The Gurney equations are a set of mathematical formulas used in explosives engineering to relate how fast an explosive will accelerate an adjacent 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 Ronald Gurney and have been expanded on and added to significantly since that time. The original paper by Gurney analyzed the situation of an exploding shell or bomb, a mass of explosives surrounded by a solid shell. Other researchers have extended similar methods of analysis to other geometries. All of the equations derived based on Gurney's methods are collectively called "Gurney equations".
Underlying physics When an explosive adjacent to a layer of a metallic or other solid material detonates, the layer is accelerated both by the initial detonation shock wave and by the pressure of the detonation gas products. Gurney developed a simple and convenient formula based on the conservation laws of momentum and energy that model how energy was distributed between the metal shell and the detonation gases that is remarkably accurate in many cases. A key simplifying assumption Gurney made was that there is a linear velocity gradient in the explosive detonation product gases; in situations where this is strongly violated, such as implosions, the accuracy of the equations is low. In the most common situations encountered in ordnance (shells surrounding explosives) this works remarkably well though. In such cases the approximations are within 10% of experimental or detailed numerical results over a wide range of metal mass (M) to explosive charge mass (C) ratios (0.1 < M/C < 10.0). This is due to offsetting errors in the simplified model. Ignoring rarefaction waves in the detonation gases causes the calculated velocity to be too high; the assumption of an initial constant gas density rather than the actual one of the gases being densest next the accelerated layer causes the value to be low, cancelling each other out. In consequence attempts to improve the accuracy of the Gurney model by making more realistic assumptions about one aspect or another may not actually improve the accuracy of the result.
Definitions and units The Gurney equations relate 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
E {\displaystyle E} - The energy per mass of an explosive that ends up as kinetic energy
2 E {\displaystyle {\sqrt {2E}}} - 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 explosive materials. For imploding systems, where a hollow explosive charge accelerates an inner mass towards the center, the calculations additionally take into account:
Ro - Outside radius of the explosive charge Ri - Inside radius of the explosive charge
Gurney constant and detonation velocity As a simple approximate equation, the physical value of 2 E {\displaystyle {\sqrt {2E}}} 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 D 2 E {\displaystyle {\frac {D}{\sqrt {2E}}}} ranges from between 2.32 for Tritonal and 3.16 for PAX-29n.
m m μ s {\displaystyle {\frac {mm}{\mu s}}} is equal to kilometers per second, a more familiar unit for many applications. The commonly quoted values for 2 E {\displaystyle {\sqrt {2E}}} are what are called the terminal values, the limiting case of acceleration in the cylinder expansion tests used to measure it (at 19–26 mm expansion). There is also a prompt value that may be measured for smaller expansion radii (5–7 mm). When no clarification is given in the literature, it is normally the limiting value.
Fragmenting versus non-fragmenting shells The Gurney equations give a result that assumes the shell or sheet of material remains intact throughout a large portion of the explosive-gas expansion such that work can performed upon it. For some configurations and materials this is true; explosive welding, for example, uses a thin sheet of explosive to evenly accelerate flat plates of metal and collide them, the plates remaining solid throughout. However, for many configurations where materials, brittle materials in particular, are accelerated outwards, the expanding shell fractures due to stretching. When it fractures, it typically breaks 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. This phenomenon allows the detonation gases to stream around the fragments or bypass them, reducing effective drive. Thus for metal shells that are brittle or have low ultimate strain, fragment velocities are typically about 80% of the value predicted by the Gurney formulas.
Effective charge volume for small diameter charges
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