1. Study on Explosive Forming of Aluminum Alloy
Multiphysics 2009, Session 2.2 Impact and Explosions, 10 Dec., University of Science and Technology of Lille, France.
Study on Explosive Forming of Aluminum Alloy
Hirofumi Iyama,　Kumamoto National College of Technology, Japan
Shigeru Itoh, 　Shock Wave and Condensed Matter
　Research Center, Kumamoto Univ., Japan
Thank you Mr. Chairman.
I’m Hirofumi Iyama, Yatsushiro National College of Technology, from Japan.
I’d like to talk about Numerical Simulation of Sheet Metal Forming Using Underwater Shock Wave.
Kumamoto National College of Technology

2. Contents 1. Introduction ・Explosive forming method
2. Experimental method
・Aluminum alloy forming
3. Experimental results
・Deformation shape of aluminum alloy
4. Simulation method
・Simulation model
・Calculation of pressure for explosive and water
・Constitutive equation of aluminum alloy
Contents of this presentation
The first, Introduction, I will explain the basis of explosive forming.
The second, experimental method, the aluminum alloy forming　was carried out. The aluminum alloy is used as the parts of car.
The third, experimental results, The forming shape of explosive forming with usual static forming, punching.
The fourth, Simulation method,
We have done the numerical simulation for this aluminum forming using Finite Difference Method.
The fifth, Simulation result
At the finally, conclusion.
5. Simulation result
・Pressure contour of water part
・Deformation process and velocity of aluminum alloy
6. Conclusion

3. Introduction Explosive Forming Underwater shock wave
This slide shows, a general explosive forming method.
A metal plate is held in place by a blank holder and a metal die is connected.
After detonation, an underwater shock wave was generated and propagated towards the metal plate and then hit it.
In this method, a bubble wave occurs after the shock wave and is also propagated towards the metal plate and hits it.
This metal plate is deformed by these loading and collides with the metal die.
In the end the deformed surface of the metal plate has come to match the shape of the metal die.
The feature of explosive forming that allows the die to sufficiently transfer its shape onto the plate is that the spring-back effect is less than with ordinary forming methods, like punching and hydro bulging.
So, we considered using this method for aluminum alloy forming. 
However, the forming of these materials as sheet metal by static methods, such as hydro bulge forming or general punching, is difficult for the aluminum forming because the aluminum alloy allow little elongation compared to conventional steel.
In order to examine the application of explosive forming, we tried free forming aluminum alloy as the basis of the study.
This slide shows, a general explosive forming method.
A metal die and metal plate set underwater, and an explosive sets upper side.
After detonation, an underwater shock wave was generated and propagated towards the metal plate and then hit it.
This metal plate is deformed by shock loading and collides with the metal die.
The feature of explosive forming that allows the die to sufficiently transfer its shape onto the plate is that the spring-back effect is less than with
ordinary forming methods, like punching and hydro bulging.
So, we considered using this method for aluminum alloy forming. 
However, the aluminum alloy forming by static methods, such as hydro bulge forming or general punching, is difficult for
the aluminum forming, because the aluminum alloy is little elongation compared with steel.
In order to examine the application of explosive forming, we tried free forming of aluminum alloy as the basis of the study.

4. Experimental equipment
Aluminum plate： A5052-O
Explosive： SEP
Detonation velocity: 6970m/s
Detonation pressure: 15.9 GPa.
This slide shows an experimental equipment.
The aluminum plate was installed between the metal die and the blank holder, and the explosive was connected to the tip of the electric detonator as seen here.
However, in the results shown by this study’s equipment, the bubble wave went off in an upper direction and there was good effect on the aluminum plate.
The aluminum plate was A5052-O
The explosive SEP is used, which was provided by Asahi Kasei Corp.
The detonation velocity was 6970m/s, and the detonation pressure was 15.9 GPa.
This slide shows an experimental equipment.
The aluminum plate was installed between the metal die and the blank holder, and the explosive was connected to the tip of the electric detonator . We used a paper container filled with water.
We used explosive SEP, this is a plastic high explosive and is provided by Asahi Kasei Chemical Corp.

5. Experimental results φ105 φ100 R4 Press forming equipment
Press type: mechanical press
Press cycle: 35[cycle/min]
Lubrication: HT103
Explosive mass: 10 [g]
Bulge depth: 39 [mm]
Sectional form
Press forming
Bulge depth: 28 [mm]
These figures show comparative forming results.
In the case of static punching, we used a punch of a 100mm diameter and a metal die has a 105mm diameter hole.
The explosive forming limit, shown in the upper figure, can be attained at 10g of explosive, and the bulge depth was 39mm.
The configuration obtained by static press forming is shown in the lower figure. The maximum bulge depth was 28 mm.
A diagram of the results using the static press method under those from explosive forming is shown in the right hand figure.
Comparison of the two configurations makes it clear that the amount of deformation by explosive forming is larger.
These figures show comparative forming results.
In the case of static punching, we used a punch of a 100mm diameter and a metal die has a 105mm diameter hole.
The explosive forming limit, shown in the upper figure, when we used 10g explosive, and the bulge depth was 39mm.
The configuration obtained by static press forming is shown in the lower figure. The maximum bulge depth was 28 mm.
A diagram of the results using the static press method under those from explosive forming is shown in the right hand figure.
Comparison of the two configurations makes it clear that the amount of deformation by explosive forming is larger.

6. Simulation method Simulation model Explosive mass：10g (A5052-O)
This slide shows a simulation model for aluminum forming.
We attempted an elucidation of the deformation mechanism of aluminum alloy forming using FDM(Finite Difference Method).
Each dimension was determined for the equipment used in the experiment.
The aluminum was in the form of a disk with a diameter of 230mm and thickness of 1mm.
So, water, the explosive, and the metal plate were divided into the mesh of a quadrilateral, and calculation was carried out.
The aluminum was A5052O.
(A water column 130mm in diameter and 50mm in height rested in the center of the aluminum plate.)
Although a paper container was used in the experiment, it wasn’t included in the numerical simulation.
(Highly explosive SEP was positioned in the water.)
The distance between the bottom of the explosive and the aluminum plate was 30mm.
(The curvature radius of the die shoulder was 15mm and the die opening diameter was 100mm.)
As a boundary condition, the die was a rigid body.
Because the blank holder presses down around the perimeter of the water, the aluminum plate’s surface is restrained in the direction z.
Since the paper vessel was disregarded in the model, the sides of the water column were considered a free surface.
The surface contact between the water and the aluminum plate was the slide boundary.
Explosive mass：10g
FDM: Finite Difference Method (Lagrangian Code)
This slide shows a simulation model for aluminum forming.
The simulation method is FDM: Finite Difference Method by self coding.
z-r coordinate defined as this figure.
Each dimension was determined for the equipment used in the experiment.
Although a paper container was used in the experiment, side wall of water is treated as free surface.
The die was a rigid body.
The surface contact between the water and the aluminum plate was the slide boundary.

7. Simulation method Pressure calculation for water
P: pressure
e : internal energy
r0: initial density
r: density
G0： Grüneisen parameter　　　
Mie-Grüneisen equation of state
Mie-Grüneisen parameter
r0 (kg/m3)
C0(m/s) S G0
Water
1000
1490
1.79
1.65
The pressure calculation for the water was solved by this Mie-Grüneisen equation of state.
Where r0 is the initial density, e is energy, G0 is the Grüneisen parameter, and h=1-r0 /r, c0 and s are material constants.
The values of those constants are given in this table.
In order to calculate the pressure of the water, Mie-Grüneisen equation of state is used.
Parameters of this equation for water are shown in above table.

8. Simulation method Pressure calculation of explosive
JWL(Jones-Wilkins-Lee) equation of state
V= r 0(initial density of an explosive )/r (density of a detonation gas)
P*: Pressure　　E : specific internal energy
A,B,R１,R２,w : JWL Parameter
JWL parameter of explosive SEP
0.28
1.10
4.30
2.31
365 w R2 R1
B(GPa)
A(GPa)
The pressure from the explosion is calculated by using the JWL (Jones-Wilkins-Lee) equation of state.
where A, B, R1, R2 and ω are the JWL parameters.
V is the ratio of the volume of the gases produced by the explosion to the initial volume of the undetonated explosive.
The JWL parameter for the SEP are shown in this table.
The pressure of the explosive part was calculated by using the JWL (Jones-Wilkins-Lee) equation of state.
where A, B, R1, R2 and ω are JWL parameters.
JWL parameter for the SEP are shown in above table.

9. Simulation method Constitutive Equation
The constitutive equation of the aluminum plate is described in the following:
σy : plastic stress, εp : plastic strain 　 : strain rate.
In order to calculate of stress and strain of the aluminum alloy, we used this constitutive equation included the strain rate hardening.
Where, sy is plastic stress, ep is plastic strain and ep is strain rate.
In order to calculate of stress and strain of the aluminum alloy, we used this constitutive equation included the strain rate hardening.

10. Simulation results Pressure contour and history
Pressure profile in water cell on the aluminum plate at r=0,10, 20, 30, 40 and 50mm.
This is an animation of a propagation of the underwater shock wave by the simulation result.
The left hand figure shows the pressure contour in the element representing the water.
This chart shows the pressure history of the element where the water touches the aluminum plate surface at r= 0, 10, 20, 30, 40 and 50mm. With the detonation calibrated at 0ms, during the time interval from 5 to 10ms the underwater shock wave produced radiates spherically
The arc of the shock wave reaches the center of the aluminum plate at about 15ms.
A wave is then instantaneously reflected off the aluminum plate.
Although the shock front has spread to the edges of the aluminum plate by 30ms, because the circumference of the water is a free surface, the shock wave arriving at the edge is soon dissipated.
Propagation process of underwater shock wave.
This slide shows a propagation of the underwater shock wave by the simulation result. The left hand figure shows the pressure contour. The right hand side figure shows the pressure history of the element where the water touches the aluminum plate surface at r= 0 to 50mm. From 5 to 10ms, the underwater shock wave produced radiates spherically. The arc of the shock wave reaches the center of the aluminum plate at about 15ms.

11. Simulation results
Comparison of pressure values of anaysis result and experimental data
Tungsten bar
(MPa)
(mm)
This figure shows the comparison of pressure values between the experimental and simulation results.
The horizontal axis is the distance through the water from the bottom of the explosive.
The experimental data was measured using a tungsten bar pasted two strain gages.
When the shock wave through inside this tangsten bar, the shock velocity was measured by strain gages.
And then, from the shock velocity, the puressure is calculated.
From this figure, it can be seen that the values are almost same and the plausability of the calculations are validated.
Experimental equipment for the measurement of pressure of water
This figure shows the comparison of pressure values by the experimental and simulation results.
The horizontal axis is the distance through the water from the bottom of the explosive.
The experimental data was measured using a tungsten bar pasted two strain gages.
When the shock wave through inside this tangsten bar, the shock velocity was measured by strain gages.
And then, from this shock velocity, the puressure is calculated.
From this figure, both pressure values of the underwater shockwave agree well.

12. Simulation results Deformation velocity
This figure shows the deformation velocity in the direction z at r= 0, 10, 20, and 50mm.
When the shock wave acting on the central part of the plate is large, the deformation velocity rises rapidly to about 280 m/s.
Then the amount of shock wave pressure lessens greatly, but the deformation velocity slows down only gradually.
Moreover, movement of the initial velocity increase is from the central part of the aluminum plate gradually toward the perimeter, with the peak value decreasing as it moves from the central area to the perimeter.
This figure shows the deformation z-direction velocity at r= 0 to 50mm.
When the shock wave acting on the central part of the plate is large, the deformation velocity rises rapidly to about 280 m/s.
Movement of the initial velocity increase is from the central part of the aluminum plate gradually toward the perimeter, with the peak value decreasing as it moves from the central area to the perimeter.

13. Deformation process Experiment: 39mm 41.8mm
This figure illustrates a simulation of the deformation process on each time.
Since the aluminum alloy hardly deformed at all after 400ms, the process was considered completed at that point.
Between 20 and 40ms deformation appears in the central part of the plate, where the underwater shock wave first begins to act.
Then at 60ms the wave has apparently approached the die shoulder, the plate shown as bending down a little around the opening as deformation progresses.
This bulge at the die shoulder continues toward the center, whereupon the central part of the plate projects all at once in a great bulge, with the aluminum plate over the opening assuming a hemispherical shape in the final stage.
The amount of deformation of the aluminum plate from top to bottom surface at 400ms was shown as approximately 41.8mm.
However, in the experimental result, it value was 39mm.
Experiment: 39mm
41.8mm
This figure shows the deformation process on 20 ms time interval . Between 20 and 40ms, the deformation appears in the central part of the plate. At 60ms the wave has apparently approached the die shoulder, the plate shown as bending down a little around the opening as deformation progresses. This bulge at the die shoulder continues toward the center, whereupon the central part of the plate projects all at once in a great bulge, with the aluminum plate over the opening assuming a hemispherical shape in the final stage. The amount of deformation of the aluminum plate from top to bottom surface at 400ms was shown as approximately 41.8mm. In the experimental result, it value was 39mm.

14. Conclusion In this research, a numerical simulation was performed on
the free forming of the aluminum alloy plate by explosive
forming method.
1. The propagation process of an underwater shock wave
and the deformation process were simulated.
2. From experimental and simulation results, both pressure
values of the underwater shockwave agree well.
3. Peak velocity at center of the aluminum plate increased
up to about 280m/s.

15. Simulation results Deformation process Time(ms) 300 280 260 320 360
380
240
340
220
080
060
040
020
100
120
200
180
160
000
140
This figure illustrates a simulation of the deformation process of an aluminum alloy.
Changes are shown at 20-microsecond intervals as far as 300ms. Since the aluminum alloy hardly deformed at all after 400ms, the process was considered completed at that point.
Between 20 and 40ms deformation appears in the central part of the plate, where the underwater shock wave first begins to act.
Then at 60ms the wave has apparently approached the die shoulder, the plate shown as bending down a little around the opening as deformation progresses.
This bulge at the die shoulder continues inward toward the center, whereupon the central part of the plate projects all at once in a great bulge, with the aluminum plate over the opening assuming a hemispherical shape in the final stage.
The amount of deformation of the aluminum plate from top to bottom surface at 400ms was shown as approximately 43.8mm.

16. Total elongation vs. Strain rate (Aluminum base alloys)
The purpose of this research is that an deformability of an aluminum alloys using for the car body is made to improve.
This figure shows total elongation vs. strain rate of aluminum alloys.
All of the aluminum alloys described in this figure, have the tendency that the total elongation increases suddenly over 1000(ten of the third power) strain rate.
The predominance of shock bulge forming is investigated by comparing with static press bulge forming.

17. Comparison of deformation shape
Numerical Simulation
Simulation model
And then, we carried out the numerical simulation.
This figure shows a simulation model as same as the experimental equipment of the open type method.
This figure shows a simulation result, the deformation process.
The aluminum plate is gradually deformed to be spherical shape.
From the comparing with the experimental result on the final shape, on the circumference part of the aluminum plate,
the deformation amount of the experimental is larger than one of the simulation.
Comparison of deformation shape
Simulation result

18. Experimental equipment
Simulation model

19. Stress distribution Simulation results
Above figure shows the distributin of normal stress along to deformation shape of the aluminum alloy at 20 and 40 ms. Black continuous line is top surface layer of aluminum and red dashed line is bottom surface layer of it. At 20ms, the inner part of aluminum alloy within approxymately 30mm, because the stress value of bottom surface layer is greater than the top surface layer, the deformation shape of aluminum allloy is formed below. At 40ms, on the part of die shoulder(from 50 to 65mm), because the stress of top surface layer is greater than the bottom surface layer, the deformation shape is formed convex to above.

20. Pressure profile Simulation results
Above figure shows the pressure history of the element of the water which touches the aluminum plate surface in r= 0, 10, 20, 30, 40 and 50mm.

21. Introduction
The feature of explosive forming that allows the die to sufficiently transfer its shape onto the plate is that the spring-back effect is less than with ordinary forming methods, like punching and hydro bulging.
So, we considered using this method for forming aluminum alloy. 
The automobile and aircraft industry has recently been manufacturing parts using aluminum alloys, either number 5000, 6000 or 7000.
However, the forming of these materials as sheet metal by static methods, such as hydro bulge forming or general punching, is difficult because these materials allow little elongation compared to conventional steel.
In order to examine the application of explosive forming, we tried free forming aluminum alloy as the basis of the study.

22. Simulation method Burn technique for explosive CJ volume burn method
Conditions in the explosion were determined by calculating its reaction rate W, which is derived by the following formula
V0：Initial specific volume
VCJ：Specific volume of C-J state
The pressure P inside the grating for the explosion is calculated by the following formula:
For the calculation of the explosion, the C-J volume burn method was used.
When the volume of the cell of the original explosive in the calculation becomes equal to the volume of the detonation products at Chapman-Jouguet (C-J) state, the solid explosive is assumed to be completely decomposed into the gaseous products.
When V0 represents the initial volume of explosive and VCJ is the volume of the detonation products at the C-J state, the reaction rate W of the explosive can be simply expressed as the upper equation.
The pressure of the explosive element are calculated by the lower equation, where, P* is given from JWL equation.
P*: Pressure calculated from JWL EOS