Gurney equations
Encyclopedia
The Gurney equations are a set of mathematical formulas used in Explosives engineering
to relate how fast an explosive will accelerate a surrounding layer of metal or other material when the explosive detonates. This determines how fast fragments are released by military explosives, how quickly shaped charge
explosives accelerate their liners inwards, and in other calculations such as explosive welding where explosives force two metal sheets together and bond them.
The equations were first developed in the 1940s by R.W. Gurney
and have been expanded on and added to significantly since that time.
Gurney made a simplifying assumption that there would be a linear velocity gradient in the explosive detonation product gases. This has worked well for most configurations, but see the section Anomalous predictions below.
For implosion systems, with a hollow explosive charge accelerating an inner mass towards their center, the calculations additionally take into account:
Values of
As a simple approximate equation, the physical value of 
is usually very close to 1/3 of the detonation velocity of the explosive material for standard explosives. For a typical set of military explosives, the value of 
ranges from between 2.79 and 3.15.
Note that
is dimensionally equal to kilometers per second, a more familiar unit for many applications.
For brittle metal shells, the fragment velocities are typically about 80% of the value predicted by the Gurney formulas.
Small explosive charges, where the explosives diameter is not significantly larger than its thickness, have reduced effectiveness as gas and energy are lost to the sides.
This loss is empirically modeled as reducing the effective explosive charge mass C to an effective value Ceff which is the volume of explosives contained within a 60 degree cone with its base on the explosives/flyer boundary.
Putting a cylindrical tamper around the explosive charge reduces that side loss effectively, as analyzed by Benham.
in which the Gurney equations misrepresent the actual physical behavior.
The range of values for which the basic Gurney equations generated anomalous values is described by (for flat asymmetrical and open-faced sandwich configurations):
For an open-faced sandwich configuration (see below), this corresponds to values of
of 0.5 or less. For a sandwich with tamper mass equal to explosive charge mass (
) a flyer plate mass of 0.1 or less of the charge mass will be anomalous.
This error is due to the configuration exceeding one of the underlying simplifying assumptions used in the Gurney equations - that there is a linear velocity gradient in the explosive product gases. For values of
outside the anomalous region this is a very good assumption. Hirsch demonstrated that as the total energy partition between the flyer plate and gases exceeds unity, the assumption breaks down, and the Gurney equations become less accurate as a result.
Complicating factors in the anomalous region include detailed gas behavior of the explosive products, including the reaction products' Heat capacity ratio
or γ.
Modern explosives engineering utilizes computational analysis methods which avoid this problem.
This configuration is a first-order approximation for most military explosive devices - Artillery shells
, Bombs
, and most missile Warhead
s. These use mostly cylindrical explosive charges.
This model approximates the behavior of military Grenades, and some Cluster bomb
submunitions.
Symmetrical sandwiches are used in some Reactive armor
applications, on heavily armored vehicles such as Main battle tank
s. The inwards-firing flyer will impact the vehicle main armor, causing damage if the armor is not thick enough, so these can only be used on heavier armored vehicles. Lighter vehicles use open-face sandwich reactive armor (see below). However, the dual moving plate method of operation of a symmetrical sandwich offers the best armor protection.
Let:
Since:
Then:
Which gives:
Open-faced sandwich configurations are used in Explosion welding and some other metalforming operations.
It is also a configuration commonly used in Reactive armour
on lightly armored vehicles, with the open face down towards the vehicle's main armor plate. This minimizes the reactive armor units damage to the vehicle structure during firing.
Unlike other forms of the Gurney equation, implosion forms (cylindrical and spherical) must take into account the shape of the control volume of the detonating shell of explosives and the distribution of momentum and energy within the detonation product gases. For cylindrical implosions, the geometry involved is simplified to include the inner and outer radius of the explosive charge (Ri and Ro).
While the imploding cylinder equations are fundamentally similar to the general equation for asymmetrical sandwiches, the geometry involved (volume and area within the explosive's hollow shell, and expanding shell of detonation product gases pushing inwards and out) is more complicated, as the equations demonstrate.
The constant
was experimentally and analytically determined to be 1.0.
The spherical Gurney equation has applications in early nuclear weapons design.
Explosives engineering
Explosives engineering is the field of science and engineering which is related to examining the behavior and usage of explosive materials.- Topics :Some of the topics that explosives engineers study, research, and work on include:...
to relate how fast an explosive will accelerate a surrounding layer of metal or other material when the explosive detonates. This determines how fast fragments are released by military explosives, how quickly shaped charge
Shaped charge
A shaped charge is an explosive charge shaped to focus the effect of the explosive's energy. Various types are used to cut and form metal, to initiate nuclear weapons, to penetrate armor, and in the oil and gas industry...
explosives accelerate their liners inwards, and in other calculations such as explosive welding where explosives force two metal sheets together and bond them.
The equations were first developed in the 1940s by R.W. Gurney
and have been expanded on and added to significantly since that time.
Underlying physics
When an explosive surrounded by a metallic or other solid shell detonates, the outer shell is accelerated both by the initial detonation shockwave and by the expansion of the detonation gas products contained by the outer shell. Gurney modeled how energy was distributed between the metal shell and the detonation gases, and developed formulas that accurately describe the acceleration results.Gurney made a simplifying assumption that there would be a linear velocity gradient in the explosive detonation product gases. This has worked well for most configurations, but see the section Anomalous predictions below.
Definitions and units
The Gurney equations use and connect the following quantities:- C - The mass of the explosive charge
- M - The mass of the accelerated shell or sheet of material (usually metal). The shell or sheet is often referred to as the flyer, or flyer plate.
- V or Vm - Velocity of accelerated flyer after explosive detonation.
- N - The mass of a tamper shell or sheet on the other side of the explosive charge, if present.
- The Gurney Constant for a given explosive. This is expressed in units of velocity (millimeters per microsecond, for example) and compares the relative flyer velocity produced by different explosives materials.
For implosion systems, with a hollow explosive charge accelerating an inner mass towards their center, the calculations additionally take into account:
- Ro - Outside radius of the explosive charge.
- Ri - Inside radius of the explosive charge.
Values of 
and detonation velocity for various explosives
As a simple approximate equation, the physical value of is usually very close to 1/3 of the detonation velocity of the explosive material for standard explosives. For a typical set of military explosives, the value of ranges from between 2.79 and 3.15.| Density | Detonation Velocity |  |
|
|---|---|---|---|
| Explosive |  |  |  |
| Composition B Composition B Composition B, colloquially "comp B", is an explosive consisting of castable mixtures of RDX and TNT. It is used as the main explosive filling in artillery projectiles, rockets, land mines, hand grenades, sticky bombs and various other munitions... |
1.72 | 7.92 | 2.70 |
| Composition C-3 | 1.60 | 7.63 | 2.68 |
| Cyclotol Cyclotol Cyclotol is an explosive consisting of castable mixtures of RDX and TNT.It is related to the more common Composition B, which is roughly 60% RDX and 40% TNT; various compositions of Cyclotol contain from 65% to 80% RDX.... 75/25 |
1.754 | 8.25 | 2.79 |
| HMX HMX HMX, also called octogen, is a powerful and relatively insensitive nitroamine high explosive, chemically related to RDX. Like RDX, the name has been variously listed as High Melting eXplosive, Her Majesty's eXplosive, High-velocity Military eXplosive, or High-Molecular-weight rdX.The molecular... |
1.835 | 8.83 | 2.80 |
| LX-14 Polymer-bonded explosive A polymer-bonded explosive, also called PBX or plastic-bonded explosive, is an explosive material in which explosive powder is bound together in a matrix using small quantities of a synthetic polymer... |
1.89 | 9.11 | 2.97 |
| Octol Octol Octol is a melt-castable, high explosive mixture consisting of HMX and TNT in different weight proportions.-Composition:Two formulations are commonly used:* 70% HMX & 30% TNT* 75% HMX & 25% TNT... 75/25 |
1.81 | 8.48 | 2.80 |
| PBX 9404 Polymer-bonded explosive A polymer-bonded explosive, also called PBX or plastic-bonded explosive, is an explosive material in which explosive powder is bound together in a matrix using small quantities of a synthetic polymer... |
1.84 | 8.80 | 2.90 |
| PBX 9502 Polymer-bonded explosive A polymer-bonded explosive, also called PBX or plastic-bonded explosive, is an explosive material in which explosive powder is bound together in a matrix using small quantities of a synthetic polymer... |
1.885 | 7.67 | 2.377 |
| PETN PETN Pentaerythritol tetranitrate , also known as PENT, PENTA, TEN, corpent, penthrite , is the nitrate ester of pentaerythritol. Penta refers to the five carbon atoms of the neopentane skeleton.PETN is most well known as an explosive... |
1.76 | 8.26 | 2.93 |
| RDX RDX RDX, an initialism for Research Department Explosive, is an explosive nitroamine widely used in military and industrial applications. It was developed as an explosive which was more powerful than TNT, and it saw wide use in WWII. RDX is also known as cyclonite, hexogen , and T4... |
1.77 | 8.70 | 2.83 |
| Tetryl Tetryl 2,4,6-Trinitrophenylmethylnitramine commonly referred to as tetryl is a sensitive explosive compound used to make detonators and explosive booster charges.... |
1.62 | 7.57 | 2.50 |
| TNT | 1.63 | 6.86 | 2.44 |
| Tritonal Tritonal Tritonal is a mixture of 80% TNT and 20% aluminium powder, used in several types of ordnance such as air-dropped bombs. The aluminium improves the total heat output and hence impulse of the TNT - the length of time during which the blast wave is positive... |
1.72 | 6.70 | 2.32 |
for some common explosivesNote that
is dimensionally equal to kilometers per second, a more familiar unit for many applications.Fragmenting versus nonfragmenting outer shells
The Gurney equations give a result which assumes that the flyer plate remains intact throughout the acceleration process. For some configurations, this is true - explosives welding, for example, uses thin sheets of explosives to evenly accelerate flat plates of metal and collide them, and the plates remain solid throughout. However, for many configurations where materials are being accelerated outwards the expanding shell will fracture due to stretching as it expands. When it fractures, it will usually break into many small fragments due to the combined effects of ongoing expansion of the shell and stress relief waves moving into the material from fracture points.For brittle metal shells, the fragment velocities are typically about 80% of the value predicted by the Gurney formulas.
Effective charge volume for small diameter charges
The basic Gurney equations for flat sheets assume that the sheet of material is large diameter.Small explosive charges, where the explosives diameter is not significantly larger than its thickness, have reduced effectiveness as gas and energy are lost to the sides.
This loss is empirically modeled as reducing the effective explosive charge mass C to an effective value Ceff which is the volume of explosives contained within a 60 degree cone with its base on the explosives/flyer boundary.
Putting a cylindrical tamper around the explosive charge reduces that side loss effectively, as analyzed by Benham.
Anomalous predictions
In 1996, Hirsch described a performance region, for relatively small ratios of in which the Gurney equations misrepresent the actual physical behavior.The range of values for which the basic Gurney equations generated anomalous values is described by (for flat asymmetrical and open-faced sandwich configurations):
For an open-faced sandwich configuration (see below), this corresponds to values of
of 0.5 or less. For a sandwich with tamper mass equal to explosive charge mass (
) a flyer plate mass of 0.1 or less of the charge mass will be anomalous.
This error is due to the configuration exceeding one of the underlying simplifying assumptions used in the Gurney equations - that there is a linear velocity gradient in the explosive product gases. For values of
outside the anomalous region this is a very good assumption. Hirsch demonstrated that as the total energy partition between the flyer plate and gases exceeds unity, the assumption breaks down, and the Gurney equations become less accurate as a result.Complicating factors in the anomalous region include detailed gas behavior of the explosive products, including the reaction products' Heat capacity ratio
Heat capacity ratio
The heat capacity ratio or adiabatic index or ratio of specific heats, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume . It is sometimes also known as the isentropic expansion factor and is denoted by \gamma or \kappa . The latter symbol kappa is...
or γ.
Modern explosives engineering utilizes computational analysis methods which avoid this problem.
Cylindrical charge equation
For the simplest case, a long hollow cylinder of metal is filled completely with explosives. The cylinder's walls are accelerated outwards as described by:This configuration is a first-order approximation for most military explosive devices - Artillery shells
Shell (projectile)
A shell is a payload-carrying projectile, which, as opposed to shot, contains an explosive or other filling, though modern usage sometimes includes large solid projectiles properly termed shot . Solid shot may contain a pyrotechnic compound if a tracer or spotting charge is used...
, Bombs
Aerial bomb
An aerial bomb is a type of explosive weapon intended to travel through the air with predictable trajectories, usually designed to be dropped from an aircraft...
, and most missile Warhead
Warhead
The term warhead refers to the explosive material and detonator that is delivered by a missile, rocket, or torpedo.- Etymology :During the early development of naval torpedoes, they could be equipped with an inert payload that was intended for use during training, test firing and exercises. This...
s. These use mostly cylindrical explosive charges.
Spherical charge equation
A spherical charge, initiated at its center, will accelerate a surrounding flyer shell as described by:This model approximates the behavior of military Grenades, and some Cluster bomb
Cluster bomb
A cluster munition is a form of air-dropped or ground-launched explosive weapon that releases or ejects smaller sub-munitions. Commonly, this is a cluster bomb that ejects explosive bomblets that are designed to kill enemy personnel and destroy vehicles...
submunitions.
Symmetrical sandwich equation
A flat layer of explosive with two identical heavy flat flyer plates on each side will accelerate the plates as described by:Symmetrical sandwiches are used in some Reactive armor
Reactive armour
Reactive armour is a type of vehicle armour that reacts in some way to the impact of a weapon to reduce the damage done to the vehicle being protected. It is most effective in protecting against shaped charges and specially hardened long rod penetrators...
applications, on heavily armored vehicles such as Main battle tank
Main battle tank
A main battle tank , also known as a battle tank or universal tank, is a tank that fills the heavy direct fire role of many modern armies. They were originally conceived to replace the light, medium, heavy and super-heavy tanks. Development was spurred onwards in the Cold War with the development...
s. The inwards-firing flyer will impact the vehicle main armor, causing damage if the armor is not thick enough, so these can only be used on heavier armored vehicles. Lighter vehicles use open-face sandwich reactive armor (see below). However, the dual moving plate method of operation of a symmetrical sandwich offers the best armor protection.
Asymmetrical sandwich equation
A flat layer of explosive with two different mass flat flyer plates will accelerate the plates as described by:Let:
Infinitely tamped sandwich equation
When a flat layer of explosive is placed on a practically infinitely thick supporting surface, and topped with a flyer plate of material, the flyer plate will be accelerated as described by:Open-faced sandwich equation
A single flat sheet of explosives with a flyer plate on one side, known as an "Open-faced sandwich", is described by:Since:
Then:
Which gives:
Open-faced sandwich configurations are used in Explosion welding and some other metalforming operations.
It is also a configuration commonly used in Reactive armour
Reactive armour
Reactive armour is a type of vehicle armour that reacts in some way to the impact of a weapon to reduce the damage done to the vehicle being protected. It is most effective in protecting against shaped charges and specially hardened long rod penetrators...
on lightly armored vehicles, with the open face down towards the vehicle's main armor plate. This minimizes the reactive armor units damage to the vehicle structure during firing.
Imploding cylinder equation
A hollow cylinder of explosives, initiated evenly around its surface, with an outer tamper and inner hollow shell which is then accelerated inwards ("imploded") rather than outwards is described by the following equations.Unlike other forms of the Gurney equation, implosion forms (cylindrical and spherical) must take into account the shape of the control volume of the detonating shell of explosives and the distribution of momentum and energy within the detonation product gases. For cylindrical implosions, the geometry involved is simplified to include the inner and outer radius of the explosive charge (Ri and Ro).
While the imploding cylinder equations are fundamentally similar to the general equation for asymmetrical sandwiches, the geometry involved (volume and area within the explosive's hollow shell, and expanding shell of detonation product gases pushing inwards and out) is more complicated, as the equations demonstrate.
The constant
was experimentally and analytically determined to be 1.0.Imploding spherical equation
A special case is a hollow sphere of explosives, initiated evenly around its surface, with an outer tamper and inner hollow shell which is then accelerated inwards ("imploded") rather than outwards, is described by:The spherical Gurney equation has applications in early nuclear weapons design.
See also
- Explosives engineeringExplosives engineeringExplosives engineering is the field of science and engineering which is related to examining the behavior and usage of explosive materials.- Topics :Some of the topics that explosives engineers study, research, and work on include:...
- Explosive velocityExplosive velocityExplosive velocity, also known as detonation velocity or velocity of detonation , is the velocity at which the shock wave front travels through a detonated explosive. The data listed for a specific substance is usually a rough prediction based upon gas behavior theory , as in practice it is...
- Table of explosive detonation velocitiesTable of Explosive Detonation VelocitiesThis is a list of the detonation velocities at specified density of various explosive compounds.The velocity of detonation is an important indicator for overall energy or power of detonation, and in particular for the brisance or shattering effect of an explosive.-References:*Cooper, Paul W.,...