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Cyclic stretch-bending: mechanics, stability and formability W.C. Emmens, A.H. van den Boogaard*. University of Twente, Enschede, the Netherlands. *Corresponding author: [email protected], tel +31-(0)53-4894785 Keywords: cyclic stretch-bending, bending-under-tension, formability, CBT test Abstract. Cyclic stretch-bending has been studied using the so-called Continuous-Bending-
under-Tension (CBT) test. This is a modified tensile test where the specimen is subjected to repetitive bending at the same time. A wide variety of materials has been tested this way. A simple mechanical model is presented incorporating non-constant bending radius and cyclic material behaviour, that is verified experimentally. The increased stability allowing high lev-els of uniform straining is discussed, a stability parameter is presented. The mechanisms of failure are discussed as are the different types of fracture that are observed. The overall con-clusion is that many observed results are determined by the test method, and are as such inde-pendent of the material tested.
1 Introduction. It is a well established fact by now that in Incremental Sheet Forming (ISF) levels of defor-
mation can be achieved well above the conventional Forming Limit Curve (FLC). In a recent paper the authors have discussed the underlying deformation mechanisms and concluded that, besides the very incremental nature of the process, four mechanisms can be distinguished that might be responsible for this enhanced formability: shear, bending, non-planar stress, and cyclic straining, see Emmens and van den Boogaard (2009a). All these mechanisms have been the subject of recent research. For example Eyckens et al. (2009) have carried out an elaborate theoretical analysis of the effect of shear in sheet forming, and concluded that in-deed this can raise the formability. Allwood and Shouler (2009) have done similar work and included thickness stress as well, and observed that thickness stress can also raise the form-ability. Yagami et al. (2009) have carried out tests on cyclic straining, albeit with some bend-ing, and also concluded that this can raise the formability. That bending can raise the form-ability has been known for quite some time, but it has gained recent renewed interest by ob-servations that formability in material tests is affected by the tool radius. For example Schleich et al. (2009) have investigated the effect of sheet thickness on the FLC (Forming Limit Curve), and concluded that indeed the FLC is raised by bending.
A special case is the occurrence of repetitive bending, not to be confused with pure cyclic straining as investigated by Yagami. This effect can raise the formability to huge levels as was first noticed by Benedyk et al. (1971) using a modified tensile test that they called Con-tinuous-Bending-under-Tension (CBT) test and that will be described in detail below. The same test method was used by student Mols (1972) of the group of prof. Jongenburger at the University of Delft to simulate the tension-leveling process, also obtaining high levels of straining, but there is only an internal report on that.
The authors have used the CBT test in their laboratory from 2007 onwards to study effects of repetitive bending in ISF. In a previous paper results have been presented that showed lev-els of uniform elongation of over 400% (Emmens and van den Boogaard, 2009b). In that pa-per many similarities in material behavior were observed, but as the tested materials had simi-lar mechanical properties it was difficult to conclude which results are material dependent,
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and which are simply determined by the testing method. In a recent paper Hadoush et. al. (2011) have presented simulations of the CBT test, and concluded that many aspects of the CBT test can be simulated successfully using only simple material models. As far as known to the authors this particular test method has not been used elsewhere, although very recently work has started by Allwood and Shouler (2011).
The present paper presents more results with the CBT test with a wide spectrum of materials that allow better founded conclusions on material behavior. With respect to the first paper the following extensions have been made:
- the number of materials tested has been increased from 5 to 15, comprising now mild steel, aluminium, dual-phase steel, stainless steel, titanium and brass;
- the range of experimental conditions has been increased - the analysis of the test has been extended considerably, including now non-constant bend-
ing radius and cyclic effects; the model is verified by the relation between force and speed; - a detailed discussion on stability and process limits is added. This main focus of this paper is to present experimental results; the model that is presented
is not intended to explain all these results, only to understand some basic aspects. The results are grouped by subject. Sections 3, 4 and 5 present and verify a mechanical model and dis-cuss: 'what is happening during the test'. Section 6 discusses stability and answers the ques-tion 'why are large levels of uniform elongation possible' (and why sometimes not). Sections 7, 8 and 9 look into fracture types, process limits and formability, answering: 'what does limit the tests'. Section 10 finally presents a specific type of roughening and instability. The pur-pose of the paper is twofold. Firstly it presents a firm background for all researchers who want to use this type of test in their laboratory. Secondly it presents a more general insight in the behaviour of material subjected to cyclic stretch-bending, and the simple model presented here can indicate the direction for (possible) future advanced modelling.
2 Description of the test method and experimental conditions.
LD
P
three-roll set
strip
vroll(const)
vc-b(variable)
Figure 1. The test equipment. The specimen is held fixed at the top and is pulled down-
wards in the tensile testing machine. Left: schematic representation, right: picture of the roll set with a specimen mounted in position.
This section presents a brief description of the test method, an overview of materials that are
used and the experimental conditions, and some basic results. The CBT test is in fact a conventional tensile test carried out on a large tensile test speci-
3
men, whilst at the same time a set of three freely rotating rolls is continuously moving up and down this specimen; this is schematically presented in figure 1. The bending of the strip by the rolls is considered to be bending-under-tension (hence the name of the test), meaning that the level of bending is so severe that at the concave side of the specimen the fibres are actu-ally in compression. The test has been described in the previous papers already, so only a brief description is given here. The main characteristics are:
- At any moment the deformation is concentrated into a number of small zones, two for each roll (bending and unbending region), unless for severe conditions where the pulling force be-comes very high.
- As a result the actual strain rate is much higher than the apparent strain rate. A detailed analysis (not presented here) taking into account both the length of the actual forming zones, and the true integrated incremental strain, showed that it can be increased by a factor of 100.
- The strain is not uniform over the entire length of the specimen, see figure 2, but there is a zone of uniform straining determined by up-down stroke of the roll set.
- The relation between the true strain ε in that uniform zone and the total elongation ∆l is given by
000
1lldl
lldl
ldl ∆
====ε ∫∫∫ (1)
where l0 is the up-down stroke of the roll-set; this has been verified experimentally. - The total strain increment per passage of the roll set is:
ud
cb
vv
=ε∆ (2)
where vud is the speed of the up-down movement and vcb is the cross bar speed. In each of the small forming zones the strain increment is one-sixth of that.
- The strain state shifts towards plane strain, and more when the depth setting P (figure 1) is larger. The relative change in strain state defined as the ratio between width strain and thick-ness strain is roughly the same for all materials.
Figure 2. Specimen after testing. The left-hand side was held stationary, the right-hand side
was attached to the moving cross-bar. The rectangle indicates the zone of uniform elongation. Due to limitations of the machine long specimens had to be used as presented in figure 3.
These long specimens presented another problem with materials that show high levels of work hardening, like stainless steel and soft brass. At high pulling forces the wider parts showed also a considerable elongation, meaning that the total elongation of the centre, thinner part was much lower than the maximum stroke of the machine (160 mm). Therefore 'elonga-tion' always means: the elongation of the thinner part, and not cross-bar displacement.
295 210 295
R = 10
30.0
Overall size : 800 x 30 mm
20.0
Figure 3. Dimensions of the specimen, not to scale.
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A wide variety of materials has been used in the test allowing thorough conclusions about influence of material. The selection comprises bcc, fcc and hcp crystal structures, single phase and dual phase structures, and variations in thickness, formability and pre-deformation. An overview of all materials is presented in table 1. All data have been obtained in standardized tensile tests, except for the conventional strain-rate hardening coeffcient m, that was deter-mined from the force maxima in three tensile tests carried out at 0.2, 1 and 5 mm/s respec-tively. Material SS1 shows a specific behaviour: carrying out separate tests with different speeds yields another result than abruptly changing the speed in a single test. The force maxima showed a negative strain-rate hardening, but the force at lower strains showed a posi-tive effect. However the amount of post-uniform elongation indicates that the momentarily strain-rate hardening is positive. A complete overview of all experimental conditions is pre-sented in table 2.
Table 1. Overview of materials tested.
Nr Description Code thick-ness
(mm)
YS (MPa)
UTS (MPa)
unif. elong (%)
r n m
MS1 mild steel DC04 0.80 161 288 22 2.54 0.21 0.017 MS2 mild steel DC04 0.97 159 296 24 2.15 0.22 0.016 MS3 mild steel DC06 0.70 143 301 23 2.51 0.23 - MS4 mild steel DC04 0.70 164 305 22 2.22 0.21 0.014 MS5 mild steel DC04 0.96 185 309 20 2.51 0.19 0.016 MS6 mild steel DC04 0.76 340 354 6 - 0.10 - DP1 DP steel; DP600 0.99 387 667 14 0.98 0.15 0.006 SS1 stainl. steel SS304 1.4301 0.80 320 695 62 0.98 0.51 ±0.011 AL1 alum. AA5182 1.15 124 287 22 0.66 0.32 -0.009 AL2 alum. AA6016-T4 0.96 109 233 18 0.64 0.25 0.005 AL3 alum. AA6016-T4 1.10 145 261 22 0.70 0.25 -0.001 BR1 brass 63/37, soft 2.0321 0.69 130 330 51 0.69 0.55 0.000 BR2 brass 63/37, medium 2.0321 0.70 250 500 35 0.70 0.36 -0.002 BR3 brass 58/39/3, hard 2.0401 0.69 520 610 8 0.71 0.08 -0.002 TI1 titanium grade 2 3.7035 0.78 400 439 10 4.0 0.15 0.023 Notes: 1: material MS6 is material MS1 pre-strained to εx = 0.18. 2: material DP1 is zinc-
coated, all other materials are uncoated. 3: material BR3 is leaded brass, materials BR1 and BR2 ordinary yellow brass.
Table 2. Overview of all experimental conditions. Material see table 1 Specimen size see figure 3 Up-down speed 66.7 mm/s (constant) Up-down stroke 140 mm (default), 120, 110, 100, 80 mm Roll diameter D (figure 1) 15 mm Roll distance L (figure 1) 35 mm Roll penetration P (figure 1) 2.3 mm (default) 0.3, 1.3, 3.3, 4.3 mm* Pulling speed 0.10 mm/s - 20 mm/s Pulling distance until fracture or until max. stroke (= 160 mm) *Note: Roll penetration P differs for material MS1. Of all tests a force-displacement curve has been recorded with a sampling rate of 20 Hz.
Figure 4 presents two representative force-displacement curves (material MS1), the indicated speeds are cross-bar speeds. The curves resemble the curves found in a conventional tensile
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test when there is no instability. The graphs show the up-down movement as periodic peaks. The detailed shape of the curves has been analysed by Hadoush et.al. (2011), but it is of less interest here. The analysis below only requires a parameter for the force level, and for that the maximum force after curve-smoothing has been used. Smoothing is carried out by the method of moving average, and averaging over a complete up/down cycle. Figure 4 shows typical results.
0 20 40 60 80 100 120 140 160elongation (mm)
0
1
2
3
4
5
pullin
g fo
rce
(kN
)
0.7 mm/s
3.5 mm/s
Figure 4. Examples of two recorded force-displacement curves (material MS1), both as mea-
sured (black) and after smoothing (grey).
Of all samples the strain has been measured after testing in the uniform section denoted by the rectangle in figure 2. This has been done by measuring both the actual width and thickness by hand, and from that both the width strain and thickness strain have been determined. These were used to determine the logarithmic length strain assuming constant volume. In most cases the strain was indeed uniform to a high degree. In some cases the strain was not entirely uni-form due to the occurrence of diffuse necks, an extreme case is shown in figure 20 in section 6; the strain outside the necks has been used in the analysis.
3 Elementary mechanics of the CBT test. This section presents a simple model of the CBT test that predicts a relation between pulling
force and pulling speed, which can be verified experimentally. Bending-under-tension implies simultaneous bending and stretching of a strip. Consider a
strip of thickness t and width w that is bent to a radius R whilst at the same time it is subjected to a pulling force F in the x-direction.
For a situation with pure bending the strain in x-direction over the thickness t is presented in figure 5, left (second order effects are neglected). The strain in the outer fibre eb is equal to:
Rte
2b = (3)
For a situation that the specimen is subjected to a simultaneous stretching over an amount e the strain distribution is presented in figure 5, right. The neutral line is shifted over a distance zt equal to:
Reeetz ⋅=⋅=b
t 2 (4)
In the following the relative position of the neutral line is described by a parameter α:
tRe
eeb
⋅⋅==α
2 (5)
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For α < 1 the neutral line is inside the material, for α > 1 it is outside the material. For a situation of perfectly plastic material with yield stress σf the stress distributions corresponding to the strain distributions of figure 5 are presented in figure 6. Note that this is only valid in situations where both bending and stretching are occurring at the same time, and not one after the other (like first bending to a radius R, and then stretching over that radius).
X
Zeb e
z t
X
Zσf
z t
Figure 5. Strain in x-direction in the bent
specimen; Left: situation of pure bending, right: situation of combined bending and stretching
Figure 6. Stress distributions corresponding to the strain distributions of figure 5, for a per-fectly plastic material with yield stress σf.
The net pulling force F and bending moment M both per unit width can now be calculated
straightforward:
1,0
1),1(4
1,1,
22
f
f
f
>α=
<αα−⋅⋅σ=
>α⋅σ=<αα⋅⋅σ=
M
tM
tFtF
(6)
These relations are presented graphically in figure 7.
1 1α
F M
σf.tσf.t
2
4
α
Figure 7. Graphic representation of the relations (6).
Figure 7 illustrates two important aspects of bending-under-tension: - if a strip is being bent, it can be stretched with a lower force than in a situation without
bending (α = ∞ > 1); - if a strip is being stretched, it can be bent with a lower moment than in a situation without
stretching (α = 0). The first observation is of relevance here, it shows that in the tests presented in this report
only material actually being bent will deform, as in first approximation the tensile force is constant over the length of the strip and material being bent requires the lowest force for elongation. Consequently the deformation is concentrated into the bending zones. The results further indicate that the most important factor that influences both force and moment is α, or better: the position of the neutral line, in other words: how much material is in compression.
When a strip is pulled over a radius under high tension it is save to assume that the bending radius of the strip is more or less equal to the die radius. However in the CBT test that is not
7
the case. The pulling force can be quite low, meaning that it is not save to assume that the strip is pulled tight against the rolls. The ‘actual’ shape will now be estimated using a balance between pulling force and bending moment, using some simplifications.
Figure 8 shows the situation of the specimen and the rolls schematically. The nominal shape is presented by the thin line, the actual shape by the thick line. A correct analysis re-quires the correct shape of the strip, and the location of contact A. Both are unknown, but it will be assumed here that the distance L is proportional to the local bending radius R at the point of contact A (this will be justified below). If the strip touches the roll at position A, and MA and FA are respectively the bending moment and the tension force of the strip there, one can simply state: MA = FP⋅L = FA⋅L/cos(ϕ) where FP is the external pulling force. The assump-tion above states: L = a0⋅R where a0 is some geometrical constant, for a constant bending ra-dius: a0 = 1-cos(ϕ). So:
ϕ=⋅⋅=⋅⋅=
cos, 0
A0PAaaRaFRaFM (7)
Note that this condition cannot have a solution for α > 1 as that requires M = 0.
ϕ
θ
R
FP A
L
nominal shapeactual shape
FA
FP ϕ
FA = FP.cos(ϕ)
Figure 8. Schematic representation of the shape of the specimen around the rolls. Right: forces at point A.
This balance of forces yields an equilibrium radius R as a function of the linear strain e.
Solving the equations (6) and (7) for α < 1:
)2(4 2
22
eeatR
+⋅⋅= (8)
So, using (5) and (6):
tFea
Fea
tFF ⋅σ=+
⋅=+
⋅⋅σ== f00fA ;12
112
1 (9)
This relation is presented graphically in figure 9 by the solid line. It may seem strange that the force only depends on the ratio e/a, and for example not on the
actual radius or thickness. However realise that the parameter a depends on both the radius and the thickness, see eq. (8).
The model is simple, based on elementary assumptions and ignores second order effects, therefore its validity is restricted. Nevertheless it is able to explain several experimental ob-servations, as will be shown below. Some conclusions will already be drawn here:
A. If a = 0 then F = F0; a = 0 corresponds to a situation without any bending and not sur-prisingly one finds the normal force-displacement curve
B. If vud = 0 then e = ∞ (eq. 2) so F = F0 (e is the strain increment per bending or unbending action); this means that if the rolls do not move one expects to find the normal force-
8
displacement curve as well. This has been verified experimentally. This is an important result as it shows that the presence of the roll set itself has no effect on the tension test, only when the (freely rotating!) rolls actually move an effect will be found.
0 1 2 3 4 5e/a0.0
0.5
1.0
F/F0
Figure 9. Graphic presentation of equation (9) (solid line) and (10) (dashed line). C. For small values of e (e<<2a) one can approximate R ≈ t/2/√(2ae), so:
aeFF
20 ⋅≈ (10)
This relation is pictured in figure 9 by the dashed line. This means that the force is a simple square-root function of e, but then also of the pulling speed vcb (see equation 2). This will be verified experimentally below. Note: this square root relation stems directly from the assump-tion that L is proportional to R, justifying that assumption. For large values of e one gets R ≈ t/2/e, and consequently F = F0 = constant.
D. The force only depends on the ratio between a (governed by the geometry of the set-up) and e (governed by the testing speed). This suggests that a change in geometry can be com-pensated by a change in speed. In other words: assuming that a is constant for a certain set-up, the effects of changing the geometry are the same as correcting the speed by a constant factor.
0.1 1.0 10.0 20.0pulling speed (mm/s)
0
1
2
3
4
5
max
imum
forc
e (k
N)
2.33.3
4.3
0.40.9
1.5
material MS1
0.3 1 3 10 20
0 2 4 6 8
pulling speed (mm/s)
0
1
2
3
4
5
max
imum
forc
e (k
N)
2.33.3
4.3
0.4
0.91.5
material MS1
Figure 10. Compilation of all results for material MS1 (mild steel). Legend indicates depth
setting in mm. For experimental verification of the relations the maximum puling force in a single test will
be used. This maximum is derived from the recorded force-displacement curves as shown in figure 4 after smoothing. Figure 4 illustrates that for a certain material this maximum occurs
9
at the same elongation, so it can be used as a parameter for the force level in general. Of the materials tested mild steel is least affected by cyclic effects, so the results for mild
steel can be used to check this model (cyclic effects will be discussed in detail in the next sec-tion). Figure 10 presents the results for material MS1. When plotted to a logarithmic speed scale (left) the results form a collection of S-shaped curves, one for each depth setting. In first approximation these curves have the same shape, the effect of depth setting indeed is simply a shifting of the curves horizontally. The right hand graph shows the same results, but now plot-ted to a linear speed scale. The shape of the curves in this way can be compared to the shape presented in figure 9, and again the agreement is satisfactory.
From each of the S-shaped curves the ‘position’ can be defined as the speed at which the force is 50% of the maximum force. This position increases with increasing depth setting. Detailed analysis showed that the relation between position and depth setting is approximately the same for all materials, despite the significant difference in mechanical properties.
The analysis above has been carried out for non-hardening material. This may seem a severe limitation as all (conventional) materials do harden, occasionally considerably. However real-ise that in the experiments that this analysis is related to, bending-under-tension occurs in stretched specimens that have been deformed already so that consequently the rate of harden-ing has been reduced. To justify the assumption some calculations have been carried out on realistically hardening material (σ(ε) = C⋅(ε0+ε)n, n=0.23) including thinning, for several lev-els of pre-straining ε0. In these calculations the stress was integrated numerically over the thickness at 50 equally distributed points, at strain intervals of 0.002. The radius has been kept constant, equal to the roll radius, but that does not affect the conclusions. Figure 11 shows the results. It is clear that already for pre-strains of 0.2 (curves B) the correct relations differ only slightly from the idealised relations that are shown in figure 7. Therefore the relations (6) can be used in the analysis.
0.00 0.05 0.10 0.15 0.20 0.25elongation
0
250
500
750
1000
bend
ing
mom
ent (
Nm
)
0
2
4
6te
nsio
n fo
rce
(kN
)AB
CD
Eforcemoment
E
D
C
B
A
pre-strain: A=0 B=0.2 C=0.5 D=0.8 E=1.1
Figure 11. Realistic force-strain and moment-strain relations for bending under tension at
constant radius. Labels denote levels of pre-strain ε0 (true strain). 4 Cyclic effects. This section discusses in detail the consequences of the fact that the material is bend cyclic;
all effects are gathered under the general description 'cyclic effects'. So far an important aspect has been ignored. In a single test the material is bent a number of
times, but this is done cyclic meaning that a bending is always followed by a reverse bending. Consequently the outer fibres exhibit cyclic straining causing special effects. The situation is visualized schematically in figure 12. The top figure presents the strain distribution (strain increment, not total strain) in a bending and consecutive reverse bending operation with si-
10
multaneous stretching, for example occurring in a situation were at A and C the strip is straight and at B the strip is curved. The figure shows that in the centre of the strip (zone I) the material is subjected to monotonous stretching only. However in the zones near the sur-face (labelled II) the material is subjected to repetitive tension and compression evoking strain reversal. The strain history for the locations 1-4 is presented in figure 12, bottom; there is no straining between the bending operations. In the outer zones II the material is subjected to repetitive reverse straining. This affects the stress level there by the occurrence of the Bausch-inger effect. Another effect is caused by the fact that the material is subjected to a momentar-ily reversal of the strain increment. This does not cause immediate reversal of stress, a certain minimal compression is needed depending on the elastic strain if the material. In extreme cases the stress may not reverse at all. A detailed analysis requires thorough knowledge of the material behaviour and is outside the scope of this paper. However the effect of both is a re-duction of the stress level in some way. The following will analyse this in general, by investi-gation what kind of effects can bb expected if the stress in the two zones II is reduced in some way.
I
II
II
A B C1234
A B C
stra
in
time
2
3 4
1
Figure 12. Top: strain distribution at a bending and consecutive reverse bending operation.
Bottom: strain history for four points. Figure 13 shows the stress distribution in the strip just as in figure 6. The sections B and C
are subjected to monotonous stretching and it may be assumed that the stress σf is simply re-lated to the net extension.. The sections A and D however are subjected to cyclic straining that will reduce the stress. A very simple correction for that can be done by correcting the stress there with multiplication factors β and γ, as shown in figure 13, right. The original situation is obtained for β = γ = 1.
A
B
C
D
σfβ.σf
−γ.σf
z t
Figure 13. Stress distribution in the strip for a perfectly plastic material; compare to figure
6. Left: without cyclic effects, right: with cyclic effects.
11
This yields the following relations for the tension force and bending moment per unit width for situations of α < 1:
2;
2
.)1(24
'
))1((2
)1('
22
f
ff
γ+β=
γ−β=
=α−⋅γ+β
⋅⋅σ=
+−⋅α⋅⋅σ=
γ−β
⋅α−+α⋅⋅σ=
qp
MqtM
ppttF
(11)
The two newly defined parameters p and q describe the anti-symmetric and the symmetric effects respectively. The original simple model corresponds to p = 0 and q = 1.
0 1 2 3 4 5q.e/a0.0
0.5
1.0
F/F0
p = 0 0.25 0.5 0.75 1.0
Figure 14. Tension force including cyclic effects. The bending moment is corrected by a factor q, which is nothing more than the average of
the two factors β and γ. The effect on the tension force F however is more complicated. In any case, for e = 0 (or: α = 0) the force is equal to σf⋅t⋅p. This means there is a non-zero minimum for p≠0, (β≠γ), or: there are non-symmetric cyclic effects.
Using these results the actual bending radius can now be calculated as before, this yields the following equation for R:
04
}{})1(2{2
22 =⋅−⋅⋅⋅+⋅+−⋅⋅⋅⋅ qtptaRqepeaR (12)
from which the tension force can be calculated by:
})1(2{0 ppReFF +−⋅⋅⋅⋅= (13)
These equations have been solved for some positive values of p, and the results are shown in figure 14. The parameter q affects the results as if a is replaced by a/q. The parameter p re-sults in a non-zero force at zero speed. A more detailed analysis showed that the presence of cyclic phenomena reduces the actual bending radius, sometimes considerably.
Of the materials tested stainless steel shows the strongest cyclic effects. Figure 15 presents the results for material SS1 in the same way as figure 10 presented the results for mild steel. It is clear that in no way the results can be interpreted as "a series of curves of the same shape, shifted horizontally". The model predicted that asymmetric cyclic phenomena can create a non-zero minimum for the pulling force. This is clearly the case here, most notably seen in the right-hand figure. Also the shape of the curves as seen in the right-hand picture show a satis-factory agreement with the theoretical curves in figure 14 (except for the 0.3 mm results around 2 mm/s, that are an anomaly).
12
0.1 1.0 10.0 20.0pulling speed (mm/s)
0
2
4
6
8
10
12m
axim
um fo
rce
(kN
)
2.3
3.3
4.3
0.3
1.3
material SS1
0.3 1 3 10 20
0 2 4 6 8
pulling speed (mm/s)
0
2
4
6
8
10
12
max
imum
forc
e (k
N)
2.3
3.34.3
0.3
1.3
material SS1
Figure 15. Compilation of all results for material 8 (stainless steel). Legend indicates depth
setting in mm. 5 Comparison of all materials. This section presents results for all materials, notably the observed relation between pulling
force and pulling speed that will be compared to the simple model developed above. To compare different materials the relative force is used, defined as the recorded maximum
force divided by the material UTS. The UTS was derived from conventional tensile tests car-ried out at 0.2, 1 and 5 mm/s, extrapolated to 10 mm/s. As the actual strain rate is much higher, the UTS is corrected using an increase of strain rate by a factor 100 (see section 2) using the m-values presented in table 1, so the UTS is corrected by 100m. The force-speed relations like the ones presented in figures 10 and 15 further showed an influence of sheet thickness: the curves are shifted to higher speeds for thicker material roughly proportional to the sheet thickness. The reason for this is not clear, but it complicates comparison of materials with different thickness. Therefore in the following the influence of thickness will be elimi-nated by correcting the speed as follows:
tV
V originalcorrected = (14)
where t is the material thickness in mm. d t All results for a depth setting of 2.3 mm are compiled into figure 16. For reasons of clarity
not all materials are shown, but only the average for the different material classes. The grey line presents the theoretical relation (9) where however the strain increment e is interpreted as the speed v using e = 1/6⋅vud/vcb, see section 2; the value for the parameter a is discussed be-low. The actual measurements deviate from this theoretical curve, notably in the lower part (low speed, low force) the results are affected by cyclic effects. Relation (9) predicts a square-root relation for low speeds; this is a straight line in a log-log graph. But the presence of cy-clic effects causes a certain non-zero force at zero speed, so a deviation from the theoretical curve is expected. This deviation apparently depends on the material class, mild steel and aluminium follow the theoretical curve much better than the other materials. In the upper part (large force, large speed) the theoretical curve seems to underestimate the forces: the meas-ured force seems to follow a square-root function to higher speeds than predicted. More im-portantly however is that in this part there is almost no difference between the different mate-rials, despite the considerable difference in mechanical properties.
13
0.1 0.5 1.0 5.0 10.0 20.0corrected pulling speed (mm/s)
0.1
0.2
0.4
0.6
0.81.0
rela
tive
max
imum
forc
emild steelaluminiumst. steelDP steelbrasstitanium
0.3 1 3 10 20
2.3 mm
Figure 16. Compilation of all results for 2.3 mm depth setting. Symbols indicates material
class; grey line shows theoretical relation (9). In the same way the results for 4.3 mm depth setting are presented in figure 17, left. Cyclic
effects are smaller than for 2.3 mm depth setting. In the upper part the deviation from the theoretical curve is larger: the measured data follow a square-root relation almost to the very end. The results for mild steel and aluminium show a peculiar behaviour. The measured curves have a slight S-shape, nevertheless in the lower part the slope equals the theoretical relation. The results for 0.3 mm depth setting are presented in figure 17, right. The results are more affected by cyclic effects than for large depth setting. Nevertheless the shape of the curves agrees reasonably well with the theoretical one, especially for mild steel and alumin-ium. The results for 4.3 mm depth setting show that the relation between force and speed can be approximated by a straight line in first approximation rather well, meaning that for larger depth settings a simple square-root relation as equation (10) may be a good approximation. In fact, the two relations shown in figure 9 can be regarded as limiting situations: the dashed line as a limit for large depth settings, and the solid line as a limit for low depth settings, the actual situation is somewhere in between.
0.1 0.5 1.0 5.0 10.0 40.0corrected pulling speed (mm/s)
0.10.10.1
0.2
0.4
0.6
0.81.0
rela
tive
max
imum
forc
e
mild steelaluminiumst. steelDP steelbrasstitanium
1 3 10 400.3
0.10.08
4.3 mm
0.1 0.5 1.0 5.0 10.0 20.0
corrected pulling speed (mm/s)
0.4
0.5
0.7
0.9
1.0
rela
tive
max
imum
forc
e
mild steelaluminiumst. steelDP steelbrasstitanium
0.3 1 3 10 20
0.3 mm
Figure 17. Compilation of all results for 4.3 mm depth setting (left), and for 0.3 mm depth setting (right). Symbols indicate material class; grey line shows theoretical relation (9).
14
Cyclic phenomena predict a possible non-zero force at zero speed. The force at zero speed has been derived for all test series. This has been done by comparing the measured curve with a theoretical one based on a square root relation and fitting the factor p according to relation (13). The results are shown in figure 18, except for material MS3 for which too little data were available for a reliable fit. This graph shows that all materials do show cyclic phenom-ena to some extent, otherwise the force at zero speed would be zero. Not all materials show the same effect, that was already concluded above. Obviously mild steel and aluminium are less sensitive to cyclic phenomena than the other materials. Another noteworthy observation is that the relation between force at zero sped and depth setting is quite similar for all materi-als.
0 1 2 3 4 5
depth setting (mm)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
rela
tive
forc
e at
zer
o sp
eed
BR
DP
AL
MS
SS
TI
Figure 18. Relative force at zero speed for all test series. Symbols denote material class. The results from figures 16 and 17 indicate that cyclic phenomena do raise the pulling force,
notably at situations of low speed. This is in agreement with simulations carried out by Hadoush et. al. (2011) that showed that kinematic material models, clearly incorporating cy-clic effects, do result in a higher force.
So far the geometry parameter a0 is unknown. This parameter can de estimated by fitting the theoretical relation (9) through measured data just as has been done by the grey lines in fig-ures 16 and 17. This has been done using the steel and aluminium results only, as these are least affected by cyclic behaviour. For this the actual contact angle ϕ is assumed to be propor-tional to the nominal contact angle θ by: ϕ = f⋅θ. where the factor f is taken constant for a cer-tain geometry. Consequently, assuming constant radius: a0 = 1 - cos(f⋅θ). The factor f has been determined by fitting and the results for 1 mm thick material are presented in table 3. The factor f is in the range 0.5-1 which is an acceptable value in all respects. Nevertheless, the results only serve as an indication. Note that for the situation in the tests described in this pa-per the difference between the geometrical parameters a and a0 is negligible.
The overall conclusion is that indeed the shape of the strip is determined by a balance be-tween pulling force and bending moment, but also that not all materials show the same level of cyclic effects.
15
Table 3. Results of data fitting for 1 mm thick material, see text. Depth setting (mm) 0.3 1.3 2.3 3.3 4.3 Nominal contact angle θ (degr) 2.7 6.2 9.8 13.7 17.7 Angle factor f 0.85 0.85 0.85 0.70 0.60 Parameters a0 0.00079 0.0042 0.011 0.014 0.017 6 Stability in the CBT test This section discusses the additional stability that allows large levels of uniform elongation,
and more in detail the possible loss of stability at situations of low speed. It has been mentioned before that in the CBT test material can be elongated (much) further
that in a conventional tensile test. Apparently the test is not limited by the instabilities that limit a conventional test, and the question is why. It is also clear that the well-known maxi-mum force condition does not apply, at least not in the conventional way. The force-displacement curves presented in figure 4 show frequent situations where the force decreases, yet this does no create an instability.
Based on elementary considerations it can be shown that when a piece of material is subjected to tension, the elon-gation is stable if and only if the pulling force increases with elongation. In this sense elongation should be read as: local strain increment (e in section 3), and not: total elon-gation, so: dF/de > 0. In other words: stability is based on the possibility of the force to increase with elongation. In a conventional tensile test this can only be due to material hardening. In the CBT test however there is another source. The force is mainly determined by the relative amount of material in a cross section that is actually in compression. If that amount can change, so will the pull-ing force. This implies that the stability in the CBT test is
based on the fact that the amount of material in compression decreases with increasing elon-gation, and stability will be lost when this property gets lost by any cause.
Figure 7 presented the pulling force as a function of the position of the neutral line in a situation of bending-under-tension. If the bending radius is kept constant this relation can also be presented as in figure 19. This graph shows that the pulling force is proportional to the elongation, so clearly satisfying the stability criterion stated above. Note that the process is only stable as long as e < eb; or: the neutral line lies still within the material, or: the fibres at the concave side are in compression. That implies that the increase in formability is restricted. But in the CBT test the material is repeatedly bent and unbent, and at each operation the in-cremental elongation will be kept stable if it is not too large. This explains the large level of possible uniform elongation.
However in the actual test the radius is not constant. So for a more realistic analysis it has to be taken into account that the actual bending radius depends on the actual pulling force, and the latter depends on the strain increment. This means that the bending radius and the strain increment are interrelated. A first expansion may be to use relation (9) that still shows that the force increases with elongation, of course as long as α < 1. A more reliable approach is to incorporate cyclic effects and other material phenomena as well but that makes the situation rather complex. Fortunately there is a short-cut. In all tests described in this paper the roll speed vrs has been kept constant. This means that the strain increment e is simply proportional to the testing speed vcb (see equation 2), therefore the condition dF/de > 0 can be replaced by dF/dvcb > 0. Experimental results presented in the previous part have shown that indeed this
eb e
F
σf.t
Figure 19. Pulling force as a
function of elongation for a situa-tion of constant bending radius
16
condition is met in all tests: the elongation is stable. Nevertheless, it will be clear that this cannot go on forever. Obviously the force cannot in-
crease unlimited, there is a maximum set by the material strength. This maximum will be reached at high pulling speeds (see for example figures 10 and 15-17), and then the stability will be lost and the test will change into a common tensile test. This will be discussed in more detail in section 8.
It may not be obvious that stability may also be lost at the other side of the range. This be-comes more clear when the condition is rewritten as: the process is stable as long as the force decreases with decreasing speed. When the force is zero at zero speed, for example in case of a proportional relation (F = C.vcb), there is no problem: there will always be stability. How-ever the situation changes when the force is not zero for zero speed. In that case a change in speed might cause only a negligible change in force, at least relative. Consequently the stabil-ity may be lost in cases of low speed. Such a situation can occur when the material shows cyclic effects as discussed in section 5, and some effects have been presented in figures 14 and 15. A non-zero force at zero speed occurs when the cyclic parameter p is non-zero. Figure 14 shows that for minor values of p the curve is shifted upwards with negligible effects on its slope. This means that for minor values of p, and a certain minimum speed, the force still in-creases significantly with speed (or: e) to create stability. Or in other words: loss of stability is only expected at large values of p. The value of p is in fact just the value of the relative force at zero speed, see eq. (13), and this value was presented in figure 18.
Figure 20. Severe case of non-uniform strain distribution prior to failure. Material: stainless
steel. When the process is fully stable the elongation will be uniform along the whole area ‘vis-
ited’ by the rolls, indicated by the rectangle in figure 2. An observation of non-uniform elon-gation therefore might indicate a situation of reduced or lost stability. An example of such a situation is presented in figure 20. This is a severe case observed with stainless steel, other materials showed less extreme cases, but only in situations of low depth setting and low speed. Observation showed that these 'necks' originate at the end of the test, immediately be-fore failure. Stage A can be considered as close to the uniform strain prior to the origination of the 'necks', and B as a sequel to that. A detailed analysis showed that the neck B is formed in a strain state that is different from that in the uniform part (A), and resembles more that of the original material. This supports the hypothesis that in the end stability is lost and the test continues more or less as a common tensile test, creating diffuse necks.
It may be helpful to define a parameter for the level of stability of the process (robustness). The process may indeed be stable when dF/dv > 0 (or: dF/de > 0), but in general, a process that is just stable will be more sensitive to external disturbing factors of any kind than a proc-ess that is very stable. The parameter dF/dv might seem a good candidate by itself, but this is based on absolute values and, for example, the value depends on the units used. A better can-didate is a dimensionless parameter based on relative changes dF/F / dv/v = d(logF)/d(logv), and this will be used here
Now there is a complication: the pulling force also depends on the strain rate, and that is af-fected by the pulling speed as well. A detailed analysis (not presented here) showed that:
17
mkedFd
vdFd
⋅+≈)(log)(log
)(log)(log (15)
where d(logF)/d(loge) is the correct stability parameter, m the conventional strain-rate hard-ening coefficient, and k a constant ranging from 1 for α << 1 to 2 for α ≈ 1.
The parameter d(logF)/d(logv) has been determined by fitting a straight line through the logF-logv data for low speeds (≤ 1 mm/s). Figure 21 presents the thus obtained value of d(logF)/d(logv) for all test series. The occurrence of severe non-uniformity as shown in figure 20 is indicted by a large symbol (all solid), this only occurred with stainless steel and tita-nium. For some materials a less severe situation of non-uniformity was observed, this is indi-cated by a medium-sized symbol. This was only noticed with DP-steel and the harder brass variants (BR2 and BR3).
0 1 2 3 4 5
depth setting (mm)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
d(lo
gF) /
d(lo
gV)
MS1
MS2
AL1
MS4
DP1
BR3
MS6
AL2
SS1
BR1BR2
TI1
DP1TI1BR2BR3
AL3
MS5
SS1
Figure 21. Value for the stability parameter for all test series. Numbers denote test series,
symbols denote material class. Symbol size refers to observed instability, see text. A first look at figure 21 indicates that indeed non-uniform elongation is related to low val-
ues of the stability parameter. A second look however shows that there is some inconsistency, there is no clear limiting value of the stability parameter below which there is instability, and above which the process is stable. The stability parameter d(logF)/d(logv) however is also affected by the strain-rate hardening of the material. To obtain the true stability parameter the values in figure 21 should be corrected for that. This means that the values for DP-steel and titanium are actually too high, as both materials show positive strain-rate hardening, and should be corrected. A correction according to equation (15) brings the results more compara-ble to the results for brass, although not in full agreement. Also, soft brass does show this in-stability despite a low value of the parameter. Nevertheless the conclusion is justified that indeed there is a relation between the stability of the process, and the way the force increases with increasing strain increment per cycle.
Instabilities of a more local character will be discussed in section 10. 7. Fracture types
18
This section describes the different fracture types, their background, and the occurrence for all materials, prior to a discussion about process limits and formability. Basically three types are distinguished: 1: fracture as in a conventional tensile test, 2: gradual fracture, and 3: sud-den fracture.
Type 1: Fracture as in a conventional tensile test. The fracture that occurs in a conventional tensile test is regarded to be well known. The fracture is at an oblique line relative to the pull-ing direction, the orientation of which depends on the material properties, notably the anisot-ropy. Fracture is initiated by a local neck that may be preceded by a diffuse neck depending on the material properties and other conditions.
This type of fracture also occurs in the CBT test in certain situations. A necessary condition for a fracture to be labelled as type 1 is an oblique angle of fracture. This is not a sufficient condition: sudden fractures of type 3 described following occasionally also are at an oblique angle. Type 1 fracture only occurs at high pulling forces, or better: high pulling speeds. Basi-cally this type occurs when the stability of the process is lost or has become very small, so that the CBT test starts to resemble a conventional tensile test, albeit with still higher levels of uniform elongation. This type is more abundant with materials that show high levels of strain-rate hardening like mild steel and titanium, which can be understood simply. Remember that only material that is being bent will actually deform. This means that the deformation is con-centrated into a number of short transverse bands, and consequently fracture will take place into such a band also. If a material shows strain-rate hardening, an onset of instability will not cause immediate failure but requires some further elongation to failure. In that time the roll set has moved to another position, and a diffuse neck can develop as in a conventional test. If however a material does not show strain rate hardening, an onset of instability will cause al-most immediate failure, and the fracture will be concentrated into the narrow transverse band of deformation. Examples of type 1 fracture are presented in figure 22. Please note that this picture and the following pictures are a composition of several photographs, and that the sam-ples are not necessarily shown to the same magnification.
Figure 22. Examples of type 1 fracture. From left to right: mild steel, aluminium, stainless
steel, titanium. Type 2: Gradual fracture. A very special fracture type is a fracture that appears gradually
and takes several passes of the roll set to complete and cause failure of the specimen. Mostly the crack starts at the centre of the specimen (in width) and grows outwards, but occasionally the crack starts at an edge, and grows towards the other edge. This type only occurs at low pulling speed because: 1. at low pulling speed the strain increment per passage of the roll set is (very) low, so that the strain increment in a single pass is simply too low to cause fracture, and: 2. at low pulling speeds the pulling force is low, so that a partially fractured cross section may still be strong enough to carry the load. The occurrence of fracture type 2 strongly de-
19
pends on the material type, as will be shown below. Examples of type 2 fracture are presented in figure 23, although the gradual development cannot be shown in a single photograph.
Figure 23. Extreme cases of type 2 fracture also showing multiple necking/fracture. From
top to bottom: stainless steel, mild steel, titanium. Although in many occasions just a single fracture occurred, multiple necking and/or fracture
occurred in extreme cases. This is a direct consequence of the type of test and should not be interpreted as a material defect. At any time the deformation is concentrated into a small zone, because in that zone the force required for deformation is lower than in the surrounding areas. If in that zone an instability occurs a neck or fracture will develop, but as the strain is only incremented by a small value, the defect may be too small to create actual failure of the speci-men. The next moment the zone of deformation has moved to an adjacent spot, that spot is now the weakest spot and the previous spot has become stronger again as there is no simulta-neous bending any longer. The result is that the defect will not grow further. As a conse-quence a second (or third, etc.) defect may develop at another location. Multiple necking has mostly been observed accompanying fracture types 2.
Type 3: Sudden Fracture. Type 3 can be regarded as the 'native' fracture type of the CBT test: the failure is sudden with a clear 'bang', the orientation is mostly at 90O relative to the strip length, and failure may be preceded by a small diffuse neck. In just a few cases and only with titanium and mild steel, the orientation was not at 90O but at an oblique angle. It will be clarified in the next section why these still have been labelled type 3 and not type 1. Examples of type 3 fracture are presented in figure 24.
Type 3 shows several variations. Figure 20 showed an example where fracture was preceded by two large diffuse necks due to reduced stability, however the fracture type is still 3. Other variations occur as well of which only two need special mentioning. With dual phase steel small cracks appeared at the outer edge that initiated fracture, see figure 25. These seemed just a case of low-cycle fatigue. With aluminium the fracture was preceded by a small trans-verse neck, this created a very straight crack edge as shown in figure 25 as well. Noteworthy, this was the case with all aluminium samples, but did not occur at any of the other materials.
20
Figure 24. Examples of type 3 fracture. From left to right: mild steel, stainless steel, dual-phase steel (2x), titanium.
Figure 25. Exceptional type 3 fractures. Top: very straight edge as seen with aluminium;
bottom: edge cracks in dual-phase steel A compilation of all experimental results showing the occurrence of the fracture types is
presented in figure 26. Note however that occasionally the difference between various types was far from clear, and some personal interpretation could not be avoided. This means for example that the transition from type 2 to type 3 can be a matter of discussion. The figure presents the normalized pulling force, defined as the measured maximum pulling force di-vided by the material UTS, as a function of pulling speed. In this way, for each depth setting a series of points is obtained that forms an S-shaped curve just as shown above in figures 10 and 15. The results will now be discussed briefly for each material class.
Mild steel: many results for mild steel have been obtained, that however present a very con-sistent image. Type 1 fracture seems dependent on the pulling force, it is occurring with for-ces above 75% of the UTS, with only a few exceptions. Type 2 fracture does not occur with the lowest depth setting, for all other depth settings its occurrence is limited by a speed of roughly 2 mm/s.
Dual Phase steel: the occurrence of type 1 fracture is similar to that of mild steel, both mate-rials having comparable strain-rate hardening. Striking is the occurrence of type 2 fracture, or more precisely: the lack of that. Only in two cases type 2 fracture has been observed, at low speed and the highest depth setting. Note that the series for the higher depth setting are not complete due to the loading limits of the set-up.
Stainless steel: the occurrence of type 1 fracture is similar to that of both low-C steel vari-ants, but with a slightly lower force limit of approx. 70% UTS. Type 2 fracture only occurred for depth settings of 2.3 mm and higher, and seems limited by a speed of 1 mm/s.
Titanium: the occurrence of type 1 fracture is similar to that of stainless steel. Titanium however shows a high level of strain-rate hardening and oblique fracture was not restricted to
21
the type 1; for example all specimens tested with 0.3 mm depth setting showed an oblique fracture, the reason why some are labelled type 3 instead of type 1 will be explained in section 8. Type 2 fracture seems more restricted by a maximum force of 40% UTS than by a speed limit.
Figure 26. Occurrence of fracture types for all material classes. Symbol type denotes depth setting, symbol colour denotes fracture type.
Aluminium: aluminium has no strain-rate hardening and consequently type 1 fracture is
more rare than for the other materials, there is no clear force limit between types 1 and 3.
22
Type 2 fracture occurs rarely, and only in situations of low speed and higher depth setting. Brass: the graph compiles results for all three grades that have been tested, but the grades do
not show exactly the same result. Brass also has no strain-rate hardening and consequently type 1 fracture is less abundant and without a clear force limit between types 1 and 3. Type 2 fracture is limited to depth settings of 2.3 mm and higher, but shows a material influence. For the soft grade (BR1) type 2 is limited by a speed of approx. 1 mm/s. The two harder grades (BR2 and BR3) show less type 2 fracture, and the occurrence seems limited by a pulling force of approx 30% UTS.
8 Fracture limits
0.1 0.5 1.0 5.0 10.0 20.0testing speed (mm/s)
0.0
0.5
1.0
1.5
leng
th s
train
at f
ract
ure
B A
1
2
3
0.3 1 3 10 20
0.0 0.5 1.0 1.5
true strain
0
250
500
750
true
stre
ss (M
Pa)
A
B
Figure 27. Length strain at fracture as a
function of testing speed, a typical example. Symbols denote fracture type as indicated by the small numbers 1, 2 and 3. Material: MS1
Figure 28. Collected stress-strain curves for the tests from figure 27. The kink at the end of the curves is caused by the smoothing pro-cedure and should be ignored. The arrow indicates the direction of increase of speed
This section investigates the process limits, and more specifically the mechanisms that do
limit this test. For a given depth setting the formability (length strain at fracture) depends on the testing
speed; a typical example of that relation is presented in figure 27. The graph shows two parts, a part where the elongation at fracture increases with increasing speed, and a part where it decreases with increasing speed; in this example the transition is at 5 mm/s. The two zones are labelled A and B. For all the experiments in this graph the recorded force-displacement curves have been converted into true-stress / true-strain curves, where the stress is the mean tensile stress over a cross-section, and the strain is the macroscopic length strain (not the cu-mulative bending strain) according to relation (1). These curves are collected in figure 28. In this diagram the zones A and B form two very distinct forming limits, indicated by the two dashed lines. This also suggests that the forming limits A and B are caused by different mechanisms. It is also clear that the maximum formability is reached at the cross section of both limits. These two limits will now be discussed separately.
Limit line A looks like the normal hardening curve of the material. More detailed investiga-tion confirms this, and the following observation will be used: the fracture strength of the ma-terial after testing depends only on the macroscopic length strain and follows the conventional hardening relation. This explains limit A. During the CBT test the pulling force cannot exceed the strength. The test is based on the condition that part of the cross-section is in compression meaning that the force is lower than the strength. If the neutral line would shift out of the ma-terial, the whole cross-section would be in tension making the test a conventional tensile test.
23
In that case fracture would occur immediately as the material is hardened considerably and the remaining hardening rate is very low. This is further illustrated in figure 29. That figure shows some force-displacement curves obtained at a condition where the fracture is expected to fall on limit line A. Test 4 was continued until fracture, but tests 1, 2 and 3 have been inter-rupted, meaning that at a certain moment he roll-set movement has been stopped so that con-sequently the test continued as a conventional tensile test. In that case the pulling force in-creases rapidly as the whole cross-section is loaded in tension, most notably seen in tests 1 and 2. After a short time fracture occurs, and as the test is now a conventional tensile test the pulling force at fracture is determined by the hardening of the material. This means that the envelope of the fracture points, as indicated by the dashed line, is nothing more than the hard-ening curve of the material converted to engineering stresses. The pulling force cannot exceed this limit, and an uninterrupted test will finally fail when the actual pulling force wants to ex-ceed this limit as can be seen in test 4.
0 20 40 60 80 100 120 140 160elongation (mm)
0
1
2
3
4
5
6
pullin
g fo
rce
(kN
)
12
31 2 4
3
Figure 29. Force-displacement curves of some interrupted tests (MS2).
0.0 0.2 0.4 0.6 0.8 1.0true strain
0
200
400
600
800
1000
true
stre
ss (M
Pa)
2.3
1.3
0.3
A
B
Figure 30. Limit stress and strain for some test series (dual-phase steel). Symbols denote
depth settings in mm. Compare to figure 28. Black symbols denote type 1 fracture. This leads to the following mechanism. During a test the specimen thins considerably. This
gradually moves the neutral line to the surface of the strip. When the neutral line shifts out of the material stability is lost, and the specimen fractures. In practice, stability will already be lost when the neutral line is close to the surface. Curve 4 shows that the specimen actually failed at the force peak that occurs during roll-set movement change. This is typical, in many
24
occasions the specimen failed at a force peak or shortly after one when a new strain increment is imposed. This mechanism predicts that there is no infinite formability. Sooner or later the specimen will always fail by this mechanism, unless of course if has already failed by another mechanism. Another consequence is that limit line A is a material limit independent of ex-perimental conditions. Basically limit A is caused by the loss of the stabilizing effect of the moving rolls. A representative example can be found in figure 30 showing the results of three test series with different depth settings, for a single material. Indeed the limit lines A of all test series coincide. Figure 30 also shows the occurrence of type 1 fracture defined as ' frac-ture as in a conventional tensile test ', and from this a second condition for type 1 has become clear: it must fall on limit line A. Oblique fractures not on limit line A are labelled as type 3, although these have been observed only occasionally with titanium and mild steel.
Limit line B is harder to analyze because it shows a clear influence of both experimental
conditions (see figure 30) and of material type, but it cannot be related to some simple me-chanical properties. The following observations are valid in general:
- for situations of constant depth setting the max. elongation increases with increasing speed;
- for situations of constant speed the max. elongation increases with decreasing depth set-ting.
These relations can be noticed clearly in figure 31 showing results for titanium as an exam-ple. Both observations point to the same: increasing speed means that the same elongation is reached with a lower amount of bending cycles, and a decreasing depth settings results in a larger bending radius, meaning less bending strain. So both hint that limit B is related to the repetitive bending of the material: less bending results in more elongation. This by itself does not explain the failure mechanism. However it can be anticipated that materials that show a high sensitivity to low-cycle fatigue, show poor performance in this test as there is a severe amount of repetitive bending. The reverse is not true, there may be other mechanisms. The fact that the occurrence of type 2 fracture strongly depends on the material type at least sug-gests that there are several mechanism indeed. Therefore a detailed microscopic examination of fractured specimens of both mild steel and aluminium has been carried out. This did reveal signs of fatigue, although the results were not fully conclusive.
0.1 0.5 1.0 5.0 10.0 20.0testing speed (mm/s)
0
40
80
120
160
elon
gatio
n at
frac
ture
(mm
)
0.3
1.3
2.3
3.3
4.3
0.2 21 5 10 20
Figure 31. Max. elongation as a function of speed (titanium). Symbols denote depth settings
in mm. Gray symbols denote type 2 or 3 fracture, indicating limit B. The actual fracture in limit B was at an oblique angle with the specimen surface. In many
samples this angle abruptly changed from positive to negative, creating a crenellated appear-
25
ance, as schematically shown in figure 32. This was most clearly the case with aluminium, but other materials showed this appearance as well.
α Figure 32. Left: schematic representation of a crenellated fracture. Right: three examples of
actual fractures, from left to right: aluminium, stainless steel, dual-phase steel. 9 Formability This section discusses the formability in this test defined as the maximum observed uniform
length strain. The main goal of this research was to determine the maximum level of uniform elongation
that can be obtained with the CBT test. Figure 28 indicates that the maximum strain is ob-tained at the intersection of limits A and B. Figure 30 shows that the maximum strain depends on experimental settings, and that presumably limit B has a stronger effect on the formability than limit A.
0 1 2 3 4 5depth setting (mm)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
max
. uni
form
leng
th s
train
MS1
MS2
AL1,2
MS4
SS1
DP1
BR2
BR1
TI1
MS3
BR3
MS6
AL3
MS5
Figure 33. Max. uniform length strain as a function of depth setting, for all materials. Sym-
bol type denotes material class, numbers denote material. For all test series, carried out at a constant depth setting, the largest measured uniform strain
has been determined for any speed. Figure 31 shows that for lower depth settings this maxi-mum is obtained at lower speeds than for higher depth settings. The results are presented in figure 33 for all materials. Realise that the maximum value is always based on a single obser-vation and as such subject to scatter; for materials that show large scatter there is more possi-bility that an accidental high value occurs than for materials that show low scatter. Within a certain band of scatter all materials show the same effect, there is a maximum in the curves.
26
The relation shown in figure 30 that the formability increases with increasing depth setting does not continue all the way. This means it has no use to go to extreme setting to obtain ex-tended formability.
Apparently mild steel and stainless steel show the highest formability. Below that are yel-low brass and titanium, and dual phase steel a little lower. Aluminium shows low to poor formability (at least the grades tested here!). The formability of leaded brass is also poor; maybe this should not be surprising, as the lead precipitates as small globules, deteriorating fatigue properties. The results for aluminium are disturbing. Materials AL2 and AL3 are both rated as 6016-T4 and their mechanical properties are comparable. By traditional standards the formability of material AL3 is excellent, yet in the CBT test it is very low. In fact. material AL3 is the only material that has been tested where in the CBT test the formability was occa-sionally lower than in a conventional tensile test, in both tests defined as the maximum uni-form strain. For all other materials the formability encountered in the CBT test was always higher than in a conventional test, despite the sometimes extreme amount of cyclic bending causing considerable heating of the specimen.
0.1 0.5 1.0 5.0 10.0 20.0testing speed (mm/s)
0.0
0.5
1.0
1.5
unifo
rm le
ngth
stra
in
prestrained-correctedoriginalprestrained
0.2 21 5 10 20
Figure 34. Effect of pre-strain on formability. Symbol type denotes material. Material MS6 is a pre-strained version of material MS1. To achieve this, samples were
strained in a conventional tensile test to a uniform length strain of 0.18. The samples where then re-cut to length and treated as a new material, the new width and thickness taken as new dimensions. The formability of material MS6 is compared to the original material MS1 in figure 34. It is clear that the process of pre-straining has reduced the formability of the mate-rial. The figure also presents the limit strain for material 6 incremented by the amount of pre-strain (0.18). This shows that on average the decrease of formability is somewhat lower than the amount of pre-strain, but otherwise comparable in size. This reduction of formability may be intuitively expected, but it is not just a simple as that. The right-hand part of the graph pre-sents limit A. It has been shown above that this limit is governed by the material's hardening behaviour, and indeed this is not changed by the pre-strain that is of tensile nature. The left-hand part of the graph presents limit B, and it has been shown that this limit is governed by fatigue or similar phenomena. It is however not clear at all beforehand how this is affected by any pre deformations. Figure 34 presents only a single test series, so one should be careful and not draw too strong conclusions from that, more research is needed.
The highest level of formability was obtained with material 2 and a depth setting of 2.3 mm. Detailed investigation showed that this is based on a very few tests that showed exceptionally high strains. Therefore it may be regarded as accidental, although it is not clear why acciden-tally very high levels of formability can occur. In the first paper it was concluded from com-
27
paring materials MS1 and MS2 that thickness affects formability: thicker is better. This how-ever is not confirmed by the results of material MS5, therefore the overall results presented in figure 33 do not allow a general conclusion about the effect of thickness on formability in the CBT test.
10 Other effects.
Figure 35. Examples of herringbone type roughening, all samples shown to the same scale,
the ruler is in cm. From top to bottom: AA5182, 1.15 mm (AL1); brass half-hard, 0.7 mm (BR2); dual-phase steel zinc coated, 1.0 mm (DP1); AA6016 1.0 mm (AL2).
This section finally present two typical effects of the CBT test: a peculiar roughening and
local instability. Some of the samples showed a peculiar type of roughening after testing resembling herring-
bone. This type of roughening has also been noticed by Benedyk et al. (1971) in their tests on aluminium and so apparently is not limited to a single test set-up. In the tests described in this report this roughening occurred with all three aluminium variants, dual-phase steel, and half-hard and hard brass, but NOT with soft brass or any other material. Representative samples are shown in figure 35.
This roughening was a side effect and has not been investigated thoroughly, yet the follow-ing observations are made:
- in a single test the level of roughening increases gradually with extension; the spacing however remains constant despite the occasionally considerable lengthening of the specimen;
- in tests with the same depth setting the level of roughening after fracture decreases with increasing speed, despite the fact that the total elongation of the fractured specimens in-creases; in tests with high speeds the herringbone effect did not appear at all.
- the herring-bone type of roughening is not present on the very side edges of the specimen;
28
- the spacing is wider for thicker materials. The effect seems less severe in specimens tested with a deep setting, but this may also be
caused by the lower general elongation of the specimens. The effect is remarkable. The observation that the effect decreases with increasing speed in-
dicates that the effect is not simply related to the elongation of the specimen. Tests at low speed have encountered much more bending/unbending operations, and hence more total cu-mulative straining of the outer fibres. Needless to say that it does not occur in tensile tests without bending, so it is clearly related to the bending operations. The variety of materials that show this roughening makes it unlikely that it is caused by a particular material property like serrated yielding. It seems more a geometrical instability caused by the cyclic bending and unbending but this remains speculative, notably in view of the material dependency. But if this is true, it should appear in simulations as well if a sufficiently fine mesh is used.
Figure 35 also shows another feature. The samples show a series of transverse bands, some more than other. This is a typical phenomenon that occurs with all materials at the end of the test to some degree. The effects are exaggerated in the figure as a result of skew lighting and contrast enhancement. These bands are not necks, they originate just as a wavy shape of the specimens, although if the test continues a neck will often initiate at one of these bands, see for example the lower specimen at 38 mm location. This also suggests some kind of geomet-rical instability. These multiple transverse bands can be the origin for multiple fracture. This band-like structure also shows up in the simulations carried out by Alan Leacock of the Uni-versity of Ulster, Belfast. The resolution is too low to show the wavy shape of the specimens, but the thinning at the very end of the operation does show up in discrete bands. These can be seen in figure 36; the greyscale is not consistent, but the darker grey areas in the narrowed part of the strip shows regions of lower thickness. Note also the difference between the sam-ple edges and the centre part that is also clearly noticed in figure 35.
Figure 36. Simulation of the CBT test showing band like thinning (see arrow). The grey
scale indicates thickness. Courtesy Alan Leacock. 11 Concluding discussion The very essentials of the CBT test can be characterized effectively as: at any moment only
the material actually being bent will deform. This localizes the deformation into a number of small transverse zones. These zones of local deformation (transverse bands) also do show up in the simulations carried out by Alan Leacock, see the light-grey areas in figure 37. In fact this also makes that test an incremental forming operation, and this was the main reason for the authors to start using this test. A departure from this rule only occurs when the pulling force becomes very high, for example when testing at high speeds. In that case other parts of the strip start to deform as well, notably parts of the tapered section that can be seen in figure 2 just to the right of the uniform section.
This essential characteristic determines to a large extent the observations presented in this paper. Based on this the mechanics have been determined and presented in sections 3-5, and these were successfully verified by experimental observations. A striking observation is that all materials that have been tested behave in a very similar way. The only exception is the occurrence of cyclic effects that are clearly material dependent. In this respect the materials
29
can be divided into two groups of which one consists of mild steel and aluminium. In many graphs these two material classes distinguish themselves from the other materials, showing a lower sensitivity to cyclic effects. It is therefore striking that the performance of these two materials in terms of formability is totally different.
Figure 37. Simulation of the CBT test showing zones of local deformation. The grey scale
indicates stress (major stress, membrane), lighter = more; the roll set is moving towards the right. Courtesy Alan Leacock.
The fracture types presented in section 7 also confirm the assumption that deformation is
concentrated into small transverse zones.. The most obvious reason is the observation that the fracture is at 90O orientation, and not at an oblique angle. Deviations only occur at high levels of the pulling force, and then mainly for materials that show a high level of strain-rate harden-ing. If the deformation is concentrated into a narrow transverse bend, so will be the fracture.
The large variety of materials used in the tests allows a better founded conclusion about the effect of material. It has already been mentioned that the behaviour of the materials as ex-pressed by the relation between relative pulling force and pulling speed is very similar show-ing little effect of material type. The results concerning stability as presented in section 6 are identical. There is only a minor effect of material. Materials that do show cyclic effects (dual-phase steel, stainless steel, brass, titanium) also show reduced stability that however only ex-presses itself in tests at low depth settings (figure 21); this is not surprising as the stability is also indirectly based on the relation between force and speed.
Contrary to this, the fracture behaviour shows strong influence of material type. Both the type of fracture, notably the occurrence of type 2, and the formability show strong material effects. So far these effects cannot simply be related to known material parameters, other than that in general materials that are prone to low-cycle fatigue are expected to show poor form-ability in the CBT test. Figure 38 shows the relation between the formability in a tensile test and in the CBT test, defined as the maximum observed uniform strain in any test. This illus-trates what is already clear: there is no relation between these two. Notably, the mild steel and aluminium variants (except for MS6) are all commercial automotive grades that show excel-lent formability in stamping. Yet the formability in the CBT test differs dramatically.
More effects of mechanical properties have been observed, that however are not understood. The two yellow brass variants (BR1 and BR2) have the same chemical composition. One is harder, but the absolute level of hardening dσ/dε is the same for both materials. Yet the softer variant does not show the instabilities described in section 6, does not show herringbone type roughening described in section 10, and shows significantly more type 2 failure than the harder variant, and it has a slightly better overall formability. Also, section 5 mentioned an effect of thickness that forced a speed correction. This effect is consistently present in all re-sults, including that of related experiments not presented in this paper, but not understood.
The overall conclusion is that, except for the fracture behaviour, all materials behave in a very similar way despite the large difference in mechanical properties. Apparently the CBT test introduces a stabilization, fairly well understood and largely independent of material properties. To conclude, the overall material behaviour is compiled into table 4, where stabil-ity is defined as: insensitive to cyclic effects.
30
Table 4. Overall performance of all material types. Stability = insensitive to cyclic effects.
FORMABILITY low medium high
good aluminium mild steel
STABILITY reduced leaded brass
dual phase steel titanium
yellow brass stainless steel
0.0 0.1 0.2 0.3 0.4 0.5 0.6tensile test
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
CB
T te
st
MS1,5
MS2
AL1,2
SS1
DP1
BR2BR1TI1
MS3,4
BR3
MS6
AL3
Figure 38. Maximum observed uniform strain in both a tensile test (table 1) and the CBT
test (figure 33) for all materials. The dashed line shows the 1:1 relation. 12 Acknowledgement The authors wish to thank Dr. Alan Leacock of the University of Ulster, Belfast, for per-
forming FEM simulation and giving permission to present the results in figures 36 and 37. References Allwood, J.M., Shouler, D.R., 2009. Generalised forming limit diagrams showing increased
forming limits with non-planar stress states. Int. J. Plasticity, 25 (2009) 1207-1230. Allwood, J.M., Shouler, D.R., 2011. Design, Analysis and Application of a Novel Test for
Sheet Metal Forming Limits under Non-Planar Stress States. Proceedings 14th Int. Esaform Conf. on Material Forming, AIP Conf. Proc. 1353 (2011), 1595-1600
Benedyk, J.C., Stawarz, D., Parikh, N M., 1971. A Method for Increasing Elongation Val-ues for Ferrous and Nonferrous Sheet Metals. J. of Materials, Vol. 6 (1971), nr 1, pp 16-29.
Emmens, W.C., van den Boogaard, A.H., 2009a. An overview of stabilizing mechanisms in incremental sheet forming. J. Mat. Proc. Technology, 209 (2009) 3688-3695.
Emmens, W.C., van den Boogaard, A.H., 2009b. Incremental forming by continuous bend-ing under tension - An experimental investigation. J. Mat. Proc. Technology, 209 (2009) 5456-5463.
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Eyckens, P., van Bael, A, van Houtte, P., 2009. Marciniak_Kuczynski type modelling of the effect of through-thickness shear on the forming limits of sheet metal. Int. J. Plasticity, 25 (2009) 2249-2268.
Hadoush, A., van den Boogaard, A.H., Emmens, W.C, 2011. Incremental forming by con-tinuous bending under tension - a numerical investigation. Submitted to J. Mat. Proc. Tech-nology, 2011.
Mols, A.J.A.M., 1972. Een mechanische techniek om het optreden van plastische instabili-teit te vertragen. Graduate thesis, Delft Technical University, June 1972 (in Dutch)
Schleich, R., Held, C., Sindel, M., Liewald, M., 2009. Investigation on the effect of curva-ture and sheet thickness on forming limit prediction for aluminium sheet metal alloys. Int. J. Mat. Forming, 2009, Volume 2, Supplement 1, Pages 411-414.
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