Investigation of dependences of stress strain state properties on metal sheet holding force at its forming/Itempiu ir deformaciju buvio priklausomybes nuo formuojamo metalo laksto prispaudimo jegos tyrimas.
Bortkevicius, R. ; Dundulis, R. ; Karpavicius, R. 等
1. Introduction
The purpose of this investigation is to determine thinning
variation (the 3rd marginal strain [[[epsilon].sub.3]) at metal sheet
forming and determination critical points on forming material. The main
mechanical properties influencing forming condition is Young's
modulus-E, density-[rho], yield stress--[[sigma].sub.y], ultimate
tensile strength-[[sigma].sub.UTS], forming velocity-v, friction
coefficient-[micro] (since we take friction coefficient as a dependence
of relative velocity we do not account it as static and kinematic
components but rather as single one). There were used a hypothesis which
states that if thickness deformation from finite element analysis (FEA)
coincides with experimental data, than stress strain relation from FEA
can be treated as correct strength state of formed metal sheet part.
To complete this task we will vary blank holding force (BHF) to
check its influence on material thinning or if it would have negative
influence on the part's performance in formability we will see
necking. Necking retardation [1-3] in metal sheet forming is one of the
major defects. This paper will present the results from FEA-simulation
of forming process of "Z" shape formed metal sheet part made
from two different materials as well as thickness distribution in real
experiment. Materials for testing were chosen by a reason: aluminium
AW6802 (brittle and very non ductile), brass L63 (soft--very ductile).
The output of simulation will be compared with physical tests.
The investigation was performed using 6 different load schemes of
changing blank holding force. Since our investigation considers
practical and finite element modeling (FEM) therefore firstly we
consider carrying out an experiment and then we do modeling with LSDyna
software. If holder uses relatively high value of suppressing force then
metal sheet is forced not to slide over the punch/die surface, but if
that relative force has value lower than tangential tension force that
sheet starts to slide and therefore arises sliding frictional forces
[4-6]. If the normal force F and tangential force T is applied onto a
punch, then the system is in either equilibrium position or into
movement with a constant velocity. The movement starts very slowly,
that's why we do not consider dynamic effect in the stamping
process. This is important because thickness distribution along stamped
part varies and as more suppressing force we do use as we get more
thinning in the particular regions we shall find those regions in our
simulation model. Our model is constructed to simulate thickness
distribution in the "Z" shape part. At the end of
investigation one shall see how thickness distribution varies in
accordance with metal sheet suppressing force. It needs to mention that
we simulate constant blank holding force. Usually in metal production
industries stamping is performed with not constant or not regulated BHF
[7]. We ignore this error through fixing the holders movement towards
blank. By doing that we assure that the gap between die and upper holder
(the same is with punch and lower holder) is always not less than 2.9 mm
and not bigger than 3.1 mm. So the gap is always within the range of 2.9
[less than or equal to] x [less than or equal to] 3.1 mm, where x is
characteristical gap value. We can fix the gap exactly to 3 mm because
by doing this we will eliminate the use of BHF [7]. In other word there
is use to regulate blank suppressing force.
2. Methods of investigation
Testing was conducting using two common models: experimental
procedures (Fig. 1) and FEA (Fig. 5). Since the investigation concerns
dynamic reaction in metal sheet forming--e FEA was conducted using
program for nonlinear dynamic analysis of structures in three dimensions
ls971 single R4.2. To simulate prestressed bolts in LSDyna we used new
feature available only from current version firstly released in ls971s
R3 beta. The keyword defining forces suppressing part is *
INiTIAL_AXIAL_FORCE_BEAM. This LSDyna card defines relative axial force
to beam. Because in original stamping kinematical scheme the pressure
objected to the material is defined by five M12 bolts which have torqued
by 20 Nm force--moment in order to get 25200 N suppressed force (full
suppressing force range defined in Table 2). According to initial set, 2
different types of materials have been examined. Main mechanical
properties were gained from uniaxial tension test where true stress
strain curves showed in Fig. 2 were determined. Initially, uniaxial test
was carried out to evaluate the mechanical and tensile properties.
Experimental procedure was carried out using different load schemes at
the same punch traveling conditions. Load was applied on a lower holder.
Force curve in LSDyna is defined with a card * LOAD_NODE_SET. For
numerical calculation we used material model 24, defined keyword as *
MAT_PIECEWISE_LiNEAR_PLASTICITY. Literature [8] recommends not to use
model 81 with damage effect since it gives unrealistic results in
bending. The model was created with Belytschko-Tsay shell element with
thickness stretch no 25 for part with 5 through thickness integration
points and Belytschko-Tsay shell element no 2 for the rest of model with
3 through thickness integration points [6, 7]. To determine
experimentally thickness distribution there computational model was
introduced. This model is shown below in the Figs. 3 and 4. In Fig. 5
the same computational model but modeled with finite element method is
shown. At the same Figure can be seen and mesh density On the FE model
there are subscriptions of the main parts acting on metal sheet forming
are set. As in real experiment we have: 1-punch, 2-metal sheet part
before deformation, 3-lower holder, 4-stationary die, 5-upper holder.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
Measurements were taken from the cross-sections along the part.
Basic position can be seen in Fig. 6. The part was divided into 2 equal
zones. Calculations were made from the middle and from right hand side
of the part cross-section. Since both outermost cross-sections are
identical there is no use to measure the third cross-section at the same
time. Boundary conditions applied to the model are described in the
Table 1: it is set, that the formed part does not have any constrained
degrees of freedom (DOF) since initially we do not know how it will
perform in stamping. Punch and blank holding holders can move only in Z
direction so we set only one DOF for unconstrained. The die has 6 DOF
constrains and it can not move or rotate.
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
Experimentally formed part was made with 25 kN power press in KTU
laboratory. In Table 2 also prescribed holders' suppressing force
in relation to bolts torque is set. Deformation speed in all examined
materials has the value of [??]-0.023s1. The described forming procedure
modeled with FEA correlates exactly with experimental model.
Metal sheet suppressing force is regulated by wrenching up some
bolts by adequate torque. Kinematics of stamping component can be seen
in Fig. 3. Material models dimension are as following: 3x30x100 mm. This
plate was initially precut from a bigger metal sheet in order to reduce
residual stresses. Characteristical data needed to simulate stamping
analysis is: stress strain curve, Poisson's ratio, and coefficients
of friction. In the figure 2 can be seen mechanical properties for
tested material L63 and AW6802. By designating R0 and R90 we mean, that
material was tested along the rolling direction and reversely of rolling
direction. Mechanical properties were used in FEA model. Analytical
calculation of thickness is based on classical mechanics of metal sheet
forming [7]. Since our investigation concerns only thickness or the 3rd
marginal strain [[epsilon].sub.3] we can calculate it as
[[epsilon].sub.3] = ln t/[t.sub.0] = -1 1/2 [[epsilon].sub.1] (1)
here [[epsilon].sub.3] is 3rd marginal strain, t is thickness after
deformation, [t.sub.0] is initial thickness, [[epsilon].sub.1] is 1st
marginal strain.
If the deformation runs on general plane stress sheet conditions
are
[[epsilon].sub.3] = -(1 + [beta])[[epsilon].sub.1], (2)
where [beta] is defined as
[beta] = [[epsilon].sub.2]/[[epsilon].sub.1]
ln([d.sub.2]/[d.sub.0])/ ln([d.sub.1]/[d.sub.0]) (3)
here [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 circles deformed to ellipses of major diameter [d.sub.1],
[d.sub.2] is the deformed state with the grid circles deformed to
ellipses of minor diameter [d.sub.2]
[[epsilon].sub.3] = -(1 + [beta])ln [d.sub.1]/[d.sub.0] (4)
From Eq. 4 we have, that current thickness is
t = [t.sub.0] exp ([[epsilon].sub.3]) = [t.sub.0] exp [-(1 +
[beta])[[epsilon].sub.1]] (5)
If volume [td.sub.1][d.sub.2] = [t.sub.0] [d.sup.2.sub.0] is in
constant, then final thickness is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
Before performing deformation of a sheet, we marked the
investigated material with square grids [9] in order to investigate
thinning effect. The dimensions of initial grid were 0.5 mm and 0.1 mm
space between grids. In case of localized necking critical strain can be
calculated [9]
[[epsilon].sub.3] = n/1 + [beta] (7)
where n is strain hardening index [7].
In case of diffuse necking critical strain can be calculated from
Swift equation [8, 9]
[[epsilon].sub.1] = 2n (1 + [beta] + [[beta].sup.2])/(1 +
[beta])(2[[beta].sup.2] - [beta] + 2) (8)
Measurement points for analytical calculation were taken as shown
in Figs. 3 and 4. Six different locations were measured. Point A and
point B were taken in the horizontal position as well as points F and G.
Only one point D was taken from vertical place of the part. And by
single one at the position were bending deformation takes part (points C
and E).
3. Investigation results
Experimental and finite element analysis revealed the following
results depicted in figures below. Full development history of thinning
in the part is shown in the Fig. 7. This is a view from material AW6082
R-90 formed with 31500 N holders suppressing force. The measurements
were taken every second time steps to get nine thinning distributions
along a part in accordance to punch travel distance. "0" point
corresponds no punch movement; there is no deformation in the part.
"16th" point corresponds the last step of punch on the
traveling curve. The part should be deformed according to forming tools
shape.
[FIGURE 7 OMITTED]
In Fig. 7, results are taken from FEA, since we can not measure
thinning in all steps. The experiment completed with the same blank
holding force, formed part experienced a crack, so this figure is only
used for theoretical examination. In the figure one can see upper and
lower punch boundaries. When the punch is at the rest in upper position
the part has it's thinning of 3 mm. When the punch starts to move
slowly downwards, deformation begins and thinning takes a part. At
holders suppressing force of 31500 N one shall see that the major
thinning upstarts at the location of 40 and end up at the location of 60
mm along the part. In metal sheet forming it is vital to secure that
thinning develops throughout the part and not only in the certain
regions as we see in the Fig. 7 where one can find thinning only in the
two regions. The highest values we found are around 2.5 mm. That is not
acceptable. From literature [1] we know that it can be true, that the
investigated sort of aluminium can not be deformed in a range of 31500 N
of suppressing force because at the end of punch traveling in the
material develops a crack. Another major concern one shall see in the
same Fig. 7 is that thinning does not develop equally in vertical region
of the part. We have part thickness value higher at the center of the
part. Following part to the left or to the right from the center the
thickness values develop lower values. Because of this phenomenon one
can be explain why the crack develops not in the center of the part but
rather a bit to the left or right from it. If left hand side differs
from right hand side in the Fig. 7 and others we can explain it as the
part initially is located on the die surface not exactly correct. The
part shall be located exactly in the center of the forming tools in
order to receive adequate results on both hand sides.
[FIGURE 8 OMITTED]
During deformation of sequence from 1 to 6 (Table 2) with adequate
holders suppressing force one can see in the Fig. 8. The following
figure is created for CuZn.
[FIGURE 9 OMITTED]
Mechanical properties of the investigated materials one shall see
in the Table 3. The most material thins out when holders suppressing
force is hypothetical--31500 N. Material changes its thickness by the
same manner throughout all suppressing force sequence--it has two peaks
at some distance from the center. Fig. 9 represents material thinning
for CuZn R0 versus Finite element data. The given data nicely fits
experimental and FEA data. From this figure we make an assumption that
the data from FEA is correct since it perfectly matches the data from
experiment observation. On the thinning curves there also is set a
characteristic points which are taken from some particular places on
deformed part. Points A, B, ..., G correspond points on undeformed part
at the Figure 3 as well as on the deformed part at the Fig. 4. As
experimental data revealed the major concern we need to account is
points C and E, because at these positions (or points) are the most
relevant to develop a crack. No cracks will occur at the location with
point D (Fig. 9). Fig. 10 corresponds adequately thinning variations for
materials Aluminium AW6082. Since we made an assumption that the 3rd
deformation from FEA perfectly matches experimental date we believe that
stress strain at the important point is also correct. Comparison of
major thickness distribution in investigated materials can be seen in
the Fig. 11. Fig. 12 is stress strain curves for materials aluminium and
brass at specific points. Needless to say that material brass has
steeper stress strain curve (Fig. 2), Fig. 12 justifies this tendency.
The more steeper the curve in monoaxial stress strain the more steeper
is stress strain at specific points at deformable part. Fig. 13 is final
results from experiment where shall be found a thickness in metal sheet
forming, when controlling blank displacement. There clearly can be seen
a tendency, that both investigated materials have linear dependency on
BHF. Since thinning distribution on left hand side and right hand side
is the same we marked this with P2 and P3 in Fig. 13 above. In the
middle of the part thinning distribution is marked with P1. Optimum BHF
is within a range of 18800 [less than or equal to] BHF [less than or
equal to] 25200 if thinning is within the range 2.6 [less than or equal
to] x [less than or equal to] 2.8.
[FIGURE 10 OMITTED]
[FIGURE 11 OMITTED]
[FIGURE 12 OMITTED]
[FIGURE 13 OMITTED]
4. Conclusions
In the paper we discovered how thinning takes part in metal sheet,
while deforming "Z" shape element. Seven main locations were
measured along the part. The optimal holder holding force was proposed
by which metal sheet least thins out. The investigations were carried
out using two common methodologies--experimental methodology and finite
element methodology. In FEA measurements were taken after spring back
calculation. Two material models were tested under the various load
ranged from 31500 N to 8800 N. The analysis was made at ambient
temperature.
The obtained thinning properties for tested materials at room
temperature showed that increasing of blank holder suppressing force
causes increasing material thinning. Experimental data revealed that
thinning is not monotonous and increases at the corners of forming
tools. The vertical part of deformed part also thinned out but not so as
at the corners. No thinning took part at the place where blank holders
touches forming material. After comparison of experimental data and the
data from finite element analysis conclusion was set, that the 3rd
strain fully coincides between methodologies. If this is true then
fraction strain calculated from FEA is suppose to be correct. In final
remark we sum up experiment data and concluded as 2 major outcomes:
1. In general, investigated materials aluminium AW6082 and brass
L63 thinning effect as function of blank holding force thins out
accordingly to linear relationship not depending of the place where
measurements were taken.
2. The highest stress points were determined and critical points on
deformed part were set where possibly a crack can be initialized.
Received May 24, 2010 Accepted October 05, 2010
References
[1.] Hutchinson, J.W., Neale, K.W. Sheet Necking -III strain rate
effect. -Mechanics of metal sheet forming. -New York, 1978, p.269-285.
[2.] Kirchfeld, M. Applicability of Casting Resin Tools in Sheet
Metal Forming. -Shemet Project Final Report. -48p.
[3.] Van den Boogaard, A.H., Huetink, J. Prediction of sheet
necking with shell finite element models.- Faculty of Engineering
Technology, University of Twente, P.O. Box 217, 7500 AE Enschede,
Netherlands, 2004, p.4.
[4.] Jensen, B. C., Lapko, A. On shear reinforcement design of
structural concrete beams on the basis of theory of plasticity. -Journal
of Civil Engineering and Management International Research and
Achievements. -Vilnius: Technika, 2009, vol.15, No.4 , p.395-403.
[5.] Lazarz, B., Wojnar, G., Madej, H., Czech P. Evaluation of gear
power losses from experimental test data and analytical methods.
-Mechanika. -Kaunas: Tech nologija, 2009, Nr.6(80), p.56 -63.
[6.] Iyama, T., Odanaka, K., Asakawa, M. Twist resulting in high
strength steel sheet hat channel product consisting of straight portion
combined with curved portion.-JSME International Journal, series A,
vol.46, No.3, 2003, p.419-425.
[7.] Hallquist, O.J. LSDyna Theory Manual. -California: Livermore
Software Technology Corporation, 2006. -680p.
[8.] Buakaew, V., Sodamuk, S, Sirevidin, S, Jirathearanat, S.
Formability prediction of automotive parts using forming limit diagrams.
-Journal of Solid Mechanics and Materials Engineering, 2007, No5, vol.1.
p.691-698.
[9.] Marciniak, Z. Mechanics of Sheet Metal Forming. -Oxford,
2002.-228p.
R. Bortkevicius *, R. Dundulis **, R. Karpavicius ***
* Kaunas University of Technology, Kcstucio 27, 44312 Kaunas,
Lithuania, E-mail: r.bortkevicius@kaispauk.lt
** Kaunas University of Technology, Kcstucio 27, 44312 Kaunas,
Lithuania, E-mail: romdun@ktu.lt
*** Kaunas University of Technology, Kcstucio 27, 44312 Kaunas,
Lithuania, E-mail: rimkarp@stud.ktu.lt
Table 1
Boundary condition applied to model
Displacement
Component x y z
Punch X X [check]
Die X X X
Upper holder X X [check]
Lower holder X X [check]
Blank [check] [check] [check]
Rotation
Component [[THETA].sub.x] [[THETA].sub.y] [[THETA].sub.z]
Punch X X X
Die X X X
Upper holder X X X
Lower holder X X X
Blank [check] [check] [check]
Table 2
Holders suppressing force sequences
Sequence Torque, Nm Axial force, N Pressure, MPa
1 25 31500 22.37
2 20 25200 17.90
3 15 18800 13.35
4 10 12550 8.91
5 8 10100 7.17
6 7 8800 6.25
Table 3
Mechanical properties of tested materials
Ultimate
Yield stress tensile stress
Density [rho], [[sigma].sub.y], [[sigma].sub.UTS],
Material kg/[m.sup.3] MPa MPa
AW6082 R-0 2700 130 225
AW6082 R-90 2700 136 230
L63 R-0 8440 165 313
L63 R-90 8440 132 293.7
Total Young's Poisson's
elongation modulus Deformation ratio
Material [A.sub.5], % E, GPa at fracture [v.sub.0]
AW6082 R-0 16 75.7 0.17 0.33
AW6082 R-90 19.8 87.6 0.169 0.33
L63 R-0 38 106.4 0.279 0.36
L63 R-90 50.13 105.1 0.262 0.36
