observations. Indeed, work-hardening was not taken into account; it would have shifted the minimum towards slightly larger reductions. But most improvements of the model may arise from the friction behaviour: as a matter of fact, since the radial stress decreases with increasing reduction, it is expected that the lubricant thickness decrease on
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J. Courbon / Scripta Materialia 48 (2003) 1519–1524
the die side; hence, the coefficient of friction should also increase. The experiments suggest that a progression of intermediate ironing reductions is better than a single large reduction in order to form the can wall, in compliance with common practice. Anyhow, in the event of residual formability needed after the ironing stages, the benefits of a maximised density are fairly low. This is illustrated by the strong effect of redraw die radius on density as well as wall elongation. The smaller die radius definitely brought a higher damage level and a lower elongation than the larger one. As pointed out by Baudelet and Grange [3], sheet bending over a small radius nucleates a large number of damage sites, ready to grow under further forming. And compressive hydrostatic stresses only reduce the void volume without wiping it out completely. An experimental test of the correlation between density and ductility could have been the ductility for the 22% reduction material, where the density is highest. Unfortunately, the tensile tests were conducted with less samples (3) as in the other cases, so that the larger interval of confidence prevents from reporting significant differences. Anyhow, further experiments should focus on more sensitive parameter, the reduction of area at fracture, as suggested by Sarkar et al. [13]. Furthermore, X-ray microtomography is a recent technique [14] that gives a chance to observe the voids in situ, without surface perturbations. Therefore, it is hoped that it can be applied to observe an interrupted ironing sequence, with scans between the inlet and the outlet, so as to characterize the competing effects of both principal stresses of opposite signs. But it may also apply to
a variety of forming stages and to evaluate the residual potential of a material of given density but with various damage history.
Acknowledgement The author acknowledges Pechiney CRV for funding the research.
References [1] [2] [3] [4] [5]
[6]
[7]
[8] [9] [10] [11] [12] [13] [14]
Sauer R. Metallurgy 1996;50:649–54. Courbon J. Mater Sci Forum 2000;331:17–30. Baudelet B, Grange B. Scr Metal Mater 1992;26:375–9. Cockroft MG, Latham DJ. J Inst Metals 1968;96:33. Raynaud GM, Meyer B, Boutin FR. In: Proceedings of the 8th international light metals congress (Leoben-Wien), Aluminium Verlag, D€ usseldorf; 1987. p. 460. Courbon J, Duval JL. Proceedings of the 3rd European conference on advanced materials and processes (EUROMATÕ93) J Phys IV 1993;3(colloquium C 7):247–50. Favier S. Contribution a lÕ etude de lÕendommagement de materiaux metalliques contenant des particules dures. Application a lÕalliage dÕaluminium 3004 H19. PhD thesis, INP Grenoble; 1995. Canova GR, Favier S, Lebensohn R, Baudelet B. Mater Sci Forum 1996;217:1441–8. Oyane M. Bull JSME 1972;15(90):1507–13. Gurson AL. J Eng Mater Technol (Trans ASME) 1977;99:16. Avitzur B. Metal forming: processes and analysis. New York: McGraw-Hill; 1968. Kenny LO, Sang H, Saez E. Scr Metal Mater 1991;25:419. Sarkar J, Kutty TRG, Conlon KT, Wilkinson DS, Embury JD, Lloyd DJ. Mater Sci Eng A 2001;316:52. Baruchel J, Buffiere J-Y, Maire E, Merle P, Peix G. X-Ray tomography in material science. Paris: Hermes Science; 2000.
Damage evolution in a compressive forming process: ironing of beverage cans Jo€el Courbon
*,1
GEMPPM (UMR CNRS 5510), INSA de Lyon, B^ atiment Saint-Exup ery, 20 Avenue Albert Einstein, 69621 Villeurbanne Cedex, France Received 8 November 2002; received in revised form 21 February 2003; accepted 10 March 2003
Abstract Density has been used to investigate the evolution of damage with strain in wall ironing of an aluminium beverage can (AA3104). The observed increase of density and the fact that it reaches a maximum are explained semi-quantitatively by the hydrostatic stress inside the material and a damage criterion for porous media. 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Damage evolution; Void growth; Forming; Ductility; Aluminium alloy
1. Introduction Ironing is the forming process by which the walls of a beverage can are being thinned: after a cup has been formed by deep drawing, its wall is pulled by a punch through a die of lower inner diameter than the cup outer diameter. The material thins and stretches. It enables one to form a container with optimal material distribution (thinner in the wall where not as much strength is needed to resist pressure). Under the die, it is often assumed that the hydrostatic stress is compressive, so that high strains can be achieved. But the critical step in the process is the flow out of the die, since the principal stress becomes a tension along the wall axis (Figs. 1–3).
*
Tel.: +33-472-43-63-81; fax: +33-472-43-85-39. E-mail address: [email protected] (J. Courbon). 1 Formerly at Pechiney Centre de Recherches de Voreppe, CentrÕAlp, BP 27, 38341 Voreppe Cedex, France.
This stress has two drawbacks: it induces a compressive hoop stress that shrinks the can on the punch and that hinders its stripping, especially for tinplate cans [1]; it may be large enough to induce fracture (‘‘tear-off’’) on aluminium cans, that are made from highly cold-worked alloy (AA3104 H19) with low residual tensile elongation [2]. The observed fracture mode is ductile, with cusps around constituent Fe-rich particles about 5 lm in size. It is then relevant to study how damage nucleates and grows, in order to minimize it and reduce the risk of failure. The relative variation of density is a powerful experimental tool in order to obtain information at the scale of a few hundreds of cubic millimetres of material. A prior study by Baudelet and Grange [3] applied it to the can making process. For the deep drawing stages, mainly depressive, density decreased and damage conversely increased with the height in cup wall, therefore with increasing forming strain; the observed density profile in the cup wall could be modelled using the Latham and Cockroft damage
1359-6462/03/$ - see front matter 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-6462(03)00131-3
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J. Courbon / Scripta Materialia 48 (2003) 1519–1524
a/ REDRAW DIE
b/ IRONING DIE
die radius
punch velocity
yield stress (MPa)
Fig. 1. Sketch of the forming process: (a) redraw die with the die radius and (b) ironing die.
360 350 340 330 320 310 300 290 280
current data previous data Ludw ik fit 0
0.2 0.4 0.6 0.8
1
decreased during the last ironing step. The authors explained this phenomenon by shear stresses in the vicinity of the die, that became dominant as the thickness was reduced. This evidence of damage healing in a compressive process is new. Damage as considered by mechanics is irreversible, whereas its microstructural cause, voids in the material, may be reduced. Yet the decohesions or fractures induced previously are not necessarily wiped out, and more generally internal interfaces created in the material may be decorated by dissolved hydrogen (for instance). So the phenomenon needs further investigations, both per se and for the consequences on ductility or formability. The purpose of the present paper is to evaluate the role of thickness reduction on density and explain the observations by the hydrostatic stress at stake in single-pass ironing. It is a complement of the former study [3], since the same forming and measurements devices have been used.
1.2
Tresca equivalent strain
Fig. 2. Strain-hardening curve of the material in the ironing process.
Np die
Td ti
B1 Nd
B4 Tp
F punch
tf O y x
Fig. 3. A slice of the wall material between the ironing die and the punch. The initial and final wall thicknesses are ti and tf . The plastic flow can be modelled in local radial coordinates with centre O, between the radii ri ¼ ti = sin a and rf ¼ tf = sin a.
criterion [4], in which damage is proportional to the integral of the main tensile stress over the strain path. But for the first two ironing stages–– after which the wall thickness successively became 280 and 200 lm (original sheet thickness: 335 lm)––they reported that density increased. The cavities developed in deep drawing were partly closed by the positive hydrostatic pressure developed by the process. On the other hand, density
2. Experimental The same sheet material has been used as in [3], on the same forming devices. The material is AA3104 H19 containing 0.99 wt.% Mn, 0.99 wt.% Mg, 0.38 wt.% Fe, 0.18 wt.% Si, 0.14 wt.% Cu was used at the sheet thickness 335 lm. The mean value of the yield stress at 0.2% proof strain in tension is 280 MPa. Cups have been deep drawn at a diameter of 85 mm. Then they have been redrawn and ironed through a redraw die and a single ironing die. Two redraw dies with different radius have been tested: 2.5 and 3.25 mm. The forming section of the ironing die was a cone of half-angle a ¼ 8. Relative thickness reductions of 11%, 22% and 35% were achieved. For damage evaluation, a 9 mm high ring was carefully cut from the can wall starting at the height of 35 mm, viz. above the transition zone between the bottom and the thin wall. It was annealed in order to wipe out the contribution of dislocations to the density. The detailed experimental protocol is given in [5].
J. Courbon / Scripta Materialia 48 (2003) 1519–1524
For can wall elongation, longitudinal tensile specimens with 45 mm length, 10 mm width were cut starting at the height of 12 mm, so that the same area covered by density underwent the tensile test. Anyhow, the tensile stress in the can wall does not vary significantly in the zone of constant thickness [6].
3. Results 3.1. Density Both the ironing strain and the redraw die radius had an influence on density. Table 1 sums up the results. They are coherent with [3]: density actually increases during the ironing process, meaning that cavities are refilled by the matrix due to the hydrostatic compression. Yet this healing effect is not perfect, since the density remains lower than that of the cold-rolled sheet, that itself is lower than the theoretical one since it already contains some decohesion and fragmentation voids [7]. Also, the variations with the ironing reduction denote that a maximum healing is reached close to the intermediate reduction, since damage is larger for the highest reduction. There is a strong effect of the redraw die radius: the smaller radius induced more damage than the large one. This is coherent with the role of bending and unbending in the nucleation of damage reported by Baudelet and Grange [3] and also with empirical practice, that suggests to use a radius at least 10 times as large as the sheet thickness to avoid trouble in sheet forming. The larger radius
1521
(ratio 9.7) is close to compliance with that empirical rule, not the smaller (ratio 7.5). As a matter of fact, the effect on density has the same order of magnitude as the ironing reduction. 3.2. Tensile tests Table 2 sums up the results obtained on wall material. The ironing operation is inducing some work-hardening of the material, as already evidenced [5]. The proof stress results have been plotted on a stress–strain curve: Fig. 2, using a macroscopic equivalent of the Tresca plastic strain ep ¼ lnðti =tf Þ where ti and tf stand for the initial and final can wall thickness. As can be seen, the present results are coherent with previous ones [6] and allow one to propose a hardening law for the wall material r ¼ Kðe0 þ ep Þ
n
ð1Þ 2
with K ¼ 348 MPa, e0 ¼ 3 10 and n ¼ 0:062. The material achieves an extreme state with very limited tensile ductility. As shown in the following Table 3, the elongation at necking As and the (total) tensile elongation Ar follow a global decreasing trend with the ironing strain. Yet small but detectable effects of damage on the average values of As and Ar are detected: the lower redraw die (maximum damage) is always associated with the lowest ductility. Altogether, the experimental results have confirmed the healing effect of the ironing process on damage and also shown that there exists an optimum reduction for maximum healing. The following mechanical analysis is going to propose an explanation for this last phenomenon.
Table 1 Effect of strain and redraw die radius on the relative variation of density of can wall State
Redraw die 2.5 mm
Redraw die 3.25 mm
Redrawn cup Reduction 11% Reduction 22% Reduction 35%
)12.0 104 )10.3 104 )7.7 104 )9.1 104
Not available )7.3 104 )5.8 104 )7.5 104
The reference density is that of the unformed sheet material. Values are the average of 3 or 4 individual measurements and are given within 0.5 104 at the 95% confidence level.
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J. Courbon / Scripta Materialia 48 (2003) 1519–1524
Table 2 Tensile properties of longitudinal specimens cut from the wall of ironed cans State (%)
rp0:2 (MPa)
rmax (MPa)
Reduction 11 Reduction 35
309 3 328 2
326 2 339 2
4. Analytical evaluation of stresses Canova et al. [8] have proposed a micromechanical damage model in which the local stress is able to open or close a cavity neighbouring a constituent particle, depending on stress triaxiality (defined by the ratio of the hydrostatic stress to the flow stress). However, the applications rely on a collection of particle shapes and orientations collected from image analysis. We simplify the problem by assuming that the major role in damage evolution is that of stress triaxiality, through an integral of triaxiality with strain. Such an expression is part of the Oyane damage criterion [9] for porous media, but it is also consistent with GursonÕs formulation [10]. Let us then determine the variation of the hydrostatic stress with the ironing reduction. The basic forces on a thin slice of material are shown in Fig. 3. The material under the die is submitted to the longitudinal tension F , the normal reactions of the punch and die Np and Nd , the friction forces Tp and Td . Friction on the punch side is a driving force, since the relative velocity of material vs. punch points backwards. Indeed, the common industrial practice uses a rough or crosshatched punch and a smooth die in order to minimize the residual tension F . Avitzur [11] proposed an upper bound method to evaluate F . We resumed the mainlines and added developments about the hydrostatic stress.
Fig. 3 summarizes the geometry. In the coordinates system of the die, the material flows at uniform horizontal speed Vi and thickness ti before the die; after the die, speed and thickness have become Vf and tf . The conservation of flowrate imposes Vf tf ¼ Vi ti . Under the die, the flow on the die side and on the punch side is parallel to the side, so it points towards O as seen in Fig. 3. The plastic volume is comprised between four boundaries: • inlet B1 : r ¼ ri ¼ ti = sin a; 0 6 h 6 a; no applied force; • die side B2 : applied forces Td and Nd ; • punch side B3 : applied forces Tp and Np ; • outlet B4 : r ¼ rf ¼ tf = sin a; 0 6 h 6 a; applied force F . The velocity field proposed by Avitzur is ~ðrÞ ¼ V ðrÞ cos h~ V ur . It is kinematically admissible since it conserves the flowrate and the normal component of velocity across boundaries. Its drawback is the fact that the slowest flow occurs close to the die and the quickest close to the punch, whereas in reality it is the reverse due to the previously described friction conditions. With the upper bound method, the coefficients of friction are of the Tresca type on boundaries B2 and B3 (the friction shear stress is a fraction of the Tresca shear flow stress rY =2): md and mp , respectively. It is suggested to use an overestimate for md (dissipative power) and an underestimate for mp (driving power) so that the calculation remains an upper bound. The balance of powers includes that of the applied forces and that of the deformation process (shear discontinuities on B1 and B4 inducing redundant work, volumic plastic power inside the forming volume). From the calculation, the residual longitudinal tensile stress, normalized by the
Table 3 Effect of strain and redraw die radius on the elongation at necking As and the (total) tensile elongation Ar of can wall Die radius (mm)
2.5
Reduction (%)
As (%)
Ar (%)
As (%)
3.25 Ar (%)
11 35
1.1 0.1 1.1 0.15
2.2 0.2 1.9 0.2
1.7 0.2 1.35 0.1
2.7 0.2 2.2 0.15
Values are the average of 15 individual measurements; intervals of confidence are given at the 95% confidence level.
J. Courbon / Scripta Materialia 48 (2003) 1519–1524
current thickness reduction 0.0% 20.0% 40.0%
rxx F ¼ 2pRtf rY ri 2ð1 cos aÞ md cos a ri þ ln ¼ ln þ sin a 2 sin a rf rf mp ri ri 1 ln 2 sin a rf rf
0
ð2Þ
The second term is the effect of the redundant shear strains on surfaces B1 and B4 . The expression can also be used to compute the longitudinal stress under the die, with the slight modification that the redundant shear strain at the outlet should not be taken into account r 1 cos a m cos a r rxx i d i þ ln ¼ ln þ sin a 2 sin a rY r r r mp ri i 1 ln 2 sin a r r for rf < r < ri : ð3Þ Conversely, since rxx ryy ¼ rY (Tresca yield criterion), the triaxiality equals rm rxx 1 ¼ rY rY 2
ð4Þ
It is highly negative (hydrostatic compression) close to the inlet but it increases with the current reduction and may reach positive zones if the reduction exceeds a critical value, close to 30% with the case shown in Fig. 4. The integral of triaxiality (here equal to the normalized hydrostatic stress shown above) over
normalized stress
thickness reduction, % 0.0% 1 0.5 0
-0.5
20.0%
40.0% xx triax. yy
-1
Fig. 4. The principal stresses under the die for a large reduction, normalized by the yield stress of the material for md ¼ 0:12 and mp ¼ 0:03. The hydrostatic stress is negative (compressive) close to the outlet, but increases with the reduction and may reach positive (depressive) values for a large reduction.
damage factor, dimensionless
flow stress rY (supposed constant: no work-hardening) is
1523
-0.02 -0.04 -0.06 -0.08 -0.1
Fig. 5. Integral of stress triaxiality over strain path for different reductions in the conditions used in Fig. 4. If such a term governs damage evolution, it is compatible with the observations.
the strain path is plotted in Fig. 5, where it is shown to present a minimum value just like the inverse of the relative variation of density.
5. Discussion The healing evidenced by density in single-pass ironing can be explained by the compressive hydrostatic stress observed for light ironing reductions. At larger reductions, the effect still exists close to the inlet but is gradually balanced by the larger longitudinal tension close to the outlet. Semi-quantitatively, the effect can be depicted by the integral of triaxiality over the strain path, that presents a minimum at a finite reduction. The reason why it depicts the evolution of density is that the (positive or negative) growth of pre-existing voids is the major feature in this forming process where nucleation does not play a large role, at least in the bulk (Kenny et al. [12] have proven that surface particles on the die side are broken and buried under the surface during the ironing operations). As such, the model does not pretend more than a semi-quantitative agreement with the experimental observations. Indeed, work-hardening was not taken into account; it would have shifted the minimum towards slightly larger reductions. But most improvements of the model may arise from the friction behaviour: as a matter of fact, since the radial stress decreases with increasing reduction, it is expected that the lubricant thickness decrease on
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J. Courbon / Scripta Materialia 48 (2003) 1519–1524
the die side; hence, the coefficient of friction should also increase. The experiments suggest that a progression of intermediate ironing reductions is better than a single large reduction in order to form the can wall, in compliance with common practice. Anyhow, in the event of residual formability needed after the ironing stages, the benefits of a maximised density are fairly low. This is illustrated by the strong effect of redraw die radius on density as well as wall elongation. The smaller die radius definitely brought a higher damage level and a lower elongation than the larger one. As pointed out by Baudelet and Grange [3], sheet bending over a small radius nucleates a large number of damage sites, ready to grow under further forming. And compressive hydrostatic stresses only reduce the void volume without wiping it out completely. An experimental test of the correlation between density and ductility could have been the ductility for the 22% reduction material, where the density is highest. Unfortunately, the tensile tests were conducted with less samples (3) as in the other cases, so that the larger interval of confidence prevents from reporting significant differences. Anyhow, further experiments should focus on more sensitive parameter, the reduction of area at fracture, as suggested by Sarkar et al. [13]. Furthermore, X-ray microtomography is a recent technique [14] that gives a chance to observe the voids in situ, without surface perturbations. Therefore, it is hoped that it can be applied to observe an interrupted ironing sequence, with scans between the inlet and the outlet, so as to characterize the competing effects of both principal stresses of opposite signs. But it may also apply to
a variety of forming stages and to evaluate the residual potential of a material of given density but with various damage history.
Acknowledgement The author acknowledges Pechiney CRV for funding the research.
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