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Homogenisation of Thickness through High Viscous Fluid Flow
Homogenisation of Thickness through High Viscous Fluid Flow 1
F. Vollertsen’(2), H. Schulze Niehoff’ Department for Metal Forming Technologies University of Paderborn, Germany
Abstract This paper describes investigations on new ways to influence sheet metal forming processes through a new parameter. The new adjustable parameter is a shear stress on the surface of the sheet applied during the forming process through high viscous fluid flows. Parallel and hourglass shaped tensile test specimen were investigated, whereby a fluid flow was applied from the ends to the middle of the specimens. The results clearly show a significant influence of the high viscous fluid flows on the specimens. For the hourglass shaped specimen it could be shown that the inhomogeneous cross section distribution could be compensated by the applied shear stresses, so that a homogenous sheet thickness distribution was reached. Keywords: Sheet metal, forming, flow
1 INTRODUCTION Sheet metal forming technologies have reached a high standard in the last decades, which was mainly driven by the automotive industry. But the demand of consumers to more sophisticated designed parts is still rising. This leads many manufacturers to the challenge to produce more and more complex shaped parts, for which conventional methods often reach their limits. Apart from the increase of complexity, another trend is the replacement of deep drawing steels by high strength steels or aluminium alloys, which are much more unlikely to reach the demanded deep drawing ratio. The limits in sheet metal forming operations are often reached, because local thinning occurs at critical areas such as radii at the bottom of a deep drawn cup. To allocate the sheet thickness at the points where it is needed, tailored blanks or flexible rolled blanks can be used [ I ] . Alternatively the stress distribution can be influenced through partial coating or the material properties can be changed locally to influence the material flow by a local heating [2]. In addition to the strategies mentioned above a new parameter was found to influence the stress distribution in sheet metal forming processes and thus to use the formability of the whole part. This paper presents a method using a high viscous fluid flow, which runs over a metal sheet while the metal sheet is formed and will be influenced by the fluid flow significantly in the forming resuIt. 2 METHOD The influence of the fluid flow on the forming process is caused by shear stresses. Figure 1 shows a flow channel with a fluid flow between the walls. This fluid flow will generate a certain velocity distribution, which is shown in the figure. If the adhesion of the fluid is big enough, the velocity at the wall will be zero and thus the velocity in the centre will be at its maximum. It was found out by fluid mechanics that under these conditions a parabolic velocity distribution of the fluid flow over the cross section
of the flow channel will be given, as it is shown in figure 1 [31.
Figure 1: Shear stress application in a flow channel This velocity distribution will again lead to a certain distribution of the applied shear stress over the cross section of the flow channel. The connectivity of the velocity distribution to the shear stress can be seen in equation 1 [3]:
Equation 1 shows that the shear stress T increases, if the shear rate 9 increases, whereby the shear rate is defined by the change of the velocity over the change of the height over the cross section of the flow channel. This means that the shear stress is at its maximum where the highest change in velocity occurs; in our case at the wall. No shear stress occurs, where the change of the velocity is zero, usually in the middle of the cross section of a flow channel. The resulting shear stress distribution has a double-triangular shape as it is shown in figure 1. Apart from the shear rate, the shear stress has a linear dependency on the viscosity q of the fluid, which is been pushed through the flow channel, whereby an increasing viscosity leads to an increasing shear stress. The surface finish does not influence the applied shear stress at the wall, if laminar flow is given. In our case Reynolds’ number is in the order of lo6, which means that the fluid flow is far in the laminar range [3].
3 CHOICE OF FLUID The number of suitable fluids for these investigations is limited to a small group due to some restrictions, which have to be fulfilled: As said before, the fluid has to have a sufficient adhesion to ensure a parabolic velocity distribution. Otherwise the velocity distribution over the cross section of the flow channel would be constant. For this condition it can generally be said, that the adhesion of the fluid has to be equal or higher than the cohesion of the fluid. This condition is often fulfilled for fluids with a low viscosity (e.g. water), but fluids with a comparable high viscosity appear as quite gluey if they fulfil this condition (e.g . honey). As denoted above, a comparable high viscosity is needed. To reach a sufficient shear stress, the viscosity of the fluid has to be very high, at least at 100 Pas. It is furthermore desirable to have a fluid, which remains constant with its viscosity while the shear rate is increasing (Newtonian fluid). But this can almost only been found for oils, which are not gluey enough. In addition to that, many gluey and high viscous fluids show a very quick core, which has to be avoided to maintain a smooth process. For the presented investigations glucose was chosen, which is not a Newtonian fluid, but shows a shear thinning, which means that the viscosity is decreasing with an increasing shear rate (see figure 2). The viscosity of the glucose is between 140 and 300 Pas in the shear rate range of 3500 to 1000 l / s , in which the experiments took place. It was measured by a double capillary rheometer and adjusted by Bargley's and WeilJenbergRabinowitsch's corrections.
The testing tool was designed to allow the application of tensile forces on the ends of the specimen, as well as a fluid flow running from top and bottom of the specimen to the centre of the specimen, as shown in figure 3, whereby the gap between sheet and wall of the tool will be the flow channel. Different gap heights could have been tried out through the interchangeable inserts, so that the geometry of the flow channel and thus the shear rate and the applied shear stress could be varied. The fluid flow was then generated by a hydraulic press, which pushed the fluid through the flow channels using a punch plunging in a cylinder, filled with the fluid. The shear rate and thus the shear stress could then also be varied by changing the punch speed.
L1(
1000
Figure 3: Experimental setup
g
100
The geometry of the used specimens is shown in figure 4, whereby the weakened spot could be varied through different values of the dimension a. ,R 10
0 0
tn
5
10 10
100
1Is
10000
Shear rate Figure 2: Viscosity of glucose depending on the shear rate. It was found out in former investigations that the product of shear rate and viscosity is increasing with an increasing shear rate, which means that the applied shear stress is also increasing with an increasing shear rate [4]. The glucose is high viscous and has still a higher adhesion than its cohesion. It can also be varied in its viscosity by just adding water. Water can also be used as a solvent to remove the glucose from the metal sheet after the process. The glucose does not harm either the environment nor the health of human beings as many other adhesives or at least their solvents do. EXPERIMENTAL SETUP For the investigations on the homogenisation of the sheet thickness during a forming process through high viscous fluid flows a flat tensile test specimen was chosen. The tensile test specimen is hourglass-shaped and is thus geometrically weakened in the centre. This weakened spot represents critical areas of a sheet metal in a forming process, where e.g. thinning occurs. If this specimen is pulled apart under conventional conditions, a higher thinning at the centre will occur compared to areas with higher initial cross section.
2ocI
40T I
- 1
1-
Figure 4: Geometry of the Specimen 5
EXPERIMENTAL RESULTS
5.1 Shear stress distribution For a closer look on the experiments on the homogenisation of thickness through high viscous fluid flows, it is helpful to know not only the amount of shear stress, which was determined in former investigations [4], but also its distribution over the length of the specimen. The experiments were realised through a rectangular specimen (40 x 100 mm), which was hold into the fluid flow and then smoothly pulled out of the flow channel, so that the length I,, of the specimen engaged with the fluid flow was decreasing, see figure 5.
4
0
n
n+l
100 mm
I b Position Figure 5: Calculation of the shear stress distribution I
I
The differences of the measured forces F then gave the shear stress in each area between position n and n+l by equation 2. Whereby the width b is constant and a factor 2 has to been taken into account due to a fluid flow on both sides of the specimen.
t
nfn a C
Y 0
.5
-
n+l -
CI
a a r
Figure 6 shows the measured shear stress over the length of the specimen given by the force, whereby the meaning of the position is explained in figure 3 and 5. This curve has a systematic error included, shown by the curve indicated as correction value. This error results from an increasing velocity Av of the fluid flow, because the resistance caused by the geometry of the flow channel is decreasing if the specimen is more and more pulled out of the flow channel, whereby the force, which pushes the fluid through the flow channel remains constant. The correction value was derived by several experiments under constant conditions apart from the velocity of the fluid flow. These experiments then gave the increase of shear stress AT caused by the increase of the velocity Av of the fluid flow. The resulting curve, shown as corrected shear stress, varies between 0.13 and 0.07 N/mm2 according to no detectable law. Reruns showed that the standard deviation is 0.03 MPa, so that it can be assumed that the values spread in the range of accuracy of measurement and that the shear stress distribution is uniform over the length of the specimen, according to equation 1.
v)
-100
-50
0
mm
100
Position Figure 7: Influence of shear stress; a = 19.5 mm, elongation: 10 mm, material: mild steel, initial thickness: 0.47 - 0.48 mm
The behaviour of specimens with a weakening of width from 20 to a = 18.5 mm compared to those with a weakening from 20 to a = 19.5 mm are shown in figure 8. The curve with a shear stress of zero and an a-value of 18.5 mm shows an even higher thinning at the weakened spot than the curve with an a-value of 19.5. This thinning could be compensated by a shear stress of 0.13 MPa and a comparable homogenous thickness distribution was reached, whereby the specimen with an a-value of 19.5 mm was overcompensated by a shear stress of 0.13 MPa.
t
fn fn
a C
Y
0
c
a
a c rn
-100
-
Position Figure 6: Shear stress distribution, standard deviation: 0.03 MPa 5.2 Homogenisation of Thickness The experimental results on the homogenisation of thickness through high viscous fluid flows during a tensile test of a geometrically weakened specimen are visualised by the sheet thickness distribution. Figure 7 shows three curves with three different shear stresses applied during the tensile test. The curve with a shear stress of zero shows a thinning of the sheet at the geometrically weakened spot and a proportional increasing thickness with an increasing initial cross section of the specimen, as expected. The other two curves show that the weakened spot could not only be compensated by the shear stresses of 0.13 and 0.14 MPa, but overcompensated, so that the thinning occurred at the point where the highest initial cross section was. The highest sheet thickness was then at the point with the lowest initial cross section, so that a homogenous thickness distribution could be reached with a shear stress lower than 0.13 MPa for a specimen with a width, weakened from 20 to a = 19.5 mm.
-50
0
mm
100
Position Figure 8: Influence of shape; elongation: 10 mm, material: mild steel, initial thickness: 0.47 - 0.48 mm
5.3 Normal and equivalent Stress distribution The experimental results of the sheet thickness distribution can be explained, if the normal and equivalent stress distribution is calculated. A normal stress calculation based on experimental data is shown in figure 9 for a shear stress of 0.13 MPa and an weakening in the width from 20 to 19.5 mm. It was assumed that the shear stress is uniform over the whole specimen. Then the normal stress was calculated, which occurs at the beginning of the experiment when the plastic deformation starts. Figure 9 shows the initial width of the specimen, which goes from 20 mm at the ends of the specimen to 19.5 mm at the centre. From this width the resulting force distribution can be calculated, which is also shown in figure 9. The force at any point of the specimen is the measured force at the ends of the specimen reduced by the force induced by the shear stress, which is the shear stress times the trapezoid area between the end of the specimen and the point where the force is calculated for. The stress can then be calculated at each point by the calculated force divided by the initial width and the initial thickness at this point.
The result shows a normal stress distribution with a minimum at the geometrical weakened spot induced by the applied shear stress, which leads to a thinning at the point of highest initial cross section.
t
5
s t
Shear stress-
al
Figure 11: Sheet thickness difference between position 0 and 75 mm and equivalent stress at position 0 over shear stress; a = 19.5 mm, material: mild steel, initial thickness: 0.475 mm, elongation: 10 mm
2
0
LL
t v) v)
al
I
m
-70
-35
0 Position
mm
70
Figure 9: normal stress distribution; a = 19.5 mm, z = 0.13 MPa, material: mild steel, initial thickness: 0.475 mm Figure 10 shows the same normal stress distribution curve and additionally the curve for a shear stress of zero MPa and 0.06 MPa, which was approximated by an interpolated tensile force. From these normal stresses the equivalent stress (also shown in figure 10) according to Tresca can be calculated by adding the amount of the pressure to the normal stress. The pressure is needed to realise the fluid flow and decreases from the ends of the specimen to the centre. The equivalent stress curves in figure 10 show a good agreement to the sheet thickness distribution in figure 7: A high equivalent stress of 438 MPa in the centre (z = 0.0 MPa) leads to a high thinning, whereby a comparable low equivalent stress of 417 MPa in the centre (z = 0.13 MPa) leads to a higher remaining thickness. 440
tI
MPa
$
425
v) v)
6 CONCLUSIONS 1. Shear stresses can be applied through fluid flows. A high viscosity and a high shear rate lead to high shear stresses, provided that the adhesion of the fluid is higher than the cohesion. 2. Glucose is a suitable fluid with high viscosity and high adhesion. The glucose can be removed with water and is a natural product. 3. Hourglass-shaped tensile test specimens were used to represent critical situations in sheet metal forming. 4. The shear stress distribution induced by the fluid flow can be seen as uniform in a certain spectrum, where measured values show a certain scatter. 5. The results of the experimental investigation show that geometrically weakened areas could be compensated by the applied shear stress induced by the high viscous fluid flows and thus a homogenisation of thickness can be reached.
7 OUTLOOK It is thought to transfer this new parameter into internal high pressure forming processes of thin walled parts, where the limit of formability should be increased. Investigations on shear stress applications on tubes with internal high pressure are currently done. Future work will also reflect the issue of dimensional accuracy.
430
420 415
-
0 mm 70 Position Figure 10: Normal stress oNand equivalent stress oE distribution; a = 19.5 mm, material: mild steel, initial thickness: 0.475 mm, elongation: 10 mm -70
The approximated equivalent stress for a shear stress of 0.06 MPa has a good uniform distribution, so that at this point a homogenous thickness distribution can be assumed. This assumption can be supported by figure 11, where the difference in sheet thickness between position 0 and 75 mm from figure 7 is plotted. A thickness difference of zero means a homogenous thickness distribution. This can be found at the point of intersection between the thickness difference curve and the straight line of zero, which is at a shear stress of 0.055 MPa, indicated as theoretical value. The corresponding equivalent stress at the geometrical weakened spot shows a decrease with an increasing shear stress.
8 ACKNOWLEDGMENTS The work reported in this paper was funded by the DFG within the project “Influence of viscous flows on the stress conditions during internal high pressure forming of thin walled parts”, VO 53013. The authors gratefully acknowledge the support by cand. Wirt.-lng. T. Meier and cand. Wirt.-lng. J. Sturmann in conducting the experiments. 9
REFERENCES [ I ] Ebert, A,, Kopp, R., 1999, Process Optimisation of Flexible Rolling and Stretch Forming of Sheet, Proceedings of 6‘h ICTP, Nurnberg, Germany, Vol. 3, 2083-2090. [2] Beckmann, M., Vollertsen, F., 2002, The influence of a Local Heat Treatment on the Drawing Properties of an Aluminium Alloy, ATTCE Proceedings, Barcelona, Vol. 4, 207-212. [3] Reiner, M., 1968 Rheologie in elementarer Darstellung, Hanser, Munchen.
[4]
Schulze Niehoff, H., Vollertsen, F., 2002, Principle of hydroforming influenced by high viscous fluid flows, Proceedings of 7‘h ICTP, Yokohama, Japan, Vol. 2, 1447-1452.
F. Vollertsen’(2), H. Schulze Niehoff’ Department for Metal Forming Technologies University of Paderborn, Germany
Abstract This paper describes investigations on new ways to influence sheet metal forming processes through a new parameter. The new adjustable parameter is a shear stress on the surface of the sheet applied during the forming process through high viscous fluid flows. Parallel and hourglass shaped tensile test specimen were investigated, whereby a fluid flow was applied from the ends to the middle of the specimens. The results clearly show a significant influence of the high viscous fluid flows on the specimens. For the hourglass shaped specimen it could be shown that the inhomogeneous cross section distribution could be compensated by the applied shear stresses, so that a homogenous sheet thickness distribution was reached. Keywords: Sheet metal, forming, flow
1 INTRODUCTION Sheet metal forming technologies have reached a high standard in the last decades, which was mainly driven by the automotive industry. But the demand of consumers to more sophisticated designed parts is still rising. This leads many manufacturers to the challenge to produce more and more complex shaped parts, for which conventional methods often reach their limits. Apart from the increase of complexity, another trend is the replacement of deep drawing steels by high strength steels or aluminium alloys, which are much more unlikely to reach the demanded deep drawing ratio. The limits in sheet metal forming operations are often reached, because local thinning occurs at critical areas such as radii at the bottom of a deep drawn cup. To allocate the sheet thickness at the points where it is needed, tailored blanks or flexible rolled blanks can be used [ I ] . Alternatively the stress distribution can be influenced through partial coating or the material properties can be changed locally to influence the material flow by a local heating [2]. In addition to the strategies mentioned above a new parameter was found to influence the stress distribution in sheet metal forming processes and thus to use the formability of the whole part. This paper presents a method using a high viscous fluid flow, which runs over a metal sheet while the metal sheet is formed and will be influenced by the fluid flow significantly in the forming resuIt. 2 METHOD The influence of the fluid flow on the forming process is caused by shear stresses. Figure 1 shows a flow channel with a fluid flow between the walls. This fluid flow will generate a certain velocity distribution, which is shown in the figure. If the adhesion of the fluid is big enough, the velocity at the wall will be zero and thus the velocity in the centre will be at its maximum. It was found out by fluid mechanics that under these conditions a parabolic velocity distribution of the fluid flow over the cross section
of the flow channel will be given, as it is shown in figure 1 [31.
Figure 1: Shear stress application in a flow channel This velocity distribution will again lead to a certain distribution of the applied shear stress over the cross section of the flow channel. The connectivity of the velocity distribution to the shear stress can be seen in equation 1 [3]:
Equation 1 shows that the shear stress T increases, if the shear rate 9 increases, whereby the shear rate is defined by the change of the velocity over the change of the height over the cross section of the flow channel. This means that the shear stress is at its maximum where the highest change in velocity occurs; in our case at the wall. No shear stress occurs, where the change of the velocity is zero, usually in the middle of the cross section of a flow channel. The resulting shear stress distribution has a double-triangular shape as it is shown in figure 1. Apart from the shear rate, the shear stress has a linear dependency on the viscosity q of the fluid, which is been pushed through the flow channel, whereby an increasing viscosity leads to an increasing shear stress. The surface finish does not influence the applied shear stress at the wall, if laminar flow is given. In our case Reynolds’ number is in the order of lo6, which means that the fluid flow is far in the laminar range [3].
3 CHOICE OF FLUID The number of suitable fluids for these investigations is limited to a small group due to some restrictions, which have to be fulfilled: As said before, the fluid has to have a sufficient adhesion to ensure a parabolic velocity distribution. Otherwise the velocity distribution over the cross section of the flow channel would be constant. For this condition it can generally be said, that the adhesion of the fluid has to be equal or higher than the cohesion of the fluid. This condition is often fulfilled for fluids with a low viscosity (e.g. water), but fluids with a comparable high viscosity appear as quite gluey if they fulfil this condition (e.g . honey). As denoted above, a comparable high viscosity is needed. To reach a sufficient shear stress, the viscosity of the fluid has to be very high, at least at 100 Pas. It is furthermore desirable to have a fluid, which remains constant with its viscosity while the shear rate is increasing (Newtonian fluid). But this can almost only been found for oils, which are not gluey enough. In addition to that, many gluey and high viscous fluids show a very quick core, which has to be avoided to maintain a smooth process. For the presented investigations glucose was chosen, which is not a Newtonian fluid, but shows a shear thinning, which means that the viscosity is decreasing with an increasing shear rate (see figure 2). The viscosity of the glucose is between 140 and 300 Pas in the shear rate range of 3500 to 1000 l / s , in which the experiments took place. It was measured by a double capillary rheometer and adjusted by Bargley's and WeilJenbergRabinowitsch's corrections.
The testing tool was designed to allow the application of tensile forces on the ends of the specimen, as well as a fluid flow running from top and bottom of the specimen to the centre of the specimen, as shown in figure 3, whereby the gap between sheet and wall of the tool will be the flow channel. Different gap heights could have been tried out through the interchangeable inserts, so that the geometry of the flow channel and thus the shear rate and the applied shear stress could be varied. The fluid flow was then generated by a hydraulic press, which pushed the fluid through the flow channels using a punch plunging in a cylinder, filled with the fluid. The shear rate and thus the shear stress could then also be varied by changing the punch speed.
L1(
1000
Figure 3: Experimental setup
g
100
The geometry of the used specimens is shown in figure 4, whereby the weakened spot could be varied through different values of the dimension a. ,R 10
0 0
tn
5
10 10
100
1Is
10000
Shear rate Figure 2: Viscosity of glucose depending on the shear rate. It was found out in former investigations that the product of shear rate and viscosity is increasing with an increasing shear rate, which means that the applied shear stress is also increasing with an increasing shear rate [4]. The glucose is high viscous and has still a higher adhesion than its cohesion. It can also be varied in its viscosity by just adding water. Water can also be used as a solvent to remove the glucose from the metal sheet after the process. The glucose does not harm either the environment nor the health of human beings as many other adhesives or at least their solvents do. EXPERIMENTAL SETUP For the investigations on the homogenisation of the sheet thickness during a forming process through high viscous fluid flows a flat tensile test specimen was chosen. The tensile test specimen is hourglass-shaped and is thus geometrically weakened in the centre. This weakened spot represents critical areas of a sheet metal in a forming process, where e.g. thinning occurs. If this specimen is pulled apart under conventional conditions, a higher thinning at the centre will occur compared to areas with higher initial cross section.
2ocI
40T I
- 1
1-
Figure 4: Geometry of the Specimen 5
EXPERIMENTAL RESULTS
5.1 Shear stress distribution For a closer look on the experiments on the homogenisation of thickness through high viscous fluid flows, it is helpful to know not only the amount of shear stress, which was determined in former investigations [4], but also its distribution over the length of the specimen. The experiments were realised through a rectangular specimen (40 x 100 mm), which was hold into the fluid flow and then smoothly pulled out of the flow channel, so that the length I,, of the specimen engaged with the fluid flow was decreasing, see figure 5.
4
0
n
n+l
100 mm
I b Position Figure 5: Calculation of the shear stress distribution I
I
The differences of the measured forces F then gave the shear stress in each area between position n and n+l by equation 2. Whereby the width b is constant and a factor 2 has to been taken into account due to a fluid flow on both sides of the specimen.
t
nfn a C
Y 0
.5
-
n+l -
CI
a a r
Figure 6 shows the measured shear stress over the length of the specimen given by the force, whereby the meaning of the position is explained in figure 3 and 5. This curve has a systematic error included, shown by the curve indicated as correction value. This error results from an increasing velocity Av of the fluid flow, because the resistance caused by the geometry of the flow channel is decreasing if the specimen is more and more pulled out of the flow channel, whereby the force, which pushes the fluid through the flow channel remains constant. The correction value was derived by several experiments under constant conditions apart from the velocity of the fluid flow. These experiments then gave the increase of shear stress AT caused by the increase of the velocity Av of the fluid flow. The resulting curve, shown as corrected shear stress, varies between 0.13 and 0.07 N/mm2 according to no detectable law. Reruns showed that the standard deviation is 0.03 MPa, so that it can be assumed that the values spread in the range of accuracy of measurement and that the shear stress distribution is uniform over the length of the specimen, according to equation 1.
v)
-100
-50
0
mm
100
Position Figure 7: Influence of shear stress; a = 19.5 mm, elongation: 10 mm, material: mild steel, initial thickness: 0.47 - 0.48 mm
The behaviour of specimens with a weakening of width from 20 to a = 18.5 mm compared to those with a weakening from 20 to a = 19.5 mm are shown in figure 8. The curve with a shear stress of zero and an a-value of 18.5 mm shows an even higher thinning at the weakened spot than the curve with an a-value of 19.5. This thinning could be compensated by a shear stress of 0.13 MPa and a comparable homogenous thickness distribution was reached, whereby the specimen with an a-value of 19.5 mm was overcompensated by a shear stress of 0.13 MPa.
t
fn fn
a C
Y
0
c
a
a c rn
-100
-
Position Figure 6: Shear stress distribution, standard deviation: 0.03 MPa 5.2 Homogenisation of Thickness The experimental results on the homogenisation of thickness through high viscous fluid flows during a tensile test of a geometrically weakened specimen are visualised by the sheet thickness distribution. Figure 7 shows three curves with three different shear stresses applied during the tensile test. The curve with a shear stress of zero shows a thinning of the sheet at the geometrically weakened spot and a proportional increasing thickness with an increasing initial cross section of the specimen, as expected. The other two curves show that the weakened spot could not only be compensated by the shear stresses of 0.13 and 0.14 MPa, but overcompensated, so that the thinning occurred at the point where the highest initial cross section was. The highest sheet thickness was then at the point with the lowest initial cross section, so that a homogenous thickness distribution could be reached with a shear stress lower than 0.13 MPa for a specimen with a width, weakened from 20 to a = 19.5 mm.
-50
0
mm
100
Position Figure 8: Influence of shape; elongation: 10 mm, material: mild steel, initial thickness: 0.47 - 0.48 mm
5.3 Normal and equivalent Stress distribution The experimental results of the sheet thickness distribution can be explained, if the normal and equivalent stress distribution is calculated. A normal stress calculation based on experimental data is shown in figure 9 for a shear stress of 0.13 MPa and an weakening in the width from 20 to 19.5 mm. It was assumed that the shear stress is uniform over the whole specimen. Then the normal stress was calculated, which occurs at the beginning of the experiment when the plastic deformation starts. Figure 9 shows the initial width of the specimen, which goes from 20 mm at the ends of the specimen to 19.5 mm at the centre. From this width the resulting force distribution can be calculated, which is also shown in figure 9. The force at any point of the specimen is the measured force at the ends of the specimen reduced by the force induced by the shear stress, which is the shear stress times the trapezoid area between the end of the specimen and the point where the force is calculated for. The stress can then be calculated at each point by the calculated force divided by the initial width and the initial thickness at this point.
The result shows a normal stress distribution with a minimum at the geometrical weakened spot induced by the applied shear stress, which leads to a thinning at the point of highest initial cross section.
t
5
s t
Shear stress-
al
Figure 11: Sheet thickness difference between position 0 and 75 mm and equivalent stress at position 0 over shear stress; a = 19.5 mm, material: mild steel, initial thickness: 0.475 mm, elongation: 10 mm
2
0
LL
t v) v)
al
I
m
-70
-35
0 Position
mm
70
Figure 9: normal stress distribution; a = 19.5 mm, z = 0.13 MPa, material: mild steel, initial thickness: 0.475 mm Figure 10 shows the same normal stress distribution curve and additionally the curve for a shear stress of zero MPa and 0.06 MPa, which was approximated by an interpolated tensile force. From these normal stresses the equivalent stress (also shown in figure 10) according to Tresca can be calculated by adding the amount of the pressure to the normal stress. The pressure is needed to realise the fluid flow and decreases from the ends of the specimen to the centre. The equivalent stress curves in figure 10 show a good agreement to the sheet thickness distribution in figure 7: A high equivalent stress of 438 MPa in the centre (z = 0.0 MPa) leads to a high thinning, whereby a comparable low equivalent stress of 417 MPa in the centre (z = 0.13 MPa) leads to a higher remaining thickness. 440
tI
MPa
$
425
v) v)
6 CONCLUSIONS 1. Shear stresses can be applied through fluid flows. A high viscosity and a high shear rate lead to high shear stresses, provided that the adhesion of the fluid is higher than the cohesion. 2. Glucose is a suitable fluid with high viscosity and high adhesion. The glucose can be removed with water and is a natural product. 3. Hourglass-shaped tensile test specimens were used to represent critical situations in sheet metal forming. 4. The shear stress distribution induced by the fluid flow can be seen as uniform in a certain spectrum, where measured values show a certain scatter. 5. The results of the experimental investigation show that geometrically weakened areas could be compensated by the applied shear stress induced by the high viscous fluid flows and thus a homogenisation of thickness can be reached.
7 OUTLOOK It is thought to transfer this new parameter into internal high pressure forming processes of thin walled parts, where the limit of formability should be increased. Investigations on shear stress applications on tubes with internal high pressure are currently done. Future work will also reflect the issue of dimensional accuracy.
430
420 415
-
0 mm 70 Position Figure 10: Normal stress oNand equivalent stress oE distribution; a = 19.5 mm, material: mild steel, initial thickness: 0.475 mm, elongation: 10 mm -70
The approximated equivalent stress for a shear stress of 0.06 MPa has a good uniform distribution, so that at this point a homogenous thickness distribution can be assumed. This assumption can be supported by figure 11, where the difference in sheet thickness between position 0 and 75 mm from figure 7 is plotted. A thickness difference of zero means a homogenous thickness distribution. This can be found at the point of intersection between the thickness difference curve and the straight line of zero, which is at a shear stress of 0.055 MPa, indicated as theoretical value. The corresponding equivalent stress at the geometrical weakened spot shows a decrease with an increasing shear stress.
8 ACKNOWLEDGMENTS The work reported in this paper was funded by the DFG within the project “Influence of viscous flows on the stress conditions during internal high pressure forming of thin walled parts”, VO 53013. The authors gratefully acknowledge the support by cand. Wirt.-lng. T. Meier and cand. Wirt.-lng. J. Sturmann in conducting the experiments. 9
REFERENCES [ I ] Ebert, A,, Kopp, R., 1999, Process Optimisation of Flexible Rolling and Stretch Forming of Sheet, Proceedings of 6‘h ICTP, Nurnberg, Germany, Vol. 3, 2083-2090. [2] Beckmann, M., Vollertsen, F., 2002, The influence of a Local Heat Treatment on the Drawing Properties of an Aluminium Alloy, ATTCE Proceedings, Barcelona, Vol. 4, 207-212. [3] Reiner, M., 1968 Rheologie in elementarer Darstellung, Hanser, Munchen.
[4]
Schulze Niehoff, H., Vollertsen, F., 2002, Principle of hydroforming influenced by high viscous fluid flows, Proceedings of 7‘h ICTP, Yokohama, Japan, Vol. 2, 1447-1452.