Expansion of Cylindrical Shells Subjected to Internal Explosive Detonations

by R. L. Martineau, C. A. Anderson and F. W. Smith

ABSTRACT--Two explosively loaded cylindrical shell exper- iments were conducted to provide experimental data for benchmarking numerical codes. Each shell was subjected to internal high-explosive detonations, which caused it to ex- pand outwardly at strain rates on the order of 104 s -1. At approximately 150 percent strain, multiple plastic instabilities appeared on the surface of these shells in a quasi-periodic pattern. These instabilities continued to develop into bands of localized shear and eventually formed cracks before caus- ing the shell to fragment. Diagnostic equipment on these ex- periments included a Fabry-Perot interferometer and a fast- framing camera. The experiments and the data obtained from the diagnostic equipment are discussed and illustrated.

KEY WORDS--Dynamic testing, fracture mechanics, im- pact/shock analysis, rate effects, viscoplasticity

I n t r o d u c t i o n

The elastic/viscoplastic behavior of materials is an area of study that encompasses a variety of scientific disciplines with industrial and military applications. Applications for such research include hypervelocity accelerators, flux com- pression generators, fragmenting munitions and explosive containment vessels for terrorist threats and power plants. When a thin-walled cylindrical flux generator is subjected to an internal explosion, a significant amount of high strain rate radial expansion occurs prior to failure by fragmentation. Documented experimental studies on the high strain rate ex- pansion and failure of ductile shells began in the 1940s and tapered off in the 1970s with the advent of the computer. This paper describes a carefully planned set of experiments whose primary purpose is to provide data, which will be used to benchmark the numerical simulations.

In 1943, Gurney 1 developed a widely used analytical model for predicting the terminal velocities of fragments from shells subjected to internal explosive detonations. The failure of cylindrical structures was again examined in 1944 as a fragmentation problem by Taylor. 2 Taylor proposed that at the start of the radial expansion, the hoop stress throughout the thickness of the shellis compressive. Eventually, the hoop stress on the outer surface becomes tensile while still being compressive on the inner surface. At this point, longitudi- nal cracks form on the outer surface of the shell. These cracks

R. L. Martineau is a Staff Member, and C. A. Anderson is a Staff Member, Los Alamos National Laboratory, Los Alamos, NM 87545. E W. Smith is a Professor, Colorado State University, Fort Collins, CO 80523.

Original manuscript submitted: May 12, 1999. Final manuscript received: February 5, 2000.

continue to grow as the transition region between the com- pressive and tensile hoop stress moves inward. The cracks will not penetrate to the inner surface until the entire shell is in tension.

In 1967, Slate and colleagues 3 concluded that thicker shells showed more surface cracking and higher strains at rup- ture. Hoggatt and Recht 4 developed a mathematical model assuming the fractures occur along lines of maximum shear. They also observed different types of fractures based on the amount of high explosive (HE) and the detonation pressures. At low detonation pressures, deep cracks formed on the outer surface before unstable shear zones began to develop. At high detonation pressures, the compressive hoop stress from the detonation retarded the growth of cracks and the unstable shear zones formed earlier, and as a result, larger shear zones were observed on the fragments. Wesenberg and Sagartz 5 analyzed the expansion of thin cylindrical shells at strain rates of 104 s -1. They discussed the radial expansion of the cylinders and their fracture by providing a numerical solution to Mott's 6 fracture equation. Wesenberg and Sagartz com- pared Mott's probabilistic analysis with experimental data, and the two were reasonably close. They also observed that the number of fragments decreases with material density and decreasing strain rate or detonation pressure.

In the early experimental literature, particularly before the 1970s, the material used in experiments was not carefully characterized or documented, and as a result, little is known about the grain size and hardness of the materials, thus mak- ing it difficult to duplicate the experimental results or use the results for comparison with numerical simulations. Sev- eral authors have examined instabilities associated with both uniaxial and biaxial stress under quasi-static conditions, but most have not considered materials subjected to multiaxial stress states or strain rates on the order of 104 s -1. The re- search presented here provides a detailed description of the material and presents data obtained from two sophisticated experiments. Data of the type presented here are limited in the open literature and are needed to verify and benchmark numerical models.

7 In this research, ABAQUS/Explicit and a newly devel- 8 oped viscoplastic material subroutine were used to numer-

ically model the expanding shell. ABAQUS is a commer- cially available Lagrangian finite element package with an HE burn model. The ABAQUS HE subroutine is based on the program burn model and the Jones-Wilkens-Lee equation of state. During the solution process, ABAQUS calculates an increment in strain based on the boundary conditions (such as an increment in load) and the previous state of stress. This in- crement in strain is passed to the VUMAT subroutine, which

Experimental Mechanics �9 219

updates the state of stress in the material. The constitutive model has several modules that are used to predict shock effects, plasticity, void growth and strength. Plots comparing the numerical predictions and experimental data are shown in this paper; more detailed discussions of the numerical model are provided in Ref. 8.

Cylinder Experiments

Two nearly identical dynamic experiments were con- ducted as part of this research, with the wall thickness of the cylindrical shells being the only variable changed be- tween the two experiments. Each shell was fabricated from alloy 101 OFE (oxygen-free-electronic) copper. OFE copper is easy to obtain in a carefully controlled high-purity form, and its high strain rate behavior and plastic deformation are well understood. PBX-9501 was used as the HE inside of each shell, and the experimental configuration was identical for each experiment.

The largest diameter of small-grain, OFE copper tubing available had a 114.3 mm outer diameter with a 6.35 mm wall thickness. This dictated the overall geometry of the experiments. After machining, the smallest inner diameter obtainable was 102.06 ram. This allowed for one cylinder to be 2.54 mm thick and the other to be 5.08 mm thick. The overall length of both cylinders was 406.4 mm. The dimen- sions of the two cylinders are summarized in Table 1.

The initial grain size in the copper tubing was 35-40 txm, with a hardness of 80 on the Rockwell F scale. A small sam- ple of the material was sectioned off and heat was treated to 350~ to determine the rate and degree of softening attain- able in the material. After 60 minutes, the microstructure of the copper was approximately the same size. The hardness of the material was now 23 on the Rockwell F scale, indi- cating the release of residual energy. The surface finish on the cylinders was carefully controlled to minimize perturba- tions and initiation sites for instabilities. Surface finishes, as defined by ANSI B46.1-1962, of 16 and 32 were maintained on the outside and inside surfaces, respectively. In addition, a concentricity tolerance of 0.05 mm was maintained during the fabrication process.

A solid circular cylinder of HE with a diameter of 101.8 mm was positioned inside of the copper cylinders. A 0.127 mm clearance was prescribed between the HE and the cylinder to provide an adequate gap for assembly. The HE was centered in the cylinder and bonded to several small shims located at each end. A plane wave lens was bonded to one end of the HE cylinder, as shown in Fig. 1. The plane wave lens, which is a cone fabricated from two types of HE, was detonated at the apex with an SE-1 detonator. The burn times of the two different types of HE in the plane wave lens produce a planer detonation wave by the time the wave front reaches the base of the cone. Once a plane wave front is established, it will propagate longitudinally through the HE and remain planar as long as there are no significant material or geometric transitions. The transition from the plane wave lens to the solid cylinder of HE was improved by extending the HE 5.08 cm beyond the end of the copper shell. This helped to ensure a stable plane wave detonation inside the copper cylinders. The final assemblies are shown in Fig. 1.

The copper cylinders have a grid with distinguishing marks on the outer surface. These marks provide a contrast- ing surface, making it easier to identify the quasi-periodic

Fig. 1--Copper shell and high-explosive assemblies

instabilities that occurred during the deformation. In addi- tion, the grid is useful for determining the overall strain from the framing camera pictures. The HE was approximately 7.3 kg, and the thick and thin copper cylinders were 6.2 kg and 3.0 kg, respectively.

Two forms of diagnostics were used in the experiments: a fast framing camera and a Fabry-Perot interferometer. Fabry- Perot is capable of recording radial velocity information at a single point on the surface of the expanding cylinder. The fast framing camera diagnostic provides 25 photographic images of the expanding cylinder at predetermined increments in time. The hoop strain and the development of the instabilities observed on the surface of the expanding shell can then be extracted from the photographs and plotted as a function of time.

Figure 2 shows the experimental setup. All dimensions are indicated in meters. A large plywood box filled with ver- miculite and asphalt sheeting was used to "soft catch" the fragments, and sandbags were used to support the plywood box and minimize blast-induced deflections. The HE-filled copper cylinder or "shot" was placed on the shot stand in an inverted configuration. The plane wave lens and detona- tor were located on the bottom so that the detonation trav- eled vertically from the bottom toward the top of the cylin- der. Thus, the shot stand helped to contain the detonation by-products.

During the explosion, the shot was illuminated using two argon flash candles. Flash candles are plywood boxes lined with detasheet, which is rubberized HE with distinct wind- ing burn patterns. The HE is ignited at one end, and the burn front follows the pattern much like a fuse, resulting in a bright light and long burn times. The timing of the illumina- tion was carefully coupled with the shutter timing on the fast framing camera. The fast framing camera was located below

220 ,, VoL 40, No. 2, June 2000

TABLE 1--DIMENSIONS OF THE TWO CYLINDERS USED FOR THE EXPERIMENTS Length Inner Diameter Wall Thick (mm) (mm) (mm)

Outer Diameter (ram)

Thin cylinder 406.4 102.06 2.54 107.14 Thick cylinder 406.4 102.06 5.08 112.22

f FL.IkSH CANDLE (2)

1,075 ='

TURNING ~ ~ .......... SHOT--~ MIRROR " ~ .305 ............ .___. N:._.._._

/ ......... ................................ /

OPTICAL PORT t /

,406 SHOT STAND

, , 1 Fig. 2--Elevation view of the experimental setup

I-, ,813

PLYWOOD BOX

~' /SANDBAGS

1 1,214

the surface in a bunker and viewed the experiment through an optical port. The turning mirror shown in Fig. 2 was used to correct the camera's line of sight.

Experimental Results

The first experiment was conducted using the thick cylin- der. In this experiment, the Fabry-Perot equipment experi- enced a hardware failure and did not record data. However, the fast framing camera performed as expected. Twenty- three images with a frame interval time of 2.257 Ixs were recorded. The second experiment was conducted using the thin cylinder, and all of the diagnostic equipment performed as expected. Again, the frame interval time of the fast framing camera was 2.257 txs, and 23 images were recorded. Ten of the 23 fast framing camera images, which were taken during the expansion of the thin cylinder, are shown in Fig. 3.

The photos in Fig. 3 provide radial displacement data as a function of time and axial position along the cylinder. They also provide data on the number of surface instabili- ties present as a function of time. Each photograph from the fast framing camera was digitized and enlarged to determine the number of instabilities for a small characteristic length on the surface of the cylinder. This length and the number of instabilities were then used to estimate the total number of instabilities on the entire circumference of the shell. The instabilities observed on the surface of the expanding cylin- der occur in a pattern that propagates down the longitudinal axis of the cylinder. An enlarged picture of the last frame is shown in Fig. 4. The image in this figure is rotated 90 deg counterclockwise for illustration purposes. The quasi- periodic pattern of instabilities is most prevalent on the black lines shown inside the dark circle of this figure. The hoop strain at the center of the dark circle is on the order of 150 percent. At any longitudinal location on the cylinder, the total number of instabilities on the surface of the thick and thin expanding shells was calculated from photos such as that shown in Fig. 4 to be 298 and 343, respectively. The error in this calculation is within -t-10 percent using the method explained in Ref. 8.

The radial displacement data were extracted at eight points along the longitudinal axis of the cylinder. Each point was separated by 50.8 mm, with the first point located 50.8 mm up from the detonated end. The data for the 2.54 mm and 5.08 mm thick cylinders are shown in Figs. 5 and 6 at 29.3 t~s and 49.7 Ixs, respectively.

The outward radial displacement of the copper cylinder experiments is plotted versus time in Figs. 7 and 8. The dis- placement curves shown in each of these figures were taken at eight different longitudinal locations on the cylinder wall. The locations, which were measured from the detonated end of the cylinder, are shown in the text on the right-hand side of the figures. The numerically predicted displacements are also shown in Figs. 7 and 8. Good agreement between the numerical results and the experimental data is evident in each figure.

Gurney 1 developed an equation for predicting the maxi- mum velocity, Vmax, of shells subjected to internal explosive detonations. Gurney's equation for a cylindrical shell is writ- ten as

Vmax = + , (1)

where M/C is the ratio of the mass of the shell to the mass of the explosive and is called the Gurney constant. The em- pirical constant, ~/2E, was determined from experiments in- volving a particular type of explosive. For PBX-9501, is equal to 2900 m/s. Table 2 gives the calculated values of the Gurney velocity for the experiments conducted in this research.

Figure 9 shows the velocity of the cylinder wall for the 2.54 mm thick cylinders. The plots shown in this figure in- clude the velocities from the empirical Gurney equation, the Fabry-Perot instrumentation for the thin-walled cylinders and the numerical model. The Fabry-Perot was positioned or- thogonal to the surface, which moves radially outward at a 19-deg angle from the longitudinal axis of the cylinder. A similar graph for the 5.08 mm thick shell is not presented here due to hardware failure on the part of the Fabry-Perot

ExpefimentalMechanics �9 221

Time (its) 0.0 6.771 13.542 20.313 27.084

Time (~ts) 33.855 38.369 Fig. 3~Fast framing camera images for the thin cylinder

40.626 45.14 49.654

Fig. 4--View of instabilities on expanding thin cylinder

70

450

. . . . . . Thick Cylinder

Thin Cylinder

. / ~ / ~ /

~* / i ~

~1) ~ 4,

/ /

/ / t

400 350 300 250 200 150

Axia l Posit ion from Detonated End (ram)

, , 0

100 50 0

Fig. 5--Deformed geometry of both cylinders at 29.3 I~S

60,--, E E

50"-"

4O E

3O ~- a

2O -~

222 * VoI. 4~ No, ~ June2000

140

- - - . - - - Thick Cylinder

- - - . - - - Thin Cylinder

. t . . . . . . . . I

. �9 '~'" " . 0 " "

. . ,1~" . r

120 .-. E E

100 "-" e,

80 E o

60 ~. 1:1

40 "~ "o

20

450 400 350 300 250 200 150 100 50 0

Axial Position from Detonated End (mm)

Fig. 6--Deformed geometry of both cylinders at 49.7 IXS

120

A E 100 E

" 80 Q) E o ~= 60

._~ cl 40 ._m "O el

20

0

50.8

. . . . . Numerical Results Experimental Data 101.6 ~-

E

152.4 ~

203.2 Q

254.0 ~

304.8 ~

355.6 ~d

406.4 i i

0 10 20 30 40 50

Time (its) Fig. 7--Radial displacement as a function of time at eight locations along the longitudinal axis of the 2.54 mm thick cylinder

TABLE 2--GURNEY VELOCITY FOR CYLINDRICAL SHELL EXPERIMENTS Mass of HE Mass of Shell

(kg/m) (kg/m) Villa1{

M/C (m/s) 2.54 mm thick shell 14.98 7.46 0.498 2902 5.08 mm thick shell 14.98 15.29 1.02 2351

instrumentation. The steps shown in the experimental data illustrate the shock reflecting through the wall of the cylinder.

The effort to "soft catch" fragments for microscopic exam- ination was unsuccessful because the setups shown in Fig. 2 were destroyed by the HE charge. A third, less sophisti- cated experiment was conducted in an attempt to obtain a bulged copper sample illustrating the development of insta- bilities prior to fragmentation. This experiment consisted of a 30.48 cm long OFE grade copper cylinder with a 2.54 cm inner diameter and 6.35 mm thickness. The copper cylinder was placed vertically on end in about 2.54 cm of sand. A

slug of composition C-4 explosive, which was 7.62 cm long and 1.27 cm in diameter, was placed inside the top lip of the copper cylinder. The C-4 was end detonated, and the explo- sive shock wave traveled down the inside of the cylinder. The shock wave reflected off the surface of the sand, resulting in an internal overpressure. The overpressure caused the cylin- der to bulge and fracture, but not fragment. A picture of the bulged cylinder is shown in Fig. 10. The arrows indicate the locations of the instabilities.

The bulged cylinder shown in Fig. 10 was cross sec- tioned through the instabilities. Examination of the section

Experimental Mechanics �9 223

90 o ~ 1 ' 50.8 80 . . . . . Numerical Results �9 101.6

E Exp E E 70 152.4 ~"

" 60 �9 203.2

50 = ~- 40 254.0 .~- a 304.8 & 30 "-

20 355.6 ~=

10 406.4 3

0 10 20 30 40 50

T ime ( i ts) Fig. 8--Radial displacement as a function of time at eight locations along the longitudinal axis of the 5.08 mm thick cylinder

3 0 0 0

2 5 0 0

2O0O

"" 1 5 0 0

N

1 0 0 0

5 0 0

............... N u m e r i c a l R e s u l t s

. . . . . G u r n e y V e l o c i t y

E x p e r i m e n t a l D a t a

0 5 10 15 20 25 30 T i m e ( m s )

Fig. 9--Radial velocity as a function of time for the 2.54 mm thick cylinder

:J

/ r-- - * * i I I

35 40

tion and a shear band was obviously connecting the instabil- ities. An exaggerated illustration of the offset instabilities is shown in Fig. 11, and a photomicrograph of the resulting microstructure from the bulged cylinder section is shown in Fig. 12. In Fig. 12, excessive plastic deformation is visible from the elongated grains, illustrating a very ductile mate- rial response. In addition, the inner surface of the cylinder appears to be rougher than the outer surface, indicating ad- ditional damage. This damage occurs possibly as a result of localized heating and shock interactions with the HE.

Fig. 10--Bulged cylinder from C-4 experiment

indicated the development of instabilities on both the inner and outer surfaces. However, unlike a uniaxial tensile spec- imen, these instabilities were offset in the tangential direc-

Summary

The experimental data presented here illustrate the defor- mation of two cylindrical shells subjected to internal explo- sive detonations. These shells were fabricated from high- purity copper, which was carefully characterized for these experiments. The configuration of the explosive and the det- onation technique results in a multiaxial state of stress, which

224 �9 Vol. 40, No. 2, June 2000

Localized Zones of Shear

/ r

R

Fig. 1 lmExaggerated illustration of quasi-periodic instabilities

\ Quasi-Periodic Instabilities

/

Outer Surface

200 gm Inner Surface

Fig. 12--Shear band from bulged cylinder experiment

is unique to the current literature. Fast framing camera pho- tographs and Fabry-Perot interferometer data recorded during the experiment provide useful information with regard to the high strain rate deformation and surface velocity of the ex- panding shells. This information is useful for benchmarking numerical codes, and the limited comparison plots given in this paper illustrate good agreement between the experimen- tal data and numerical predictions, thus lending credibility to the numerical model developed in Ref. 8. In addition, the fundamental work by Gurney provides a supplemental verification of the experimental data and numerical results.

The fast flaming camera photographs and photomicro- graph shown here illustrate excessive plastic deformation and shear localization, thus providing insight into the develop- ment of the multiple plastic instabilities that are associated with rapidly expanding shells. These instabilities appear in the photomicrograph to be localized zones of plastic strain and are related to the thickness of the expanding cylinders. The experiments conducted in this research provide useful information for further experimental, analytical and numeri- cal work with regard to shell expansion and the development of the quasi-periodic plastic instabilities.

A cknowledgmen ts

The authors gratefully acknowledge Dynamic Experimen- tation and Engineering Sciences and Applications divisions for providing support for this research. This work was per- formed under the auspices of the U.S. Department of Energy by Los Alamos National Laboratory in cooperation with Col- orado State University.

References

1. Gurney, R., "The Initial Velocities of Fragments from Bombs, Shells, and Grenades," BRL Report No. 405 (1943).

2. Taylor, G.L, "Fragmentation of Tubular Bombs," Scientific Papers of G. I. Taylor, 3 (44), Cambridge University Press, Cambridge, 387 (1963).

3. Slate, P.M.B., Billings, M.J. W., and Fuller, P.ZA., "The Rupture Be- havior of Metals at High Strain Rates," J. Inst. Metals, 95, 244-251 (1967).

4. Hoggatt, C. and Recht, R., "Fracture Behavior of Tubular Bombs," J. Appl. Phys., 39, 1856 (1968).

5. Wesenberg, D.L and Sagartz, M.J., "Dynamic Fracture of 6061-T6 Aluminum Cylinders," J. Appl. Mech., 44, 643 (1977).

6. Mort, N.R., "Fragmentation of Shell Cases," Proc. Roy. Soc. London, 189, 300 (1947).

Z ABAQUS Version 5.7, Hibbit, Karlsson, andSorenson Inc. (1998). 8. Martineau, R.L, "A Viscoplastic Model of Expanding Cylindrical

Shells Subjected to Internal Explosive Detonations," PhD dissertation, Col- orado State University (1998).

Experimental Mechanics �9 225