Investigation of dependences of metal sheet variable blank holding force/ Kintamos metalo laksto prispaudimo jegos charakterizavimas ir vystymas.
Bortkevicius, R. ; Dundulis, R. ; Karpavicius, R. 等
1. Introduction
The purpose of this investigation is to characterize and latter
make a full development of variable metal sheet blank holding force
(VBHF). Permanent blank holding force has been used so far (Fig. 1). The
present paper is a continuation of authors' papers. Until present
day optimal dependence of minimal blank holding force on punch travel in
order to avoid the wrinkles has not been identified, even in theory
[1-3]. All theoretical formulations of blank holding force (BHF)
describe only the maximum of BHF, which does not describe nonlinearity
in metal forming operation. From literature [4] we know the holding
maximum pressure to the sheet should be around 1-3 MPa. Several
analytical equations have been invented or derived in order to
characterize BHF. For example scientists [1, 2, 5] describes the
following equation for needed BHF
[F.sub.BHF] = [bF.sub.DR] sin([pi]/2 h/[h.sub.max]) (1)
The given equation describes BHF as the dependence on punch
traveling distance h, mm. The same authors change travel h into time t,
s and get BHF as the dependence on punch traveling time.
[F.sub.BHF] = [bF.sub.DR] sin([pi]/2 t/[t.sub.max]) (2)
where b is empiric coefficient; [F.sub.DR] is drawing force,
[h.sub.max] and [t.sub.max] are maximal piece depth and suitable process
duration [1]. Another two authors [4, 6] give the same equation for
constant blank holding force (CBHF), N
[F.sub.BHF] = [pi]/4([D.sup.2] - (d + 2[gr.sup.2]))p (3)
Or simplified
[F.sub.BHF] = [pi]/4([D.sup.2] - [d.sup.2])p (4)
where D is outer diameter of the workpiece, d is diameter of formed
part, r is corner radius, p is pressure. The comparisons of VBHF to CBHF
will be provided. By our opinion [F.sub.BHF] falls below zero line or
the equation gives linear dependency of BHF vs. time, and that is not
acceptable with the results provided in literature [7], where relative
VBHF shapes are investigated. One of the reasons why BHF should be
controlled is described in the articles [2, 5, 8], were clearly exists a
description. This paper formulates nonlinear dependency of BHF versus
time or punch traveling distance. Pulsating blank holding force [9] does
fall into our view and stays behind the scope of our investigations
though its shape can be also described as nonlinear variable blank
holding force. The experimental results will present data extracted from
finite element analysis (FEMA) and analytically formulated VBHF.
Validation of the given results can be found in the paper [5].
Mathematical formulation [10] of variable blank holding force (latter
VBHF) presented here represent nonlinear relation of holder force versus
punch displacement (or time).
[FIGURE 1 OMITTED]
[F.sub.vbhf] = q + b [[Lt.sup.2.sub.0] [T.sub.tension]/W] sin
[alpha] (5)
where q, b are empirical coefficients, depending on chosen
material, L is the length of forming material [11], [t.sub.0] is
thickness of the sheet, [T.sub.tension] is tension of the material to
the principle direction, W is die opening clearance, a is angle of
change of vertical part of material, it represents the shape function of
the blank holding curve. The most important factor in the equation is
its coefficients. We determined these coefficients by choosing to
examine the punch force needed to withdraw the correct stamped part. The
examination of this force for different material mechanical properties
is given in the article [12]. The results expected here we hope will
allow to employ given equation into industrial manufacturing of metal
sheet part. Since we investigated 3 most common material models, we
believe that given equation meets the highest standards for requirement
of choosing blank holding force. Our analysis is based on numerical
formulation and calculation. Since we equalize mechanical properties
given from Finite element modeling analysis (FEMA) with those from real
researches in nature [2, 3, 13], we state, that results from FEMA are
correct and correlate with real researches. Therefore FEMA analysis
results can be treated as reliable. Development of VBHF theory consists
of a plenty of experiments made to determine metal sheet stiffness,
thickness, spring back angle and so forth. Also big attention must be
paid to gain correct stress strain properties of the investigated
materials. We build up our model reliance on strength proportionality
constant or simply strength coefficient K [12, 13] and coefficient n
from stress strain curves in order to get a curve slope (angle). The
place, where the change of phase should begin or end up is determined
from metal sheets stiffness curves. The first and the last points of
VBHF are determined empirically. The calculated strength coefficient K
[12, 13] and investigated materials are given in the Table 1 [6].
Another important factor in equation is f [sin [alpha]] which describes
shape function of the deformed part. This shape function is very
important, because it shows the separation of drawing part and bending
part and describes when it begins and ends.
Initial blank holding force was calculated with Eqs. (6), (7). This
equation gives the first standpoint of initial BHF. Since this equation
gives needed force [F.sub.L63] to withdraw material to a needed shape,
BHF must be higher.
[F.sub.bhf L63] = [Lt.sup.2.sub.0]([[sigma].sub.ut]/W] (6)
[F.sub.bhf L63] > [Lt.sup.2.sub.0]([[sigma].sub.ut]/W] (7)
where [F.sup.bhf L63] is BHF for investigated materials, L is the
length of supporting place, [t.sub.0] is initial thickness,
[[sigma].sub.ut] is ultimate tensile strength [14], W is the gap between
the punch and matrix. For the investigated materials initial BHF is
[F.sup.aw6082] [congruent to] 10350 N, [F.sub.L63] [congruent to]
13216.5 N, [F.sub.AISI304] [congruent to] 30015 N.
2. Methods of investigation
Testing was conducting using two common models: analytical
equations and Finite element analysis (FEMA) described in the literature
[7, 5, 12].
The overall description of the investigated model including
boundary conditions, mechanical properties of investigated materials can
be found in the [12]. Schematically the model is shown in Fig. 2. Here
computational model is presented with three punch displacement steps.
During deformation of metal sheet deformation was measured, since we
need the highest value of plastic strains. Measurements were taken from
the top surfaces of the deformed part. Since our model was divided into
many of equal rectangulars, after the deformation these rectangulars
were changed into rhomboids. The dimensions of initial grid were 0.5 and
0.1 mm space between grids [9, 15]. If the deformation runs on general
plane stress sheet conditions are measured by coefficient [beta] [13],
which is defined as
[beta] = [[epsilon].sub.2]/[[epsilon].sub.1] =
ln([d.sub.2]/[d.sub.0])/ln([d.sub.1]/[d.sub.0]) (8)
where [d.sub.0] is the undeformed state with circle and square
grids marked on an element of the sheet, [d.sub.1] is the deformed state
with the grid squares deformed to rhomboid of major diameter [d.sub.1],
[d.sub.2] is the deformed state with the grid squares deformed to
rhomboids of minor diameter [d.sub.2] [3, 13].
[FIGURE 2 OMITTED]
3. Geometrical variable
Since we investigate geometrical variables in metal sheet forming,
serious consideration must be taken into account while measuring punch
displacement in accordance to the angle change of vertical part place.
By the investigation described here--angle to displacement ratio is
almost linear only until the 10th step, i.e. 10 mm downward, latter
ratio R changes keeping linear correlation to angle and until punch
reaches its maximum downward point (14 mm). Correlation of the described
angle versus displacement can be seen in Fig. 3. In the same figure
there are clearly divided linear and nonlinear phases of the change of
angle versus displacement. It is clear that linear to nonlinear phase of
ratio change starts at 10 mm of the punch displacement. On another hand
it can happen than bending of the metal sheet plate is finished and its
deep drawing begins. So we must emphasize that we are developing VBHF
for the combination of bending and deep drawing operations of metal
sheet forming. This can also be proven by investigating material [11]
stiffness curve of the same experimental model. In the Fig. 4 clearly
can be seen that when material experiences a bending and when its phase
changes from bending to deep drawing--the same is the punch displacement
10 mm. the described figure is metal sheet stiffness curve for the
chosen material stainless steel grade AISI 304. The mentioned stiffness
curves were gained using constant metal sheet holding force. The
familiar metal sheet stiffness curve can bee seen in [12]. The previous
figure brings us proof that the change of angle has non linear
dependence on displacement, and it can be considered straightforward in
the investigation of VBHF. Serious consideration must be taken in the
interpretation of aspect ratio R
R = [alpha]/h (9)
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
Latter we develop an equation of VBHF for the chosen analysis
model. This equation has to be created from two parts. One part shall
describe linear or bending processes (since we know that until 10 mm of
punch displacement only bending deformations) occur and the second part
of equation shall describe deep drawing processes (since we know that
after 10 mm of punch displacement deep drawing deformations occur).
Linear and non linear parts of bending and deep drawing are universal
and they do not depend on material properties and fit to all kinds of
metal sheet parts. The full development of the proposed variable blank
holding force is defined below
[F.sub.vbhf] = q + b [[Lt.sup.2.sub.0][T.sub.tension]/W]sin [alpha]
(10)
Our model should be deformed according to equation
[[sigma].sub.1] = K [[epsilon].sup.n] (11)
[T.sub.tension] = [sigma]t (12)
[T.sub.tension] = [[sigma].sub.1][t.sub.0] exp(-[mu][theta]) (13)
[mu] = [[mu].sub.D] + ([[mu].sub.S] - [[mu].sub.D])exp(-[??]) (14)
[??] = v/h (15)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
where q, b are empirical coefficients, depending on chosen
material, L is the length of forming material, [t.sub.0] = f(h) is
thickness of the sheet as a function of displacement, [T.sub.tension] is
ultimate tensile tension of the material, W is die opening clearance,
[alpha] is the angle of change of vertical part of material. [sigma] is
tension properties of material, [mu] is friction coefficient,
[[mu].sub.s] is static friction coefficient, [[mu].sub.d] is dynamic
friction coefficient, [??] is the speed of deformation. The graphical
representation of Eq. (16) can be seen in the Fig. 6 and Fig. 7 below.
In the figures four curves for the investigated materials (AISI 304,
L63, and AW6082-T6) can be seen. The mentioned curves represent VBHF
versus punch displacement. In the beginning of deformation of metal
sheet the blank holding force begins to acquire the value from zero
until it reaches the highest possible value and suppresses the sheet.
Later on VBH force acquires much lower values it follows thence that the
stiffness of the curve also reduces its value accordingly.
The angle of the sheet between the punch changes its vertical
position while the punch moving downward by the values written in the
Table 2. The main idea of VBHF is to reduce maximum material stiffness
in order to avoid excessiveness of strength of material fatigue. The
part, while being deformed, must always slide over the punch surface and
must if possible avoid tension deformation.
4. Investigation results
According to Eq. (1) and (2) the calculated variable blank holding
force is plotted in Fig. 5, where VBHF for investigated material AISI
304 can be seen. Needless to say that the mentioned VBHF does not fall
under the expected VBHF shape. Expected or wanted shape should be within
the range of 31500 N and 8800 N with some tolerance of course. In the
beginning the onset of the shape can be few times higher than the one
which acts while the sheet is being deformed. In Fig. 5 VBHF shape at
the beginning rises very high up to 100 t--and that is not correct,
because this value can seriously damage the forming sheet. Deformation
of the metal sheet ends without any inclination of the curve in order to
depress suppressing force--and that is also incorrect, because this
value can influence significantly spring back angle and final shape of
the part. Metal sheet stiffness (for investigated materials AISI 304,
L63, AW6082-T6) calculated with VBHF (calculated with (16)) can be
plotted in Fig. 6 and Fig. 7. There can be seen four curves for the
investigated materials. The final plastic strain value does not depend
on initial metal sheet place on the die, i.e. the metal sheet can be
pushed in any direction and that does not have any influence on plastic
strain state. Plastic strain begins besides bending radius though
plastic strain originated at the one of the radii does not have any
influence on plastic strains originated at the other bending radius
while the part is being deformed. Plastic strain value of course grows
while the part is being deformed and moves towards that direction in
which is the place where stress grows. In the Fig. 8 below can be seen
the relation of coefficient b with to punch displacement for material AW
6082-T6. As coefficient b changes (grows) its phase from value 0.1 to
0.5 the growth of curve steepness can be seen clearly in the place where
the part is being bend (from 0 to 10 mm). Then the part is being
stretched over the punch surface (from 10 to 14 mm) the coefficient b do
not really have any influence on material deformation properties. In
Fig. 9 below can be seen the relation of coefficient q on punch
displacement for material AW 6082-T6. The value was changed from 1000 to
5000 in order to investigate how the curve changes its shape. The result
reveals that the change of coefficient value has linear correlation to
variable blank holding force. If we change coefficient q by the value of
1000 the whole curve changes its position by the same amount on ordinate
(VBHF) axis. In Fig. 10 two forming limit diagrams for material AISI 304
can be seen. Fig. 10, a is for metal sheet formed with constant holding
force and Fig. 10, b is for metal sheet formed with variable holding
force. As mentioned, FLD consist of two axis of deformation: on the axis
of deferred minor strain is given and on the axis of ordinate is
deferred major strain is given. The ratio of these strains is formulated
as coefficient [beta] [1, 5, 7, 16].
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
[FIGURE 9 OMITTED]
[FIGURE 10 OMITTED]
5. Significance of coefficients q and b
The main meaning of the coefficients q and b is to correct or to
control certain value of Variable blank holding force, in order to
minimize principal stress [[sigma].sub.1] and plastic strain
[[epsilon].sub.pl]. In the figures below empirical experimental analysis
for investigated materials: AW 6082-T6, L63, and AISI 304 is provided.
Fig. 11Fig. 16 represent the significance of coefficients q and b for
the investigated materials. In the Fig. 11,Fig. 13 and Fig. 15 we have
changed coefficient b in respect to q, when q is in constant. Similarly
in the Fig. 12, Fig. 14 and Fig. 16 we changed coefficient q in respect
to b, when b is in constant. On the first attempt the results revealed
that increasing value of b and leaving the same q gives linear relation
of stress-strain state, so we resume that increasing coefficient b from
0.1 to 0.5 increases stress strain state value from [[[epsilon].sub.pl]
= 0.355, [[sigma].sub.1] = 188.4] to [[[epsilon].sub.pl] = 0.462,
[[sigma].sub.1] = 603] if [q = 1000] for material AW6082-T6. If
increasing value q and leaving coefficient b constant we are get several
states of stress-strain values.
[FIGURE 11 OMITTED]
[FIGURE 12 OMITTED]
[FIGURE 13 OMITTED]
[FIGURE 14 OMITTED]
[FIGURE 15 OMITTED]
[FIGURE 16 OMITTED]
6. Conclusions
In the paper we introduced variable blank holding force which can
be significantly decreased final values of strains, there fore following
conclusion can be derived:
1. An equation describing variable blank holding force [F.sub.VBHF]
has been proposed.
2. It's been determined, that the change of the angle [alpha]
relative to unchanged position of the matrix vary by the linear function
before the 10th step of the punch displacement h.
3. After the 10th step of the punch displacement h the change of
the angle [alpha] changes its linear dependence to nonlinear.
4. Angle [alpha] nonlinearity has straight forward influence to
proposed variable blank holding force function.
5. Initial coefficients q and b values for investigated materials
were proposed.
6. Coefficient b has an influence on the material deformation
properties only while the part is being bend. It does not have any
influence on the material deformation properties while the part is being
stretched.
7. The change of coefficient q value has linear correlation to
variable blank holding force.
8. Increasing coefficient b from 0.1 to 0.5, when q is 1000,
increases stress strain state value from [[[epsilon].sub.pl] = 0.355,
[[sigma].sub.1] = 188.4] to [[[epsilon].sub.pl] = 0.462, [[sigma].sub.1]
= 603] for material AW6082-T6, [[[epsilon].sub.pl] = 0.367,
[[sigma].sub.1] = 1488.4] to [[[epsilon].sub.pl] = 0.469,
[[sigma].sub.1] = 2364] for material L63, [[[epsilon].sub.pl] = 0.4,
[[sigma].sub.1] = 712] to [[[epsilon].sub.pl] = 0.565, [[sigma].sub.1] =
2609] for material AISI 304.
9. Increasing coefficient q from 1000 to 5000, when b is 0.1,
increases stress strain state value from [[[epsilon].sup.pl] = 0.355,
[[sigma].sub.1] = 188.4] to [[[epsilon].sub.pl] = 0.402, [[sigma].sub.1]
= 790] for material AW6082-T6, [[[epsilon].sub.pl] = 0.367,
[[sigma].sub.1] = 1488.9] to [[[epsilon].sub.pl] = 0.379,
[[sigma].sub.1] = 1600] for material L63, [[[epsilon].sub.pl] = 0.4,
[[sigma].sub.1] = 712] to [[[epsilon].sup.pl] = 0.42, [[sigma].sub.1] =
1279] for material AISI 304.
10. Values of coefficients b and q give a basic knowledge of how
the strain can be controlled.
Received December 14, 2010
Accepted May 27, 2011
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R. Bortkevicius, Kaunas University of Technology, Kestucio 27,
44312 Kaunas, Lithuania, E-mail:
[email protected]
R. Dundulis, Kaunas University of Technology, Kestucio 27, 44312
Kaunas, Lithuania, E-mail:
[email protected]
R. Karpavicius, Kaunas University of Technology, Kestucio 27, 44312
Kaunas, Lithuania, E-mail:
[email protected]
Table 1
Strength coefficients
Material Coefficient K Coefficient n
AW 6082-T6 590 0.3
L63 1600 0.6
AISI 304 1400 0.45
Table 2
Angle versus punch displacement
Punch Change of
displacement, mm angle, [degrees]
0 90
1 83
2 76
3 68
4 60
5 50
6 41
7 31
8 22
9 14
10 9
11 6
12 5
13 4
14 3