Detonation Threshold Calculations for Insensitive Munitions Tests

by Daniel J. Vavrick, Ph.D.,
Dahlgren, Virginia

Pages 1 | 2 | 3 | 4 | 5

Probability of Proposed NATO Fragment
In early 2001, discussions and analyses were undertaken to evaluate Insensitive Munitions (IM) fragment impact testing procedures. The objective was to critically evaluate current procedures and, if necessary, recommend changes to the MIL-STD-2105B [Ref. 5] fragment testing procedure. Consensus resulted in a procedure that formed the basis of a U.S. position for ongoing negotiations for NATO STANAG 4496, Fragment Impact, Munitions Test Procedure. The major issues of the then-current procedure were repeatability, impact fragment orientation, the proximity of multiple fragment impacts, and the impact velocity. This effort was conducted under the immediate direction of the Impact and Shock Induced Reactions Panel of Joint Army-Navy-NASA-Air Force Propulsion Systems Hazards Subcommittee (JANNAF PSHS).

One major recommendation of the effort was to change the impact fragment to a single conical-tipped right-circular-cylindrical fragment with a mass of 18.6g [Ref. 6]. Similar to the current MIL-STD-2105B test, the recommended procedure has primary and alternate test methods with velocities of 2530±90 and 1830±60 m/s, respectively. These revised test procedures are currently spelled out in the NATO STANAG 4496 [Ref. 7] and MIL-STD-2105C [Ref. 8] documents.

Orientation of the fragment at impact is an important issue of the current IM test fragment, a 16g steel cube, and is also an important issue of the proposed one. Under the assumption of a random tumbling, the probability of the impact orientation can be calculated analytically with a method outlined in Ref. 2.

The impact angle of a flat-faced fragment with a surface is the angle between the normal to the nearest face and the normal to the impacted surface. For the conical-tipped, right-circular-cylindrical fragment of interest here, the definition of the impact angle must be modified. This fragment has three "faces." The impact angle of the rear circular face and the side cylindrical face to the impacted surface is, as before, the angle between the normal to that face (at the impact point) and the normal to the impacted surface. The impact angle of the front conical face to the impacted surface is defined as the angle between the axis of the cone and the normal to the impacted surface. The probability that this angle will be in a particular range can be predicted by calculating an area ratio similar to that discussed in Ref. 2. This ratio is area swept out by a particular impact angle interval on a sphere, divided by the total area of the sphere. The sphere is the smallest diameter sphere that will contain the impacting fragment when the center of the sphere and center of mass of the fragment are coincident. The angle swept out by a particular impact angle interval is the area on the surface of the sphere that is penetrated by a radius of the sphere as it is oriented, so that it assumes all positions with the given impact angle interval.

Given the conical-tipped, right-circular-cylindrical fragment shown in Figure 9, if the impact angle was varied randomly, the probability of the fragment impacting at a given angle is the ratio of the number of times it impacts at that angle over the total number of times that the fragment impacted the surface. The random number of impact occurrence needs to be large for the probability to be accurate.

Figure 9 — Shape of the NATO IM Test Fragment.

The probability of the fragment impacting at a given angle can also be viewed in terms of areas on a sphere that contain just the fragment and have center and center of gravity (CG) coincident. For the fragment of cylinder length l = 14.3 mm, cone height h = 1.26 mm and radius r = 7.15 mm, the volume is given by the equation:

(13)

The position X_CG of the CG is located along the axis of the fragment at a distance from the end given by the equation:

(14)

Then, the radius R of the containing sphere is given by the equation:

(15)

Consider the radius that coincides with the normal to one of the faces of the fragment or with the axis for the front cone "face." The radii that are oriented at an angle θ to that coinciding radius represent possible impact surface normals. Therefore, θ can be considered a possible impact angle. For a given impact angle interval Δθ, the radii, which have orientation angles in that interval, swept out an area A_i for that i^th face. The area A_i for radii between 0 and θ are shown in Figure 10.

Figure 10 — Area A_i Swept Out by Radii Within the Angle θ.

The conical front or rear side would have an equal area for the same angle. The side face is cylindrical so an impact angle between 0 and θ would be the projection of a cylindrical ring onto the spherical surface. Because these areas represent possible impact orientations and the total area of the sphere represents all possible orientations, the probability P_i (θ) of impacting at an angle less than θ for a given i^th face is given by the expression:

(16)

The faces are not identical, so the combined probability of hitting any face with an impact angle less than θ is less meaningful. However, as in the case of a polyhedron, it could be calculated by adding the areas of all three faces for a given angle θ. The probability of hitting at any orientation relative to the axis of the projectile can also be calculated by letting the area swept out by an angle relative to the axis grow to 180 degrees and thus eventually encompass the entire surface area of the sphere. The area A_i can be found by integration using Equation 7.

Determining the area is now straightforward until the radii orientations are such that the radii, which were passing through one face, begin to pass through an adjacent face. However, one can still determine an analytic expression for the area. For a conical-tipped, right-circular-cylindrical fragment, there are two angles for each of the two ends that denote transition from one expression for the area to another. The two angles correspond to a radius reaching the rear (θ₁) or front (θ₂) edges of the right circular cylinder as shown in Figure 11.

Figure 11 — Critical Angles for Transition of Area Calculation.

The transition angles for this fragment are as follows:

(17a,b)

The expression for the area, as a function of the angle θ, for the front face of cone dimensions r and h, is given by the following expression:

(18)

The expression for the area, as a function of the angle θ, for the rear face of radius r, is given by the following expression:

(19)

The expression for the area, as a function of the angle θ, for the side face of the cylinder of radius r, is given by the following expression:

(20)

(21)

Using these expressions for areas swept out by radii having (impact) angles of θ and the equation for using area to calculate probability (Equation 6), one can calculate the probability of having an impact angle less than or equal to any θ for any face. Subtracting the probabilities for two different angles will give the probability that the impact angle will lie in the interval θ_I to θ_II.

The probability for this fragment can be computed rather easily using the given equations, and the computation is straightforward.

For instance, one can easily compute the probability of impacting any face. The probabilities of hitting the front, rear or side face are 0.1518, 0.1413 and 0.7069, respectively. As a check, the sum of the three probabilities equals one, as it should.

The probability of interest is that for impacts with the front face. The above equations were used to generate these values and are listed in Table 5 under the heading "Equation."

Table 5 — Probability for Impact Angles with the Front Face for the U.S. Proposed NATO IM Test Fragment.

Angle (deg.)	Equation	Angle (deg.)	Equation
0	0.0	30	0.0670
5	0.0019	35	0.0904
10	0.0076	40	0.1170
15	0.0170	45	0.1464
20	0.0302	45.866	0.1518
25	0.0468

This data is plotted in Figure 12 for comparison. Besides illustrating how impact orientation can be calculated for an arbitrarily shaped fragment, this approach indicates the likelihood of impacting in a particular orientation range for this fragment under the assumption of random tumbling. If the tumbling is not completely random, the result can still be obtained by consideration of a slightly modified area ratio.

Figure 12 — Minimum Impact Probability for Front Face.

Summary
Analytic equations have been derived to predict the probability of impact angles for one or several tumbling fragments. This analytic method is based on the area ratio of a subsurface of a sphere just containing the fragment to the total area of that sphere. The subsurface is that swept out by the radius of the sphere as it assumes orientations with angles in a given impact angle. These equations are simple and exact.

These expressions, when used in conjunction with hydrocodes or the Jacobs-Roslund, can be used to estimate the probability of detonation from fragment impact. Consequently, the relationship of a munition to the detonation threshold for the fragment impact IM test can be estimated.



About the Author

Dr. Dan Vavrick has worked in the Lethality and Weapons Effectiveness Branch at the NAVSEA Dahlgren Laboratory since 1995. He came to Dahlgren from NSWC White Oak Laboratory, Reentry Systems Branch, where he began his government career in 1985. Prior to that, he was an engineer in the Explosive Dynamics Group of Honeywell, Inc., a physics teacher with the Peace Corps in Belize, and a combat engineer with the U.S. Army in Vietnam. He is a Registered Engineer in Maryland.

Dr. Vavrick is an active officer or member in a number of engineering societies and community organizations. He received his Ph.D. in Solid Mechanics in 1977 from the University of Minnesota. He has published a number of papers and holds several patents.



References

1. Vavrick, D.J. "Analysis for Critical Velocity for Detonation from Multi-Fragment Impacts on Bare Explosive and Composite Plate Covered H-6 Explosive." Proceedings, International Workshop on New Models and Predictive Methods for Shock Wave/Dynamic Process in Energetic Materials and Related Solids. University of Maryland Conference Center, College Park, Maryland, July 5-9, 1999.
2. Vavrick, D.J. "Analytic Expression for Predicting the Probability of Impact Angles for a Tumbling Polyhedron." International Conference on Computational Engineering and Sciences (ICCES '03). Corfu, Greece, July 24-29, 2003.
3. Myruski, B. and D.J. Vavrick. "Fundamental Improvements in the Jacobs Equation for the Prediction of Shock Initiation of Cased Munitions." Warheads and Ballistics Classified Symposium. Monterey, California, June 2003.
4. Dickinson, D.L. "Predicting the Probability of Impact Angles for a Tumbling Polyhedron." NSWC TR 80-152. Naval Surface Weapons Center, Dahlgren, Virginia, May 1982.
5. MIL-STD-2105B, Military Standard, Hazard Assessment Tests for Non-Nuclear Munitions. AMSC N6037. Department of Defense, 1994.
6. Baker, Patrick. "Summary of JANNAF Review of MIL-STD-2105B Fragment Impact Test Procedure." Weapons and Materials Research Directorate, U.S. Army Research Laboratory, Aberdeen Proving Ground, Maryland, 21005-5066, 2002.
7. Fragment Impact, Munitions Test Procedure. NATO Standardization Agreement, NATO STANAG 4496, Draft 12, Edition 1. July 30, 2003.
8. MIL-STD-2105C, Military Standard, Hazard Assessment Tests for Non-Nuclear Munitions. AMSC N6037. Department of Defense, July 14, 2003.

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