End effect in the explosive forming of tubes

Journal of

ELSEVIER

J. Mater. Process. Technol. 42 (1994) 431-441

Materials Processing Technology

End effect in the explosive forming of tubes M . M . M o s h k s a r , * S. Borji Department of Materials Science and Engineering, School of Engineering, Shiraz University, Shiraz, Iran

(Received November 23, 1992; accepted in revised form October 5, 1993)

IndustrialSummary The expansion of short thin-walled tubes under internal explosive pressure has been analyzed in order to achieve a better understanding of the process of forming hollow parts using an explosive charge. For uniform expansion of the tubes, where forming should take place over the whole length of the tube, a linear shape of the detonating charge (such as is achieved when employing PTEN) is the best. Even under this condition, the deformation of the tube is not uniform, the ends of the tube not deforming as much as the mid sections. Increasing the mass of charge at the ends will not alter the situation, as the so-called "end effect" will not disappear. The effect of the mass of charge on the shape of the tubes and on the "end effect" have been investigated in this work, a new approach to eliminate the "end effect" being suggested. The relationship between the mass of the charge and explosion factors such as strain distributions and deformed shapes has been studied also. Key words: Explosive forming; Tubes;

I. Introduction The history of the explosive forming of metals goes back to the nineteenth century. However, the method was not developed until about 1950, when it began to be investigated extensively by the aero-space industry [1]. Progress in explosive forming was due to several industrial necessities: (i) close tolerance, quick response, and limited production requirements; (ii) unusual and unsymmetrical shapes; (iii) large size workpieces; and (iv) unusual material properties [2]. All explosive forming operations have three main aspects: the explosive charge; the medium of energy transfer; and the workpiece [3]. The behavior of the metal workpiece during deformation depends strongly on the above aspects, a slight change in them causing a huge variation in the metal behavior and complicating the manner

* Corresponding author 0924-0136/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0924-0136(93)E0118-Z

432

M.M. Moshksar, S. Borji/Journal of Materials Processing Technology 42 (1994) 431-441

of the treatment. This is why more understanding of the method is obtained by means of experimental investigation than by means of analytical and theoretical study.

2. Propagation of the hydrodynamic shock wave The pressure produced from the detonation of an underwater spherical charge propagates with a spherical shock front through the transition media. The pulse, is usually shown by a pressure-time curve at any point of the media (Fig. 1), the "peak pressure" being the maximum pressure at a point as the shock wave passes through that point [4]. The area under the pressure-time curve expresses the "impulse", which is the momentum of the wave per unit area. As the shock wave expands and increases its shock front through the media, the peak pressure and impulse decrease and result in a change of the impulse curve. The pressure of the first portion of the "impulse" curve (which is almost linear in a logarithmic scale) is expressed by: P(t) = P m Exp ( --

t/O)

where P(t) is the pressure, Pm is the peak pressure, t is the time during which the wave has passed through the point and 0 is a constant that depends to the type, the amount,

PSl

6 000 ------

THEORY

6 000 EXPERIMENT

2000

1000 800

\\

600 400

200

O

0.5

\

1.0

15

2.0

25 MICRO SEC.

Fig. 1. Pressure-time curve (after [4]) (1 PSI = 6.89 kN/m2).

M.M. Moshksar, S. Borji/Journal of Materials Processing Technology 42 (1994) 431- 441

433

and distance of the charge. The impulse is given by [5]: t

I = f P (t) dt 0

3. Shock wave energy flux density With some modification in the equation of the energy flux of the acoustic waves, the energy flux of an explosive shock wave can be obtained as:

Ef = (1/fc) ~p2 dt t

w h e r e f i s the density of the water, c is the wave speed in the water, and P(t) is the pressure function. It has been shown that, for the pressure ranges that are usually generated in explosive forming, acoustic wave theories are well applicable, the error being less than 5%, [6]. Bebb [7] reports that in addition to shock-wave energy, there is some "after-flow" energy (media momentum energy) that is produced in an explosion and has an effect on the deformational efficiency.

4. Linear explosive charge Charlsworth and Brayant [8] assumed that a linear charge is identical to the conjunction of a number of spherical "unit elements" that each can produce a pressure Po = exp ( - kt)/R at a distance R from the unit element charge, where Po and k are constants and t is time. Now consider such a linear charge with a length of L which is laid along the x-direction, its detonation beginning from x = 0 at t = 0 with a detonation velocity of Vd, Fig. 2. If dp (xl, t) is the incremental pressure due to the charge element dx at a point Q (Xl, d) within the water, then d P (xl, t) = Po exp ( -

kt) (dx/R)

from which the pressure and energy flux at the point Q will obtain as: L2

P (x,, t)= 1- {(Po/R)=exp [ - k ( t - x , / c - (R/D))]} dx LI

to+t

E (x1)

=

1/fc | p2 (x,, t) dt lo

where L1 and L2 are the limiting values of the linear charge that upon detonation have an effect on point Q, to is the required time for the wave to reach point Q, and t is its incubation time.

434

M.M. Moshksar, S. Borji/Journal of Materials Processing Technology 42 (1994) 431- 441 x: 0

dx

x: x

x:L I

I

'

R\

i

t

Id

",lxl

I

Xl

O(xlld) =-

I

Fig. 2. Representation of a linear charge.

5. Calculation of strain For a work-hardening metal, the flow stress a for a given deformation e is expressed by a = Ke", is the required ideal work of deformation in a volume V being given by:

W=V

+. j ~.a d e - -VKel l-+ n 0

Therefore the value of strain for work of deformation of value W becomes: e = [(1 + n) W / V R ] 1/(1+") Thus, having the flux energy E as the required work, the strain can be measured. Using a computer program for the given equations, together with the boundary conditions, the theoretical calculations of P (x, t), E and e can be obtained for any point on the surface of the workpiece, consequently establishing curves of theoretical results for comparison with those of experimental results.

6. Experimental procedure 6.1. Explosive material All experimental field tests have been run using detonating fuse with an explosive charge core of PTEN. The detonation velocity of the explosive fuse is 6700 m/s, the quantity of the explosive charge is 10.45 g/m and the release energy is about 59 300 J/m with about 30% mechanical efficiency. Detonation is initiated by a standard No. 6 electric detonator and a magnetic dynamo. 6.2. Tube specimens In all experiments, seamless low-carbon steel tubes with dimensions of 300 mm length, 112 mm. o.d., and 4 mm thickness were used, the true stress-strain behavior of the tube material being determined as: a = 554e °'12~ (MPa)

M.M. Moshksar, S. Borji/Journal of Materials Processing Technology 42 (1994) 431-441

~ELECTIC DETONATOR

DETONATOR._..... CORD

I

!-2 : : -

I.-7.-

"

435

,/WORK PIECE -"

. . . . .

--q

i

.

.

.

.

.

i

.

.

.

.

.

i

. . . .

WATER. .--::---~=,

.i

0 0 0 0 0 0 lO^O^O^O^O~O,~O 0 0 0 0 0

(a)

(b)

•

TIHCK WALL STEEL CYUNOER

(C)

t

STEEL END CAP

-L -

..

•:- ~ P A P E R TUBE

L-

PLASTIC TUBE

WORK PIECE

\

-L T--

--

.._

~LI I t I / ~0~0~0~0~0 0 0 0 0

0

_J

~ 0

I Co_ O o O

(d)

o

o

o~

Io~o~o~o~o~o~

I~oe~oe~o~o~o~

(e)~

0

0

0

\

(f) " \

WOOD OR POLYSTYRENE

WOOD OR POLYSTYRENE

Fig. 3. Schematic diagram of the test arrangements employed.

6.3. Dies, Fixtures, and test arrangements

Fig. 3 shows the test arrangements, Fig. 3(c) illustrating the using of a fixture with steel end caps. For the extensions of the charge and of the corresponding energy transfer media, thick-wall steel cylinders, plastic tubes, and paper tubes were employed, respectively Figs. 3(d)-3(f).

436

M.M. Moshksar, S. Borji/Journal of Materials Processing Technology 42 (1994) 431- 441

7. Results and discussion In order to predict the relationship between the amount and location of the charges to that of strain distribution along the whole length of the tubes, a modified computer program was used. Table 1 represents some of the experimental tests, the results of which were compared to those of the computer calculations, the outline of the test procedure and the information on each test, together with the specimens numbers and corresponding figures, being given also in this table. Figs. 4 and 5 show, respectively, the measured and numerically-calculated values of hoop strain along the whole length of the specimens of Table 1, for simplicity of the results, the hoop strains along the vertical axes of these figures being normalized with respect to their maximum values. As can be seen in most of these figures, the computational results are in good agreement to those of experiment, except for the Table 1 lnformations on the experimental tests Specimen No.

Test procedure

Length of strands (cm)

No. of charge strands

Figs.

S-103

Free forming with the charge along the whole length of the tube

30

3

3 (a) and 4 (a)

S-26

Free forming with a short charge at the top of the tube

10

13

3 (a) and 4 (b)

S-104

Top end extended by a thick-wall steel cylinder

12

7

3 (b) and 4 (c)

S-107

Using a fixture with end steel caps

30

3

3 (c) and

4 (d) S-105

Both ends extended by thick-wall steel cylinders

54

3

3 (d) and 5 (a)

S-102

As S-105, but no extension of the charge strands

30

2

3 (d) and 5 (b)

S-135

As S-105 but extension by both paper and steel tubes

42

3

3 (e) and 5 (c)

S-108

As S-105 but extension by plastic tubes

54

3

3 (f) and 5 (d)

M.M. Moshksar, S. Borji/Journal of Materials Processing Technology 42 (1994) 431-441

!"°I

!10~-

. . . . . . .

= -'~

'

Distance

.

.

.

.

.

from

top

end

~'~"A Expenmenlal

\\

-''°'

.

2o

,o

~

437

0

3o

10

20

30

Distonce from top end [cm,)

(cm.)

(a)

(b)

(xpenmeta[ hlumeeca[ 0.'

i

g~ fO Distance

3~ from

top

end

10

20

30

Distance from top end (cm.)

(cm,]

(c)

(d)

Fig. 4. Distribution of experimental and numerical normalized hoop strain along the length of the tube, for specimen no.: (a) S-103; (b) S-26; (c) S-104; (d) S-107. ,£

~ 1.0 --

.E

&

o~

-~

3:

Experimental I~mericat

E

z 0

ii

i,iii

, I . i , i i

i li,J

|e

~ -, i l i l , a

lO 20 Distance from top end (cm,)

30

.° 0

(a)

.,a,**

, , l i , , , i

(xpeeimental Nwnefical t,li

I , l i , , , l i i

In 20 Distance from top end (cm.)

30

(b)

c

c

;1,0

g

o o

o

:o.s| ExpehQ1ev)lat NumeTicoJ o

J 1o 2O Distance from top end (¢m.)

(c)

30

nO

f0 20 Distance from top end (¢m.)

30

(d)

Fig. 5. Distribution of experimental and numerical normalized hoop strain along the tube length of the tube, for specimen no.: (a) S-105; (b) S-102; (c) S-135; (d) S-108.

438

M.M. Moshksar, S. Borji/Journal of Materials Processing Technology 42 (1994) 431-441

1

2

3

Fig. 6. Condition of wave reflectionat the water surface. end sections of the tubes. In most cases the deformation of the tubes ends is clearly less than that of the mid-sections of the tubes: this is known as the "end effect" and it can be seen that this effect is altered by change in the test arrangement. Fig. 4(a) shows the results for specimen number S-103, where the lengths of the charge, the tube and the transferring water media are the same. As can be seen, the computational strains at the tube ends are less than those at the mid part, but in the case of experimental results, no strain appears at the ends. Generally similar results can be seen for test specimen number S-26 (Fig. 4(b)). In the case of specimen number S-104, where the media of the top end of the tube is extended by a thick-wall steel cylinder with the charge remaining unextended, the computational results are almost in exact agreement with the experimental results (Fig. 4(c)). The test of specimen number S-107 was performed using a fixture and steel end caps, the result (Fig. 4(d)) showing more uniformity of the strain distribution. In order to cancel the end effect, a set of new tests was arranged by extension of the tubes by thick-wall steel cylinders, paper tubes, and plastic tubes, the results of extension of the specimen tube by thick-wall steel cylinders (specimen No. S-105 and S-102) being shown in Figs. 5(a) and 5(b). As can be seen, the ends are deformed to about 20% more than the mid part. For these cases the computational curves show uniform strain with less strain at the ends. Fig. 5(c) shows the results of the test on specimen number S-135, where the tube is extended at the top end by a paper tube and at the bottom end by a steel cylinder, a difference in the end effect being observable in this figure. In the last test (specimen number S-108), the charge and media of both ends are extended by about 12 cm by means of plastic tubes, the result of this test being shown in Fig. 5(d). In this case, the computational strain remains constant over the whole length of the tube, whereas with the experimental results, although there is some improvement, the "end effect" is still observable. Fig. 6 indicates clearly why the "end effect" in the former tests are so different. Case one of this figure shows an explosive at the free surface of the water media. In this case the gaseous products of explosion would expand freely through the air: in fact, the compressive shock wave on the free surface of the water would reflect in tension. Liquids cannot resist tensile stress, therefore the result would be the throwing out of water from the container, and relaxation of the explosive energy at the free surface of the water. In the case of using steel cups, the reflected shock wave would be

M.M. Moshksar, S. Borji/Journal of Materials Processing Technology 42 (1994) 431-441

439

compressive, but still the experimental energy distribution at the tube ends is less than that of the mid part. In the case of extending the specimen by plastic or paper tubes, the reflected wave is again in the mode of tension, the plastic tubes rupturing and the end effect still appearing. When the extension of the tube is effected by steel cylinders, the "end effect" essentially disappears. For these results, two main points can be made, the first relating to the condition of compressive wave reflections from thick wall cylinder, and the second to the form of the state of stress at the ends with respect to that of the mid part of the tube specimen. The condition of the stress state at the ends is approximately similar to simple tension, whereas at the mid part it is restricted to the plane-strain condition. These different conditions would cause a change in the yield situation of the metal, the yield stress for simple tension being about 12-14% less than that for the plane-strain condition.

8. Strain measurement

In order to study the condition of the state of strain along the length of the tube, a set of tests was performed on the tubes using a fixture and steel end cups. For accurate measurement of the strain the specimens were originally marked along their length into a number of ring elements, after deformation the measurements being performed on each element independently. A linear relationship between the average hoop strain and the number of strands was obtained: eo = 0.4195 + 2.038 N 0.25

0.20

._c

OJ5

o

OJO

0,05

0.00

, . , , , t i t

i l l l l l l , =

2

.°,

/, NO. of

,,.,.

,,

6 charge

.

.

.

.

.

8 strands

= , , . l | , o .

10

12

Fig. 7. Effect of the mass of the charge on the average hoop strain.

440

M.M. Moshksar, S. Borji/Journal of Materials Processing Technology 42 (1994) 431- 441 0.9 o

~x 'Y

0,8

x A x ~ A

x

~

A

x

~

A

c

X

0.7 o

-r

x

A

0.6

o

~0 chorge strands B charge strands

xxxxx

0.54 cl

"6 o

r~

0.4

~1

0

i

i

I

i

I

I

,

l

I

I

5 No,

,

,

,

i

,

,

,

i

I

10 of Etements

,

,

i

i

i

i

I

i

I

15

Fig. 8. Distribution of the ratio of radial-to-hoop strain along the length of the tube.

where N = is the number of strands, the results being shown in Fig. 7. The experimental results of Johnson et al. show a similar linear relationship between strain and number of strands [9] Fig. 8 shows the distribution of the ratio of radial-to-hoop strain (er/ee) along the length of the tube. As is shown, this ratio is about 0.5-0.6 at the ends and increases to more than 0.8 at the mid sections. Volume constancy (i.e. e0 + e, + e~ = 0), indicates that at the tube ends e, = ez = - ½e0 so that the condition is as simple tension, but at the mid sections the axial strain ez approaches zero so that the plane-strain condition dominates.

9. Conclusions A computer program has been written to predict the distribution of strain over the length of explosively formed specimen tubes. When the numerical results are compared to those of experiment, good agreement is obtained. The tube "end effect" in explosive forming is the result of tensile reflection of the explosive shock-wave from the free surface of the water. The use of steel caps and appropriate extensions of the tube and charge decrease the effect. A linear relationship between deformation and mass of explosive charge has been found. The state of strain at the tubes ends is found to be similar to that of simple tension, whereas the mid sections approach the plane-strain condition.

References [1] H.G. Baron and E. de L. Costello, Explosive forming, Metall. Rev., 8 (1963) 369. [2] L. Zernow and I. Lieberman, Explosive Metal Fabrication, a Technical-Economic Tradeoff, 12th Annual Symp. Proc., Behavior and Utilization of Explosives in Eng. Design and Biomechanical Principles Applied to Clinical Medicine, Albuquerque, ASME, (Mar. 1972) pp. 85-108. [3] J. Pearson, The Energetics of Explosive Operations, ibid, pp. 68-84. [4] R.H. Cole, Under Water Explosions, Dover, New York, 1965.

M.M. Moshksar, S. Borji/Journal of Materials Processing Technology 42 (1994) 431-441

441

[5] E.J. Bruno, High-Velocity Forming of Metals, ASTME, Philadelphia, (1968) p. 20. [6] W. Scherder, High energy forming with explosives in a water medium, Int. R. Prod. Eng., (1963) 32. [7] A.H. Bebb, Under water explosion measurements from small charges at short ranges, Phyl. Trans. Royal Soc., series A., 244 (879) (1951) 153-175. 1-8] G. Charlesworth and A. Brayant, An approximate calculation of the pressure produced by detonating line charge under water, Under Water Explosion Research. Vol. 1, O. N. R. Washington, 1950, 301-309. [9] W. Johnson, E. Doege and F.W. Travis The explosive expansion of unrestrained tubes, Proc. Instn. Mech. Engrs., Vol. 179, Pt. 1 (7) (1964-1965) 240.