Extension of the Gurney Equations to Two Dimensions

for a Cylindrical Charge

by Benjamin A. Breech

ARL-TR-5467 March 2011

Approved for public release; distribution is unlimited.
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Army Research Laboratory

Aberdeen Proving Ground, MD 21005-5066

ARL-TR-5467 March 2011

Extension of the Gurney Equations to Two Dimensions

for a Cylindrical Charge

Benjamin A. Breech

Weapons and Materials Research Directorate, ARL

Approved for public release; distribution is unlimited.
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ARL-TR-5467

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September 2010-October 2010

Extension of the Gurney Equations to Two Dimensions for a Cylindrical Charge

Benjamin A. Breech AH80

U.S. Army Research Laboratory

ATTN: RDRL-WML-A

Aberdeen Proving Ground, MD 21005-5066

Author email: <benjamin.a.breech@us.army.mil>

We provide a two-dimensional extension to the traditional Gurney equations describing the velocity of fragments from an

exploding charge. Our approach differs from the recent Tie-peng et al. publication by applying different underlying

assumptions that may be more physically realistic. We demonstrate the procedure by computing the fragment velocity from an

detonating cylindrical charge surrounded by a thin shell with metal plates on top and bottom. We also demonstrate that under

specific conditions, the the results of Tie-peng et al. ’s approach and ours are identical.

Gurney equations, fragment velocities

UU 21

Benjamin A. Breech

Unclassified Unclassified Unclassified 410-278-3976

ii
Contents

List of Figures iv

1. Introduction 1

2. Background: Derivation of the Gurney Equations 1

2.1 Cylindrical Charge ......................................................................... 2

2.2 Sandwich Charge .......................................................................... 4

2.3 Limitations and Discussion................................................................ 6

3. Extension to the Gurney Equations 7

3.1 Relating v1 to vs............................................................................ 9

3.2 Tie-peng et al. Method.................................................................... 10

3.3 Discussion of Differences and Limitations ............................................... 12

4. Conclusion 13

5. References 14

Distribution List 15

iii
List of Figures

Figure 1. Diagram of a cylindrical explosive charge surrounded by a thin shell of mass Ms. ... 2

Figure 2. Sandwich charge configuration. If M1 = M2, the charge is referred to as a ‘symmetric sandwich charge’. Otherwise, the charge is an ‘asymmetric sandwich charge’

. .................................................................................................... 4

Figure 3. Cross section of the cylindrical charge considered for this work. The charge has

a radius of R and height of H. Two plates with masses M1 and M2 attach to the top and

bottom of the charge. Additionally, the charge is surrounded by a metal shell with mass

Ms. ................................................................................................ 7

iv
1. Introduction

The Gurney equations (1) provide an estimation of the velocities of fragments from an detonating

charge. The equations derive from the principles conservation of energy and momentum applied

to the gases produced by the detonation and the shell surrounding the charge. The equations

assume the fragments travel in one dimension (e.g., radially in the case of a cylinder).

Tie-peng et al. (2) recently demonstrated an extension to the Gurney equations for the case of a

cylindrical charge surrounded on the sides by a metal shell and with metal plates on top and

bottom. The Tie-peng et al. result assumes the detonation wave reaches the top and the

surrounding shell at the same time regardless of the radius and height of the cylinder. This

necessitates the assumption that the acceleration of the gases due to the explosion is not uniform

in all directions.

In this work, we adopt a different approach by assuming the acceleration of the gas due to the

explosion is uniform. We then work out a different version of the Gurney equations for the

cylindrical charge described. The resulting equations may be simpler to use than the Tie-peng et

al. as they do not rely upon quantities that are difficult to measure, such as the pressure on the

plates and sides.

Section 2 provides some background material on the Gurney equations. Section 3 then describes

the extension to the equations. We finish by providing some concluding remarks in section 4.

2. Background: Derivation of the Gurney Equations

This section provides background material on the Gurney equations. In particular, we show the

derivation for specific geometries of interest. Readers who are already familar with the Gurney

equations may skip this section.

The Gurney equations are based on some simple assumptions and modeling. Initiating the charge

results in a detonation wave that moves outward generating gaseous combustion products that

flow along behind it. Upon contact with the casing, the gases (e.g., combustion products) force

the casing to fragment and then push the fragmets outward with some velocity. The Gurney

equations ignore the work done on the casing to cause the fragmentation. Instead, the casing

1
(and all fragments) are assumed to go outward at the same velocity as the gases when contact was

made.

We obtain the velocity of the fragments by applying conservation of energy and, where

applicable, conservation of momentum to the gases and casings. Importantly, we assume that all

of the chemical potential energy in the explosion goes into the kinetic energy of the fragments and

combustion products. This ignores the work done on the casing to cause fragmentation, as well

as the increased internal energy of the explosive, but allows for a closed form estimate.

2.1 Cylindrical Charge

Let us now consider a simple cylindrical charge with radius R surrounded on the sides by a thin

metal casing. The charge has a mass per unit length of C. The casing has a mass per unit length

of Ms. Figure 1 shows the example charge.

Figure 1. Diagram of a cylindrical explosive charge surrounded by a thin shell of

mass Ms.

To simplify the geometry, we require the detonation wave to start uniformly along the center line

of the cylinder (e.g., r = 0) at the same time. The detonation wave will then expand outward in

all directions, enforcing azimuthal symmetry. We make the additional assumption that the

detonation wave has no significant effects in the ˆz direction. In section 3, we show how to relax

this assumption. With these simplifications, the problem reduces to the radial direction only.

To find the velocity of the fragments, we enforce conservation of momentum and energy. Due to

the azimuthal symmetry, conservation of momentum provides no useful information as

momentum is trivially conserved. We thus turn our attention to conservation of energy by

computing the total kinetic energy and equating to the chemical energy stored in the explosive.

The blast wave accelerates gases outward. We take the velocity of the gases when they reach the

2
shell to be vs

. The blast then fragments the shell and pushes the fragments outward. We will

assume the fragments move at the same velocity as the gases, i.e., vs. The total kinetic energy of

the fragments is then (1/2)Msv

2

s

.

The gases also have kinetic energy, which must be accounted for. The blast begins at the center

axis, where the velocity of the gases would be 0. The gases reach a maximum velocity of vs at

the shell. We therefore assume the velocity of the gases varies linearly with position, e.g.,

v(r) = r

R

vs (1)

where R is the radius of the cylinder. Taking ρ = C/πR

2

to be the density of the charge, we can

compute the kinetic energy (per unit length) of the charge products, Tc, as

Tc =

1

2

Z R

0

2πρrv2

(r)dr

=

1

2

2πρv

2

s

Z R

0

r

3

R2

dr

Tc =

1

2

C

2

v

2

s

(2)

Let E denote the chemical energy per unit mass for the particular explosive being used. The total

chemical energy per unit length of the explosive is given by EC. Following Gurney (1), we

assume the total chemical energy available in the explosive converts entirely to kinetic energy.

This provides us with the required relation to determine vs,

EC =

1

2

Msv

2

s +

1

2

C

2

v

2

s

2E = v

2

s



Ms

C

+

1

2



vs =

√

2E

√

α α

−1 ≡

Ms

C

+

1

2

(3)

The constant √

2E, which has units of velocity, is commonly referred to as Gurney’s Constant and

varies with the explosive used. All of the Gurney equations take the form of equation 3. The

equations differ only with the definition of α.

If we substantially increase Ms, then equation 3 predicts the fragments would move outward with

smaller velocity, which agrees with intuition. Interestingly, equation 3 predicts an asymptotic

maximum speed of √

2

√

2E for fragments as the amount of explosive is increased.

3
2.2 Sandwich Charge

Another useful example is a sandwich charge, where two flat plates, with masses per unit area of

M1 and M2, are separated by a charge mass per unit area C. If M1 = M2, then the entire package

is referred to as a symmetric sandwich charge. If the two masses are not equal, the package is a

asymmetric sandwich charge. The general case is shown in figure 2.

Figure 2. Sandwich charge configuration. If M1 = M2, the

charge is referred to as

a ‘symmetric sandwich

charge’. Otherwise, the

charge is an ‘asymmetric

sandwich charge’ .

For convenience, we orient the charge and plates so that the blast would send the plates in the

vertical (“ ˆy”) direction. Again, we reduce the problem to one dimension by assuming the

velocity of gaseous combustion products are insignificant in the ˆx direction.

As the blast wave goes outward, the gases imparts a velocity v1 to the M1 plate and a velocity −v2

to the M2 plate (minus sign added because the M2 plate will move in the opposite direction of M1).

We assume the velocity of the explosion products varies linearly in the ˆy direction. We thus have

v(y) = (v1 +v2)

y

H

−v2 (4)

where H is the height of the charge.

The problem is to determine the velocities, v1 and v2. As before, we apply conservation of

momentum and energy to obtain the required relations

From conservation of momentum, we have

0 = M1v1 − M2v2 +

Z H

0

ρv(y)dy, (5)

4
where ρ = C/H is the mass density. From this, we obtain the first relation,

v2 = Av1 A ≡

1+2(M1/C)

1+2(M2/C)

. (6)

The kinetic energy relation is

EC =

1

2

M1v

2

1 +

1

2

M2v

2

2 +

1

2

Z H

0

ρv

2

(y)dy (7)

Carrying out the computation, using equation 6 and noting that 1−A+A

2 = (A3 +1)/(A+1),

we arrive at the Gurney equation,

v1 =

√

2E

√

α1 (8)

α

−1

1 ≡

M1

C

+A

2 M2

C

+

1

3

A

3 +1

A+1

We can compute some interesting limits from equation 8. If the masses of the plates are equal,

then we obtain the symmetric sandwich equation with

α

−1

1 = 2

M1

C

+

1

3

. (9)

Another interesting limit occurs when one of the plates is not present (the “open face” sandwich).

For convenience, we pick M2 = 0 so there is no bottom plate. In this case, v2 then becomes the

velocity gas exiting from below the charge. We obtain

α

−1

1 =

(1+2

M1

C

)

3 +1

6(1+

M1

C

)

+

M1

C

(10)

The last limit of interest occurs when we allow one of the plates, say M2, to have substantially

large mass, e.g., M2 → ∞. If we apply this limit to equation 8, we find that v1 =

√

2E

√

α1 with

α

−1

1 =

M1

C

+

1

3

(11)

The result is correct, but only by coincidence as there is a subtle inconsistency. The problem lies

in applying conservation of momentum. For a plate of infinite mass, the bottom plate does not

move (v2 = 0), nor do gases from the explosion escape from the bottom. As a result,

conservation of momentum, equation 5, would imply that v1 = 0 for this case, which is

5
inconsistent with equation 11. Since the bottom plate is not moving, momentum is not conserved

which gives rise to this inconsistency (the bottom plate basically provides an external force on the

system). The proper way of handling this limit is to take v2 = 0 in the kinetic energy relation,

equation 7. We would then obtain equation 11.

2.3 Limitations and Discussion

Generally, the Gurney equations consider one dimension only by considering “ideal” charge

configurations. The “ideal” configurations impose some symmetry condition and have infinite

length allowing edge effects to be ignored. Some of these limitations can be relaxed. Edge

effects can be accounted for by adjusting the M/C ratio (3) or by introducing correction terms (4).

Symmetry conditions could be relaxed, but at the expense of dramatically increased complexity.

All of the Gurney equations depend on the constant √

2E, which must be measured for each

explosive. √

2E can be measured either by empirically fitting the Gurney equations or through

calormetric means (5). The computated velocities obviously depend on the quality of the

measurement of the constant.

The main advantage of the Gurney equations lie in their simplicity. The equations can quickly be

derived and can provide good estimates of the fragment velocities. There may be compensating

errors present in the analysis. The energy required to fragment the casing is ignored, which

should cause the equations to over estimate the fragment velocities. On the other hand, using

simple velocity profiles that vary linearly can underestimate the actual velocities of the

combustion products. Actual detonations cause shocks to develop which will cause combustion

products to move faster than a simple linear fit.

6
3. Extension to the Gurney Equations

Here we discuss our extension to two dimensions for the cylindrical charge described in section 1.

The geometry of the problem incorporates both the cylinder and the asymmetric sandwich

problems discussed in section 2.

We now move to two dimensions. The charge configuration is shown in figure 3. The charge has

a radius of R, and height of H. At z = 0, we attach a plate of mass M2. A second plate of mass

M1 is attached at z = H. A metal shell of mass Ms surrounds the charge.

For this case, we assume the detonation point is located along the center axis (r = 0). This

provides azimuthal symmetry. The height of the detonation point is determined by velocity

profile imposed on the combustion products.

Figure 3. Cross section of the cylindrical charge

considered for this work. The charge

has a radius of R and height of H. Two

plates with masses M1 and M2 attach to

the top and bottom of the charge. Additionally, the charge is surrounded by

a metal shell with mass Ms.

7
Our problem is to compute the velocity of fragments, v1, v2, and vs

, created from the plates and

shell given the masses of the plates, M1 and M2, the mass of the shell, Ms, the mass of the

explosive, C, and the Gurney Constant for the explosive, √

2E.

As before, we assume the blast pushes gases outward with a velocity that varies linearly. In the

radial direction, the velocity varies with distance from the center line. Along the height of the

cylinder, the velocity varies linearly between −v2 and v1. Combining these, we have

v(r,z) = r

R

vsrˆ+

h

(v1 +v2)

z

H

−v2

i

zˆ (12)

With this velocity profile, we can examine conservation of momentum. In the radial direction,

momentum is trivially conserved and provides no useful relations. In the ˆz direction, however,

we obtain

0 = M1v1 − M2v2 +

Z R

r=0

Z H

z=0

2πρr

h

(v1 +v2)

z

H

−v2

i

dr dz

which gives our first relation,

v2 = Av1 A ≡

1+2(M1/C)

1+2(M2/C)

(13)

The kinetic energy relation is

CE =

1

2

M1v

2

1 +

1

2

M2v

2

2 +

1

2

Msv

2

s +

1

2

Z R

r=0

Z H

z=0

2πρr|v(r,z)|

2

dr dz (14)

which, combined with equation 13, gives the second relation among the velocities,

2E =



M1

C

+A

2 M2

C

+

1

3

A

3 +1

A+1



v

2

1 +



Ms

C

+

1

2



v

2

s

(15)

We clearly see that the first term on the right hand side of equation 15 corresponds to the Gurney

equation for an asymmetric sandwich, while the second term corresponds to the cylinder charge.

If we take Ms → ∞, which requires vs → 0, then we obtain the Gurney equation for the

asymmetrical sandwich. If we repeat the process for the limits of M1,M2 → ∞, we recover the

Gurney equation for the cylindrical charge.

8
If we empirically know either v1 or vs

, then equation 15 may be used to obtain the other.

However, we can make no further analytic progress without additional assumptions.

3.1 Relating v1 to vs

One approach to relate v1 and vsis to assume the detonation wave accelerates the gases uniformly

in all directions. The gases will, in general, reach the shell and the plates at different times. If a

is the acceleration, we can compute the velocity and distances the gases travel as

vs = ats and R =

1

2

at2

s

(16)

for the radial direction. This implies

a =

1

2

v

2

s

R

. (17)

In the axial direction, the distance traveled is dependent on the location of the detonation point.

For symmetry reasons, we assumed the detonation point was located along the r = 0 line. We can

locate the axial location by computing where vz = 0 from the velocity profile given in equation 12,

vz(z0) = 0 = (v1 +v2)

z0

H

−v2 (18)

z0 =

v2

v1 +v2

H =

A

1+A

H (19)

where equation 13 was used.

The gases cover a distance of H −z0, which can then be used to write down

v1 = at1 and 1

1+A

H =

1

2

at2

1

(20)

which implies

a =

1+A

2H

v

2

1

. (21)

Equating equations 17 and 21, we obtain the needed relation between v1 and vs,

v

2

1 =

H

R

1

A+1

v

2

s

. (22)

9
Substituting equation 22 back into equation 15 provides us the equation for vs

,

vs =

√

2E

√

αs α

−1

s ≡

1

A+1



M1

C

+A

2 M2

C

+

1

3

A

3 +1

A+1



H

R

+



Ms

C

+

1

2



(23)

Using this with equations 13 and 22 gives the other velocities,

v1 =

√

2E

√

α1 α

−1

1 ≡



M1

C

+A

2 M2

C

+

1

3

A

3 +1

A+1



+ (A+1)



Ms

C

+

1

2



R

H

(24)

v2 =

√

2E

√

α2 α

−1

2 ≡

1

A2



M1

C

+A

2 M2

C

+

1

3

A

3 +1

A+1



+

A+1

A2



Ms

C

+

1

2



R

H

(25)

We may take limits of these equations, but doing so requires careful interpretation. All the

speeds depend, in some fashion, on the height to radius ratio, H/R. When H ≪ R, the problem

geometry becomes two plates sandwiched around a layer of charge. We may expect that the

equations would then reduce to the sandwich equations shown in section 2. However, equations

23, 24, and 25 emphasize the cylindrical charge equation instead. The reason for the seemingly

contradictory limit lies in the assumptions used to obtain a relationship between v1 and vs. We

assumed that the blast wave accelerates the gases uniformly. If H ≪ R, then the gases in the ˆz

direction have far less time to be accelerated before they encounter the plates than the gases in the

radial direction. As a result, most of the available energy will go into the fragments from the

shell. Hence, the cylindrical charge should be emphasized. When considering limits of this

type, it may be better to apply the limits to the kinetic energy relation shown in equation 15.

3.2 Tie-peng et al. Method

Recently, Tie-peng et al. (2) examined this identical charge configuration, but with different

underlying assumptions. In particular, they obtain relations between v1 and vs by assuming the

detonation wave reaches the top plate and the surrounding shell at the same time. This

necessarily implies the acceleration of the gases due to the blast wave is not uniform in all

directions, but rather accelerates the gases differently in the ˆr and ˆz directions.

We can obtain the accelerations by introducing the pressures on the top plate, P1, and the

surrounding shell, Ps. The definition of pressure immediately provide

P1 =

M1a1

πR2

Ps =

Msas

2πRH

(26)

Since Tie-peng et al. assumed the detonation wave reaches the plate and shell at the same time, we

10
can immediately write

v1

vs

=

a1

as

=

Ms

M1

P1

Ps

R

2H

(27)

Using this relation in equation 15, we obtain the Tie-peng et al. result,

v1 =

√

2E

√

αt

α

−1

t =

1

3

A

3 +1

A+1

+A

2 M2

C

+

M1

C



16

P

2

s

P

2

1

M1

Ms

H

2

d

2

+1



+8

P

2

s

P

2

1

M2

1

M2

s

H

2

d

2

, (28)

where d = 2R is the diameter of the cylinder.

The Tie-peng et al. approach and our approach utilize different underlying assumptions.

However, we note that when H = 2R, and M1 = M2, the two methods should produce the same

result. M1 = M2 implies that v1 = v2, and therefore the detonation point is at the center of the

cylinder. In the Tie-peng et al. procedure, H = 2R implies that the acceleration must be uniform

in all directions in order for the blast wave to hit the top and side at the same time. Within our

approach, the blast wave also hits the top plate and side shell at the same time for this

configuration.

When H = 2R (and recalling that, for this case, as = a1), we immediately find from equation 26,

Ps

P1

=

Ms

4M1

(29)

Due to the assumption that M1 = M2, we must also have that A = 1.

Putting all this into equation 28, we find

α

−1

t =

1

3

A

3 +1

A+1

+A

2 M2

C

+

M1

C

"

16

Ms

4M1

2 M1

Ms

H

2

d

2

+1

#

+8



Ms

4M1

2 M2

1

M2

s

H

2

d

2

=



1

3

+

M2

C

+

M1

C



+



Ms

C

+

1

2



. (30)

which is identical to our result shown in equation 24 when H = 2R and A = 1.

11
3.3 Discussion of Differences and Limitations

The differences in the two procedures lie in how the third and final relation among the velocities

is determined. The Tie-peng et al. result assumes the blast wave accelerates gases in such a way

as to make the gases reach the top plate and the surrounding shell at the same time. This may not

be physically realistic as the blast wave should emanate from the detonation outward causing an

acceleration of the gases that is the same in all directions. The assumption of the same

acceleration in all directions underlies our approach. Furthermore, our approach may be simpler

to use as it does not rely upon the pressures on the plate and shell, which are difficult to measure

quantities.

At the same time, however, we note that both procedures ignore more complicated physics that

can happen when considering two dimensional flows from a detonation. In particular, once the

gases impact on the surface closest to the detonation point, the surface will cause some of the

gases to reflect back, or deflect into the other direction. Such a process would invalidate the

assumptions of both procedures. A more thorough investigation of the complex physics involved

is beyond the scope of this work.

Another limitation to note is that neither method handles cases where the radius is “much

different” than the height of the charge. For example, let us consider a charge whose height is

four times the radius. We (somewhat) arbitrarily place the detonation point r = 0,z = R (e.g., one

quarter of the height). As the blast proceeds, the detonation wave will “fill” the bottom half of

the cylinder. That is, the detonation wave will reach the bottom plate and the lower half of the

surrounding metal shell at the same time. However, the top portion will remain unaffected at this

time. Effectively, the bottom portion of the charge will explode well before the top portion even

recognizes a detonation has occured. This case is not accounted for in the Gurney style methods

and would likely require the use of hydrocodes for a proper analysis.

The most applicable cylindrical charge configuration for this method is one where the height is

twice the radius, a detonation point located at the center (e.g., r = 0,z = H/2) and whose top and

bottom plates have the same mass. This configuration should avoid the limitations discussed in

the previous paragraph. The velocities of the fragments will then be given by equations 23, 24,

and 25. The velocity estimates may also be applicable when H is not “too different” from 2R.

That is we may expect the velocity estimates given above will still be reasonable even if H is

slightly larger than 2R. We feel that the estimates will certainly break down once H gets near to

4R. Similarly, the estimates may hold if H is decreased to values less than 2R, but will break

down again at some lower limit (perhaps H = R). Comparisons with hydrocodes would be useful

12
to determine the exact range of applicability.

4. Conclusion

In this work, we examined extensions to the Gurney equations for fragments that can travel in two

dimensions. Our work was based somewhat in the work of Tie-peng et al. , but with different

underlying assumptions. In particular, we assumed that the blast wave causes an accleration that

is the same in all directions. This results in a simple equation for the velocity fragments that

bears similarities to the usual cylindrical and sandwich Gurney equations.

13
5. References

1. Gurney, R. The Initial Velocities of Fragments from Bombs, Shells, Grenades; U.S. Army

Ballistic Research Laboratory: Aberdeen Proving Grounds, 1943.

2. Tie-peng, L.; Jian-peng, Q.; feng Niu Gong-jie, X. E. An Engineering Model for Calculating

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Charge. 25th International Symposim on Ballistics 2010, 659.

3. Charron, Y. J. “Estimation of Velocity Distribution of Fragmenting Warheads Using a

Modified Gurney Method”, Master’s thesis, Air Force Institute of Technology, 1979.

4. Chanteret, P. Velocity of HE Driven Metal Plates with Finite Lateral Dimensions. 12th

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to Driven Metal; SC-RR-7-790; Sandia Laboratories 1970.

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