through-thickness hardness measurement for pure …through-thickness hardness measurement for pure...
TRANSCRIPT
Through-thickness hardness measurement
for pure bending sheet metal specimens Bachelor Final Project by: C.W. Verberne MT 07.14
Supervisors: C.Cem.Tasan, R.H.J.Peerlings
1
Contents
1. Introduction 2
2. Theory of Free-bending 3
2.1 Stress-strain………………………………………….. 3 2.2 Moment……………………………………………… 5 2.3 Springback…………………………………………… 6 2.4 Edge effects………………………………………….. 8 2.5 Core movement………………………………………. 9
3. Experimental Methodology 11
3.1 Bending tests……...………….…………………………….. 11 3.2 Specimen preparation for Nanoindentation and ESEM......... 13 3.3 Nanoindentation……………………………………………. 14 3.4 ESEM analysis……………………………………………... 14
4. Results and discussion 15
4.1 Bending test results…..……………………………………. 15 4.2 Nanoindentation…………………………………………... 17 4.3 ESEM analysis…………………………………………….. 20
5. Conclusion 22
6. Recommendations for the future 23
7. Appendix 26
A: Wrong sample preparation………………………………… 26 B: Calibration …...……………………………………………. 27 C: Radius measurements....…………………………………… 31 D: Result graph form nanoindentation…...…………………… 32
8. Reference 35
2
1. Introduction
In most types of sheet metal forming processes, like drawing, stretching and bending operations, necking is the main failure mechanism that defines the safe working strains. However, the new and non-traditional materials that were recently introduced, such as some aluminum alloys and high strength steels, have shown failures occurring without any significant necking. This phenomenon occurs especially in materials with relatively low formability such as the dual phase (DP) and transformation induced plasticity (TRIP) steels.
It is known that these kinds of failures are due to the occurrence of internal damage in the material. The micro mechanisms that cause damage, i.e. void nucleation, growth and coalescence, are quite well known and understood. Continuum damage models proposed in the literature aim to predict such failures, however experimental data is necessary to tune damage growth relations and parameters. Several methods have been proposed to provide these relations. The microindentation method has provided the most promising results so far. In this approach, which was proposed by Lemaitre et. al. [1], damage is quantified by comparing the micro hardness of damaged material, H, to the micro hardness of the undamaged material, H’
)(')(1)( εεε HHD −= (1.1)
Here D is the amount of damage. Note that it has been defined such that 0 § D § 1, where D = 0 represents virgin material and D = 1 a complete loss of strength (hardness). By doing a tensile test and microindentation experiments along the specimen the hardness is compared to the strain for an IF steel in fig 1.1. Until 35%-40% strain the hardness increases by strain hardening. At higher strains damage occurs. The red dotted curve in figure 1.1 is the extrapolated result without any damage occurring, according to the procedure proposed by Lemaitre et. al. [1].
Figure 1.1 result from tensile test where the strain and Hardness are compared [2]
Another way of possibly obtaining this curve is by carrying out compression tests. In compression no voids and thus no damage is expected to occur. However, compression tests are difficult to carry out for sheet metals. One possibility to impose compression is pure bending. In bending, in one sample the tensile side and compression side coexist, and can be compared. In this work, the bending tests are done by a free bending device. Further analysis is carried out by Nanoindentation and Electron microscopy analysis.
The general aim of this project is to compare the hardness for different strains in compression and tension to verify if there is any damage in compression, or not.
3
2. Theory of Free-bending
2.1 The Stress-Strain Behavior in Free-Bending
For understanding what is happening in the metal plates in free-bending, the stresses and strains are examined. In a simple bend without any overall tension applied and where the thickness of the plate is small compared to the radius of curvature, R, the strain distribution is as sketched in figure 2.1.
Figure 2.1 strain distribution in a bent plate
The strain plotted here is the engineering strain. It is given as a function out of the plane coordinate y and the bending radius R
R
yengx =ε (2.1)
The corresponding true (natural) strain is given by
)1ln(R
ytruex +=ε (2.2)
At small strains, the stress is elastic and thus linear. The relation between the elastic stress and engineering strain can be obtained by using the generalized Hooke’s Law [3]
( )[ ]zyxx
Eσσνσε +−=
1 (2.3a)
( )[ ]xzyy
Eσσνσε +−=
1 (2.3b)
( )[ ]yxzz
Eσσνσε +−=
1 (2.3c)
Here a plane stress state is assumed and therefore 0=yσ . From equation (2.3c), xz νσσ =
is obtained, because 0=zε plane strain is assumed. Substituting xz νσσ = in equation
(2.3a), a relation between the engineering strain and the elastic stress, for small curvature, is obtained
xxE
σν
ε21−
= ,or xx
Eε
νσ
21−= (2.4)
4
At large strains, the rigid-ideally plastic model is assumed. i.e. the elastic strains are neglected and no hardening is assumed. This assumption results in a stress distribution as sketched in figure 2.2.
yσ3
2
yσ3
2−
Figure 2.2 rigid-ideally plastic stress distribution
The values for xσ in figure 2.2 are calculated by the Von Mises relation
dd
vmy σσσσ :2
3== (2.5)
Where yσ is the Yield stress and σd is the deviatory stress tensor
)(3
1σσσ tr
d−= , with zzzxxx eeee
vvvvσσσ += (2.6)
Again with the assumption of plane strain in the z-direction the relation between zσ and
xσ can be obtained: xz σσ2
1= . Now the deviatory stress tensor can be derived as
( )yyxxx
deeeevvvv
−= σσ2
1 (2.7)
Substituting Eq (2.7) into Eq (2.5), the relation between xσ and the yield stress, yσ , can
be obtained;
xyield σσ 32
1= , or yx σσ
3
2±= (2.8)
As shown in figure 2.1, the highest strains occur in at the outer surfaces of the plate whereas in the middle of the plate it is minimal or even zero. This means that for the goals of this project the outer fibers of the plate are the most interesting areas to examine.
5
2.2 Moment (torque)
For a better understanding of what happens during bending, it is useful to have an analytic expression for the moment that is needed to bend the plate. This moment can be calculated by taking the stress integrated over the thickness, y:
∫−
⋅⋅=
2/
2/
t
t
x dyybM σ (2.9)
Here xσ is the stress, b is the width of the sample and t is the thickness of the plate.
In an elastic bend the moment then becomes,
Rbt
Edyy
R
yEbdyybM
t
t
t
t
x
1
112
1
1
3
2
2/
2/
2
2/
2/νν
σ−
=−
⋅=⋅⋅= ∫∫−−
(2.10)
At large strains the moment becomes,
2
2/
0
2/
2/ 32
1
3
22 btdyybdyybM y
t
y
t
t
x σσσ =⋅⋅⋅=⋅⋅= ∫∫−
(2.11)
6
2.3 Springback Calculation
When a plate is bent an important phenomenon occurs. When the acting moment is released, the plate relaxes elastically. This is called springback. When a plate is only bent elastically it will completely relax until it is flat plate again. At higher strains the plate will springback a few degrees of its initial bend, till the applied moment is zero. The release will look as sketched in figure 2.3:
The total moment in figure 2.3c is zero. This means that the total moment of springback is
as in Eq (2.11): 2
32
1tbM yσ⋅=∆
The difference in moment also can be obtained from Eq (2.10)
∆
−=∆
Rbt
EM
1
112
1 3
2ν (2.12)
Now the following relation between the current angle, arc length and curvature radius is employed
Rl θ=
or
lR
θ=
1 (2.13)
where l is the arc length and θ the angle before springback. During the unloading the arc length will remain constant so that we have
θ
θθ
RlR
∆=
∆=
∆
1 (2.14)
Figure 2.3a stress distribution in a rigid perfectly
plastic bend
Figure 2.3b the amount of springback
Figure 2.3c stress distributions in a bend after
springback
7
Combining Eq. (2.11), (2.12) and (2.14) results in
23
232
1
112
1bt
Rbt
EM yσ
θ
θ
ν−=
∆
−=∆ (2.15)
or after rewriting
t
R
E
y θνσθ
)1(32
2−
−=∆ (2.16)
The total expected springback for the bending tests then becomes:
MPay 0.92=σ
GPaE 65.44= ( yσ , E and ν are from [2])
ν = 0.27
(Calculated with angle o115 and arc length 5.3mm)
mEt 37.0 −=
o115=θ
o87.2−≈∆θ
mER364.2 −
=
8
2.4 Edge effects
When the width of a bent plate is small compared to the thickness, transverse stresses are produced and undesired deformations may occur (fig 2.11).
To minimize this effect to the edges, the specimen needs to be wide compared to the thickness (fig 2.12).
In the middle of the specimen show in figure 2.12 the transverse curvature is zero and the bending occurs under plane strain conditions. To have good conditions with minimal edge effects, the b/h ratio (width vs. thickness) should be over eight (figure 22.11 in [5]). The samples that are used in this report are: 100xLx0.7 (ratio≈143). The ratios of all samples are all high enough for not having any influence of the edge effect. Nevertheless all the results that are gathered from the specimen are measured away from the edge.
Figure 2.11 edge effect in thick or narrow plates
Figure 2.12 Negligible edge effect in thin or wide plates
9
2.5 Core movement
The core or the neutral plane at the start of the bending is logically in the middle. However with increasing amount of deformation, it moves away from the middle. To calculate the amount of core movement, the state of stress on a typical element in plane strain is examined [6] (fig 2.13).
rr dσσ +
θσ θσ dr
rσ
θd
The equilibrium in the vertical direction for plane strain deformation then becomes:
( ) 01
=−+ θσσσ
r
r
rdr
d (2.17)
Now using the same Von Mises relation as in Eq. (2.5), (2.6), (2.7) and (2.8) (now in term
of the coordinates r, θ and z instead of x, y and z), here the following equation can be
derived
( )rσσσ θ −=2
3 (2.18)
Substituting Eq (2.18) into Eq (2.17) and setting yσσ = we obtain.
y
r
rdr
dσ
σ
3
21= (2.19)
The general solution of this equation reads
BrAr += )ln(σ (2.20)
Where A and B are constants.
Figure 2.13 typical element in plane strain bending
10
Using the boundary conditions 0=rσ at 0Rr = and iRr = , Eq (2.20) results in
on
oy
r RrRforr
R≤≤
−= ln
3
2σσ (2.21a)
ni
i
y
r RrRforR
r≤≤
−= ln
3
2σσ (2.21b)
On the neutral axis Eq (2.21a) and (2.21b) must be equal, which implies
i
n
n
o
R
R
R
R=
So that,
ion RRR = (2.22)
Thus from the geometric properties after the bending the neutral axis can be calculated when the inner and outer radii are known.
11
3. Experimental Methodology 3.1 Bending Test
Figure 3.1 shows a sketch of the bending device which was used in our experiments. A zoom of the specimen and its clamping is shown in figure 3.2.
1. Linear guide 2. Linear guide 3. Hydraulic clamp 4. Hydraulic clamp 5. Specimen 6. Elastic joint 7. Base 8. Cylinder 9. Cylinder
Figure 3.1 free-bending machine schematic setup. [7]
Figure 3.2 close up of the clamping [7]
The idea of this machine is to bend a plate without any shear or axial forces. Therefore a plate specimen (5) is clamped at both sides by hydraulic clamps (3& 4). The left clamp (3) cannot rotate around the bending axis, while the right one (4) is rotated to a prescribed angle. At this clamp the rotation angle is also measured. Due to the two air suspended linear guides (1& 2) the clamps can translate frictionless along and around the two axes of the two cylinders. With the guides at a relative angle of 90 degrees there consequently is no transfer of lateral forces between the clamps. The applied moment is measured by an elastic joint (6) [7]. The bending device has three different clamps (fig. 3.3), one for plates with thickness up to 1mm, another for up to 2 mm and another for up to 3mm.
12
3mm 2mm 1mm
Figure 3.3 sizes of the clamps
In figure 3.3 it can be seen that the thicker the plate is, the thicker the clamps must be. The free bending device is able to bend to an angle of 180˚. The free bending device also enables limited amounts of negative bending. In this work this has not been done. The free bending device is linked to a computer which controls the machine and records all the data from it. This is done by a program called LABVIEW, which controls different parameters such as moment, angle and angular speed. This makes it possible to bend a specimen to whatever angle is desired, up to 180˚. The angle can be controlled to a precision of 0.01˚ and it is measured in 0.001˚. The force is controllable up to 0.02 Newton and is measured in 0.001 Newton. The material that is used in our experiments is IF steel (Interstitial Free steel) with a Young’s modulus of 44.65 GPa and Yield stress of 92.0 MPa [2]. The plates used are approximately 100x25x0.7mm (bxlxh). Specimens are cut with a crude cutting device. Deformation due to this cutting is not significant at the surfaces that are going to be examined. The rolling direction however is taken into account by bending all specimens in the rolling direction.
13
3.2 Specimen preparation for ESEM and Nanoindentation
After the specimen are bent they need to been cut into several pieces to be able to examine the surfaces that are interesting. Figure 3.4 shows the geometry after the specimen is cut.
Because the analysis on these surfaces is delicate, the cutting must be done without changing the material microstructure. Therefore the cutting is done by spark erosion. For ease of preparation the specimens are molded in a polymer resin after cutting(fig 3.5). For the Nanoindentation experiments epoxy was used, which hardens by curing. For the ESEM analysis the sample is pressure- and temperature molded in a conducting resin. This gives some problems for the grinding and polishing. After the cutting is done the surfaces should be perfect without any scratches. This surface preparation is done by grinding the surfaces (1200) to remove the resin cover and then polishing (1µm grain size diamond). The polishing needs to be done for quite a long time, to keep surface hardening to a minimum. It is wise to look every half an hour at the specimen. It is not desirable to remove more material when the specimen surface is already good. The surface is good when under a microscope no scratches are seen.
Figure 3.5 molded samples: Left sample for
Nanoindentation and right for ESEM-analysis
Figure 3.4 the green area designates both the cutting surface
and interesting area
14
3.3 Nanoindentation
The method of nanoindentation has been developed to directly measure material properties from indentation load displacement data obtained from one cycle of loading and unloading.
In figure 3.6 schematic data are shown from one cycle. During the unloading, it is assumed that only elastic deformation occurs. This elastic unloading curve gives the information needed to analyze the material. Three values are measured from the P-h
curve: the maximum load, Pmax, the maximum displacement, hmax, and the elastic unloading stiffness S =dP/dh which is the slope at the start of the unloading [8]. The maximum load on the specimens in this report is 700 mN. The maximum displacement is 8000 nm. The loading is stopped when one of these values is reached. Indents have been made over the thickness of the plate as shown in figure 3.7.
Figure 3.7 the place of all indents are shown. Every dot is an indent.
3.4 ESEM analysis
Following the bending and indentation, specimen surfaces are examined for any visible damage formation in an environmental scanning electron microscope (ESEM).
Figure 3.6 schematic load-displacement data
15
4. Results and discussion 4.1 Bending test results
Before using the bending device it should be calibrated. The calibration of the free-bending machine is done by comparing the measured moment and theoretical results obtained for a stiff elastic specimen. Table 4.1 is obtained by calculations that are shown in Appendix B.
Table 4.1 corresponding angle, voltages and moment for calibration
Angle α 0.125 0.25 0.375 0.50 0.625 0.75 0.875 1.00
Voltage 0.149 0.195 0.241 0.287 0.333 0.380 0.429 0.479
Moment M 1.875 3.75 5.625 7.5 9.375 11.25 13.125 15
This table is entered in the calibration screen in labview.
After the calibration the bending itself can be done. For the best comparison of the compression and tensile side it is desired to have the highest amount of strain possible. This can be done in two ways: first the bending angle should be as high as possible; second the radius of curvature should be as low as possible. With the geometry of the bending device and clamps the two requirements (high angle; low radius) can possibly be met by two configurations as sketched in figure 4.2 a and b
Figure 4.2a Figure 4.2b
Figure 4.2a has the highest angle and figure 4.2b has the lowest radius. With the specimen used (thickness = 0.7mm), the method in figure 4.2b results in the highest strains.
While bending there is a high risk of the two clamps touching each other. To avoid this, a safety margin is taken. Then using the other geometric properties of the clamps (fig 3.3) and the thickness of the sheet, the maximum strain, minimum length and maximum angle can be calculated (table 4.2).
Table 4.2 Maximum Strain, angle and minimum length
without safety with safety
εmax 0.231 0.120
Minimum length 2.67 mm 5.00 mm
Max. α 118˚ 115˚
The safety margins that have been taken are an angle reduction of three degrees and a distance safety of 2 mm between the two clamps. Both strains in table 4.2 are the maximum at the compression side. The strains at the tensile side are slightly higher (0.27 and 0.14). This is due to the neutral axis movement to the compression side. The strains in table 4.2 are without the springback. The total springback of the samples is a little over three degrees. Thus the plastic strains are lower.
16
The real length of the specimen is 25.33mm. The bending part of the specimen is 5.33mm. The bending test resulted in the deformed geometries shown in figure 4.3. The measured inner and outer diameters have been indicated in the figure.
Due to the fact that the other four radius measurements (Appendix C) have variations in thickness of the plate in the bend, and this measurement is done manually and has some error, it is assumed that the thickness remains 0.7 mm during the entire bending operation. The thickness is also measured in the nanoindentor and is indeed equal to 700 µm (0.7 mm) everywhere. The radius measurements have therefore been adjusted to obtain this thickness. The adjustment is done by equally adjusting the inner and outer diameter so that the thickness is 0.7mm. This result in the data given in tables 4.3 and 4.4. Table 4.3 Results of radius measurements and predicted neutral radius Rn
Sample Inner radius [mm] Outer radius [mm] Neutral radius [mm]
1 (fig 4.3 left) 2.4 3.1 2.73
2 (fir 4.3 right) 2.49 3.19 2.82
Table 4.4 Maximum strains at both compression and tensile side
Sample Max. compression strain [-] Max. tensile strain [-]
1 0.120 0.136
2 0.116 0.132
Damage starts occurring from strains higher than 35% [2]. The maximum strains that are reached with this experiment are lower than 35%. Therefore an adjustment to the machine or the bending specimen should be made to get the higher strains. Possible adjustments are discussed in chapter six.
Figure 4.3 The radius measurements of the two samples used in the following Nanoindentation and ESEM analysis
17
4.2 Nanoindentation results
The nanoindentation tests were carried out on a total of four samples. The first two samples were taken from specimen 1 and the last two samples from specimen 2 (see Tables 4.3 and 4.4). After the nanoindentations have been done, the specimens look like in figure 4.5.
Every indent affects the surrounding material within a certain radius. Therefore all indents should have a certain minimum spacing. To reach this and still have a sufficient number of indents across the thickness, the indents were made in a diagonal pattern on the samples. These diagonal rows consist of 30 indents. Since the bending of the plate is homogeneous, for each y-position, every x-position has the same strain. The lower-right part of figure 4.5 shows some rows of dots close to each other. The sample, on which these indents were made, after polishing, still had a part that wasn’t sufficient for nanoindentation. Thus all the rows of indents were made on a smaller region. Different results occurred at the last row. For al the rows of dots it is important to know the exact y-position of each dot. Therefore the distances from the first and last dot to the corresponding edges are measured and the y-positions of all other dots are interpolated between them.
Figure 4.5 Pictures, made with a microscope, of the samples after nanoindentation
18
Figure 4.6 shows the measured hardness versus y-position for sample 2 (specimen 1). The definition of the Y-position is made in figure 4.7
Y-position vs. Hardness Sample 2
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
0 70 140 210 280 350 420 490 560 630 700
Compession side Y-position [µm] Tensile side
Hard
ness
Figure 4.6 hardness measurements across the thickness of sample 2
Figure 4.7 definition of the Y-position in a sample
From figure 4.6 it can be seen that at the lowest strains (in the middle) the hardness is also to lowest. This corresponds with the theory that the hardness increases when the strain increases. It can also be concluded that there is no damage in the material since the hardness continues to increase towards the top and the bottom of the specimen. This corresponds with the expectation that damage towards the tension side starts at strains of 0.40 or higher. Another thing that can be concluded out of figure 4.6 is that the neutral axis indeed moves during the bending test. According to the hardness measurements shown in figure 4.6 the neutral axis after bending (i.e. the minimum hardness) is at a y-position of 330 µm. This corresponds with the prediction of table 4.3. The neutral axis calculated there is 2.73mm minus the inner radius 2.40mm for which the y position becomes 0.33mm. Several indents had an error during the indentation; these are the zero hardness dots in Figure 4.6 (and also 4.8). The most interesting questing is: Do the hardnesses from the compression and tensile sides correspond to each other? Figure 4.8 shows the hardness in both compression and tension as a function of the absolute value of strain, so that the hardnesses of the same strain values in compression and tension can be compared. The strain at each indent
19
position has been determined using the y-position, the position of the neutral axis as determined above and the bending radius
Compression vs. Tensile side Sample 2
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
0,0 1,0 2,0 3,0 4,0 5,0 6,0 7,0 8,0 9,0 10,0 11,0 12,0 13,0
Engineering strain [%]
Hard
ness [
GP
a]
Compression
Tensile
Figure 4.8 Comparison between compression and tensile sides
It is seen that there is no significant difference in hardness between compression and tension. There are more dots in the tensile regime than in compression, because the compression region is smaller due to the movement of the neutral axis. Another point that can be seen in figures 4.6 & 4.8 is that at the edges the hardness levels out somewhat. This may be caused by some sort of edge effect. However this effect is quite small and the hardness increase can still be assumed to be linear in this strain range. The other samples (App. D), except sample three, give similar results as the sample used in this chapter. However sample three shows some deviation. This may be caused both by non-sufficient sample preparation and a wrong placement of the indents (lower-right part of figure 4.5).
20
4.3 ESEM results
Figures 4.9-4.13 show electron micrographs of the samples taken after nanoindentation. The orientation of the pictures is as follows. The top of the pictures is the tension side and the bottom of the pictures is the compression side.
In figures 4.9 and 4.10 a normal resin is used and in figures 4.11 and 4.12 a special resin is used. The special resin is used to minimize the charging, which it does by absorbing more of the electron activity. The normal resin does not absorb much of the electron activity, but instead it reflects electrons into all directions. This causes some noise in the pictures for which the normal resin is used.
In figures 4.9 and 4.10 it can clearly be seen that there seems to be no difference between the top and bottom of the specimen. Both faces are not perfectly flat due to the fact that during polishing some damage was induced. The dots in both figure 4.9 and 4.10 are not the result of nanoindentation, but undesired damage of the material due to the polishing
Figure 4.9 Sample2 ESEM-picture
Figure 4.10 Sample2 ESEM-picture
21
If there was any damage in the material, it would be expected on the tensile side of the sample. Therefore figure 4.11 gives a closer look at the top of the sample. This figure suggest the same information as figures 4.9 and 4.10: there is no visible trace of damage in the samples. To indicate the importance of using a good resin, figures 4.12 and 4.13 are shown. In figure 4.12 the special resin is used. The resin itself is not a very good quality. It does not completely solidify when molded. It therefore shows some irregularities. But the edge of the metal surface is very clear. And the surface of the metal is accurate. On the other hand, in figure 4.13, where the normal resin is used, the edge is barely visible. And it is hard to distinguish which part of the picture is the steel and which part is the resin. No conclusions can be drawn from this picture.
Figure 4.11 Sample 4 ESEM-picture
Figure 4.12 Sample 3 ESEM-picture Figure 4.13 Sample 2 ESEM-picture
22
5. Conclusion
In the introduction the question was raised if the hardness without any damage could be determined, by comparing the compression and tensile side of a uniformly bent specimen. However the problem with the current bending setup is that only a limited amount of strain can be reached. Figure 5.1 schematically shows the region which was examined by our tests. In chapter six some recommendations are made to increase the bending strain so that a larger strain regime can be examined.
In the region examined some conclusions can be made; the hardness at the compression side is the same as at the tensile side. Thus in this region there does not seem to be any damage. Before a sample undergoes the nanoindentation or ESEM-analysis, surfaces must be very carefully prepared and examined. The surfaces must be flat and polished for a long time (5 hours minimal is preferred). The results of a bad preparation are shown in Appendix A. These results can cause wrong conclusions. The conclusion of this is that the sample preparation should not be underestimated! Another point that can be concluded is that the special resin mentioned in the ESEM analysis, is an improvement to the previous one. However this resin is not perfect; because the solidifying process is also based on pressure the samples can possibly undergo a deformation while being molded. This possible deformation depends on the geometry of the sample. Also the resin cannot be completely solidified for some sample geometries, because the molding machine cannot create pressure in all holes, corners or edges.
Figure 5.1 Hardness vs. strain, the
colored region is the region that has
been studied in this report.
23
6. Recommendations for the future One finding in this report is the fact that only a limited amount of strain could be reached. To increase the available strain range some adjustments could be made to either the machine or the samples that are bent. Ideally, one would like to reach significantly higher strains in the bend, preferably close to 1.0 (=100%). A strain of 1.0 means that the y-
distance is equal to the Rn. This gives the situation sketched in figure 6.1.
However the problem that occurs is that there is no space for the clamps. Thus first of all the clamps of the bending device should be as small as possible to get the highest possible strains. But even with very small clamps 100% strain will never be reached. When the sample is bent like in figure 6.2 there is some space for the clamps; however to have the angle a high as possible the clamps should be as far away from the bend as possible: in point B there is a lot more space than in point A. So what would be desirable is that the clamps are far away while the radius of curvature remains small. This can possibly be done by including a thin part in the geometry of the sample so that this part accommodates the entire bending. The thick part may be as long as desired. Possible solutions are shown in figure 6.3.
Another way to have the clamps far away form the bend is by not thinning the sheet plate but making a smaller cross-section by decreasing the height, as sketched in figure 6.4 on the next page.
Figure 6.1 bent sample with 100%
strain
Figure 6.2 bent sample with less than
100% strain
Figure 6.3 Three possibilities for a thin part in the middle of the Metal plate sample; the green part is
where the clamps are holding the sample
24
The small part in the middle needs less torque to be bent than the rest of the sample. This way the small part will have a high angle and the length of the big parts can be made bigger. This way the clamps can be put far away from the bend. Even fir this geometry the maximum angle is limited by the clamps touching each other. A limited additional gain can therefore be obtained by an asymmetric geometry as sketched in figure 6.5, in which the clamps can never touch each other. The bent sample will then look like figure 6.6, and clearly shows that only one clamp will limit the angle.
Figure 6.4 plate sample with a small cross-section in the middle
Figure 6.5 Thin part at the end of he sample close to the clamp; non-bent
Figure 6.6 Thin part at the end of he sample close to the clamp; bent
25
Another possibility to let only one clamp limit the angle is by applying the profile of figure 6.5 to the height as sketched in figure 6.7.
It is possible to combine some of the previous possibilities to make the angle as high as possible and the radius as small as possible. Another option is to use different sorts of materials in one specimen. An example is a metal plate with a low E-modulus and Yield strength which on both sides is glued together with a metal (or other material) with high values and only a small part of the soft metal is not covered with the strong material, see figure 6.8.
The different geometry parameters can be adjusted as wanted. And of course the thickness should not be thicker than 3 mm, or new clamps should be made for thicker plates. New clamps can also be a solution for the strain problem. When using a thicker sheet the strains will also be higher when bending it like in figure 4.2a. Now the maximum thickness of the plate is 3 mm maybe this can be increased to a maximum of 5-10 mm. One disadvantage is that; the thicker the plate is, the bigger the required moment is. This means that the clamps should withstand greater forcer and thus be thicker also. Redesigning could also increase the amount of force that the clamp can withstand. With a plate of 10mm strains above 25% are possible. Another thing to improve is the resin that is used. It would be ideal to have a resin that is cured by drying and behaves like the new resin. The polishing and grinding done in this research took a lot of time. This can be improved. There is already an automatic grinder/polisher in the lab and it is recommended that the next person that needs to polish or grind any specimen does that by using that machine.
Figure 6.7 plate sample with a
small cross-section close to the
clamp
Figure 6.8 Plate sample (yellow) glued together with stronger material (green)
26
7. Appendix Appendix A: Wrong sample preparation
Picture when the specimen preparation is not done the right way or time.
All figures on this page do not have proper sample preparation. On all top sides of the pictures a very rough edge can be seen.
27
Appendix B: Calibration
Because the theoretical results are preferred to be linear the Calibration takes place in the linear and thus elastic area. The moment then can be written as Eq. (2.9).
nRbt
EM
1
112
1 3
2ν−=
M = Moment E = Young’s Modulus ν = Poisson’s ratio b = Plate width t = Thickness Rn = Middle Radius of curvature The region of calibration must be as big as possible, therefore a very tough and thick
material is used: Worksteel (E= 210 GP; 0yσ = 400MPa)
The machine itself outputs voltages. Therefore a table must me made that show the relation between the angle and voltage. The angle is not part of Eq.(2.10), but can be retrieved from Rn, neutral radius of curvature
3602
1020360
2
' 3
⋅⋅⋅
⋅−=⋅
⋅⋅=
−
nn R
l
R
l
ππα (B.1)
The length l is the total length of the sample but due to the clamping of 10mm on both sides the sample is only bent over the length l’. Furthermore it is important that bend stays
in the elastic region. Therefore with l x t x b = 75 x 3 x 100 [mm] minnR is calculated
mE
EE
t
Ey
ERyy
n 7875.0400
35.02102
6
39
00
min =⋅
=⋅=⋅=
−
σσ (B.2)
From which the maximal angle is obtained
oo
n
EE
R
l40016.4360
7875.02
2075360
2
1020 33
min
3
max ≈=⋅⋅⋅
−=⋅
⋅⋅
⋅−=
−−−
ππα (B.3)
To be in the safety zone the bending will be carried out to o1 . With Eq. (2.10), (B.1), (B.2), (B.3) and the Calibration itself here below, table 4.1 can be obtained.
Figure 4.1 dimensions of the calibration
specimen
28
Appendix C: Radius measurements
29
Appendix D: Result of the nanoindentations
Sample 1
Y-position vs. Hardness Sample 1
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
0 70 140 210 280 350 420 490 560 630 700
Compession side Y-position [µm] Tensile side
Hard
ness [
GP
a]
Compression vs. Tensile side Sample 1
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
0,0 1,0 2,0 3,0 4,0 5,0 6,0 7,0 8,0 9,0 10,0 11,0 12,0 13,0
Engineering strain [%]
Hard
ness [
GP
a]
Compression
Tensile
30
Sample 3
Y-position vs. Hardness Sample 3
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
0 70 140 210 280 350 420 490 560 630 700
Compession side Y-position [µm] Tensile side
Hard
ness [
GP
a]
Compression vs. Tensile side Sample 3
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
0,0 1,0 2,0 3,0 4,0 5,0 6,0 7,0 8,0 9,0 10,0 11,0 12,0 13,0
Engineering strain [%]
Hard
ness [
GP
a]
Compression
Tensile
31
Sample4
Y-position vs. Hardness Sample 4
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
0 70 140 210 280 350 420 490 560 630 700
Compession side Y-position [µm] Tensile side
Hard
ness [
GP
a]
Compression vs. Tensile side Sample 4
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
0,0 1,0 2,0 3,0 4,0 5,0 6,0 7,0 8,0 9,0 10,0 11,0 12,0 13,0
Engineering strain [%]
Hard
ness [
GP
a]
Compression
Tensile
32
8. Reference [1] Lemaitre, J; A course on damage mechanics, Springer-Verlag, 1996 [2] C.Tasan, Quantification of ductile damage by nanoindentation; Mate Poster 2007 [3] Roget T. Fenner; Mechanics of solids, Blackwell Scientific Publications 1989 Pages 103-105 [4] Z. Marciniak and J Duncan, Mechanics of Sheet metal forming. British Library Cataloguing in Publication data. St. Edmundsbury Press, Bury St., Bending
Pages 68-81 [5] Dieter, G.E., Mechanical Metallurgy, McGraw-Hill 1961 [6] ASMH Handbook 8: Mechanical testing and Evaluation ASM International Handbook committee 2000, Stress-strain behavior in bending. P.Dadras Pages 109-114 [7] S.H.A. Boers. Optimal Forming Strategies with a 3D Reconfigurable Die.
Universiteitsdrukkerij TU Eindhoven, Directional hardening for complex strain path changes. Pages 81-84.
[8] W.C. Oliver and G.M. Pharr. Measurement of hardness and elastic modulus by
Instrumented indentation: Advances in understanding and refinement to
methodolygy. Material Research Society 2004. [9] Y.C. Fung and Pin Tong. Classical and computational solid mechanics. World
scientific publishing Co. Pte. Ltd. 2001. Linearized theory of elasticity, bending beams, pages 224-229.