Use of Internal Shear Lines to Predict
Gun Cartridge Pressures,
as a Function of Time
By Smalley, 2004

A series of cartridges has been designed and tested using the parametric techniques disclosed in patent #6,523,475 B2.  These cartridges are configured to maximize ballistic performance within the capability of existing geometry.  These are designated by bullet diameter, in either English or metric units, and case volume in grains of water when the case is filled to the neck-shoulder junction, where, ideally, the bullet base is located in the finished cartridge.  The cartridge designation also includes our registered trademark; SMc.  A cartridge designation therefore would be; 22/40 SMc ™ where 22 is the nominal bullet diameter, 40 is the nominal case volume and SMc defines the cartridge configuration.

These cartridges operate as follows: The pyrotechnic blast from the primer initiation compresses the propellant into the front of the cartridge case.  This compression heats the interstitial air trapped between the propellant granules, which transfer heat to the granule surfaces, preparing them for later ignition.  The patented shoulder design focuses the sonic compression wave slightly rearward of the bullet base, so heat loss to the bullet is reduced and bullet movement is retarded.  Heat is concentrated in the propellant, just behind bullet base.  This increases the burn rate of this portion of the charge, that is normally the last to ignite and burn.

Bullet movement begins when combustion pressure reaches approximately 2800 to 3700 psi, depending upon bullet retention characteristics of the cartridge case.  In most rifle cartridges, propellant pushing on the bullet base initiates bullet movement.  That is the situation when the cartridge is long enough that the primer does not ignite all the propellant in the case.

At initiation of bullet movement, a shear line is set up in the propellant.  This is approximately the diameter of the bullet.  It extends back, to the burning propellant interface.  This shear surface ignites much more rapidly than the normal propellant burn rate, due to heating by infrared radiation.  Shear lines in solid rockets have ignited as rapidly as 120 inches in two milliseconds at 500 to 1200 psi.  Shear lines in gun cartridges would be expected to ignite even more rapidly in the pressure range of 3000 to 30,000 psi.  As the shear line burns radially both inward and outward and the bullet accelerates into the barrel, maximum burning area is reached within the cartridge and peak pressure is generated.

Propellant burning continues within the cartridge and barrel, further accelerating the bullet until it reaches the muzzle.  Best efficiency occurs when the propellant mass accelerating down the barrel behind the bullet is minimized and propellant mass retained within the cartridge is maximized.  Propellant trapped in the case burns at higher pressure, whereas propellant moving down the barrel burns more slowly, due to acceleration and lower surface pressure (lower pressure means lower temperature).

Tests performed by M.L. (Mic) McPherson, published in Metallic Cartridge Reloading, Third Edition, published by DBI Books Inc, 1996, pages 45 through 49 give propellant compression for most smokeless powders.  Most rifle propellants compress between 15% and 22% at 3320 psi, which is about average for initial bullet movement.  Further testing by McPherson and me has proven that ignition of propellant in cartridges does not extend beyond 0.5 to 0.6 inches beyond the compressed surface described above.  This occurs because the interstitial gaps between the propellant granules are reduced or partially closed at pressure and pyrotechnic gas quickly loses heat to the igniting propellant surface.

Tests and Results are Described Hereafter

An inert propellant simulant was obtained from the Nexplo division of Bofors Munitions in Sweden.  This material was designed to have the same mechanical properties and heating characteristics as their normal smokeless powders.  A primed 45-70 case was filled with this simulant and a lead bullet was seated over it.  The cartridge was then fired and bullet movement into the barrel measured.  Upon disassembly, the inert simulant was carefully removed and examined.  It was welded together to a depth of 0.5 inches from the primer flash hole and soot was noted on the granule surfaces where exposed.  Scratches were noted in the soot deposited on the case inside surface, verifying normal propellant compression and movement.  Sectioned cases fired with regular propellants also exhibit these scratches.

This test was then repeated but 5 grains less simulant was used and 5 grains of normal propellant was added on top of the inert material.  Primer ignition did not result in ignition of the live propellant as it was still present at disassembly, and bullet displacement was the same as with the all-inert load.  The base of the lead bullet was noted to have been impressed with the shape of the propellant granules.  This test was repeated with less inert simulant and more live propellant in five-grain increments until ignition occurred.  The height of the inert propellant column was then measured at 0.6 inches.  This test was repeated with several different live propellants and primers.  All ignitions occurred between 0.5 and 0.6 inch inert column height.  It is interesting to note that magnum rifle primers gave ignitions tending toward 0.5 inches and weaker primers such as the match type tended toward 0.6 inches.  This is believed to have been caused by the higher pressures generated by the magnum primers closing the interstitial air conduits within the inert simulant faster.

Generation Of Shear Lines In Gun Cartridges

Use of the generated shear line areas to predict gun cartridge peak pressures and other aspects of cartridge performance has not been previously disclosed or utilized.  This is therefore considered an advancement of the state of the art.

Cartridges which have internal lengths measured from flash hole to bullet base less than about 0.6 inches, in general, do not have a discernable shear line formed behind the bullet because nearly all the propellant is ignited by the primer.  Thus, cartridge configurations described by Alexander, Patent #6293203 B1 are excluded.  (Most pistol propellants have compressions in excess of 20% at first bullet movement).  That propellant in contact with the case is excluded from ignition because the thermal conductivity of brass is up to 400 times (280 times higher for chrome-moly steel and 240 times higher for stainless steel) higher than nitrocellulose.  That propellant in contact with the brass case is either consumed by turbulence in the barrel or exits the muzzle unignited.

Cartridges that are longer
but have a shoulder angle less than about 35 degrees

(Jameson, patents # 5970879, 6595138, 6550174) or double radiused shoulders, per Weatherby, do not have a well-defined shear line as the shoulder angle is insufficient to trap the propellant in the cartridge case.  A substantial portion of the sheared propellant follows the propellant plug down the barrel.  In longer cases with mild shoulder angles, all propellant not initially ignited can follow the bullet down the barrel, as is the situation with straight-walled cases.  This reduces efficiency, increases recoil, and slows propellant combustion.

As case design changes toward shorter and fatter, and shoulder angle becomes steeper (greater than the 35-degrees claimed by Jamison) the shear line acting at bullet diameter becomes more pronounced between the propellant plug pushing the bullet and the propellant trapped by the shoulder.  This sheared surface ignites more quickly than the normal propellant burn rate, as previously described.

The resulting double burning surface area of the sheared surface adds greatly to the pressure being generated and can be added to the semispherical burning surface originally ignited by the primer to determine peak pressure.  Peak pressure is achieved when total area reaches a maximum, early in bullet movement into the barrel, and by bullet acceleration rate and propellant progressivity.  The use of this additional surface area to explain the pressure-time curve in gun cartridges has not previously been postulated or disclosed, except in our prior work.

Previous techniques used progressivity, regressivity, and progressivity-regressivity rollover coefficients for each propellant to explain the burn front progression.  Naturally these coefficients are cartridge specific and not usable for any cartridge except the one for which the coefficients were generated.  Performance predictions based on these coefficients for new cartridges are, in general, not acceptably accurate.

Utilizing the additional double burning area defined by the shear line caused by bullet movement makes possible a reasonable prediction of peak pressure.  In fact, iterative solution of the equations given below make it possible to calculate the entire pressure time curve for any cartridge of length greater than 0.6 inches and shoulder angle greater than about 35 degrees.

We can predict cartridge propellant burn rates from the classic solid rocket burn rate equation:

  • Br2=Br1 [P2/P1]exp N
  • Where: Br2 is local burn rate inside the cartridge or barrel.
  • Br1 is test burn rate at test pressure determined by Crawford Bomb.
  • P2 is local static pressure.
  • P1 is test pressure.
  • N is burn rate exponent (less than one)
    over range of pressures being considered.
  • Initial burning surface area is calculated by:
  • A=T[4 pi D2/4]
  • Then by A=T[2 pi D2/4] + 2 pi dO[lO-lOC]+2 pi dI [lI-lIC-mb]

Where:

  • A is burn area at time t
  • T is a “texture” term defining width of burn front and a constant for each propellant type.  This is always greater than unity and is
    controlled by granule configuration, inhibition layer etc.
  • D is internal diameter of the brass case
  • dO is diameter of the outer shear line
  • dI is diameter of the inner shear line
  • lO is length of outer shear line
  • lOC is compression factor for the propellant at first bullet movement
  • lI is length of inner shear line.  This term disappears when
    bullet movement exceeds inner shear line length.
  • lIC is compression factor for propellant at inner shear line
  • mb is bullet movement at time t

Peak pressure is reached approximately coincident with the burning surface area reaching a maximum in the cartridge, keeping in mind that the plug of propellant following the bullet can only burn from the chamber side because of the quenching action of the barrel or brass case neck.

Use of this burn front model for parametric cartridge design has maximized cartridge performance and efficiency beyond anything heretofore achieved.  This was done by setting D (diameter) between 2 and 2.3 times bullet diameter and L (Length) to more than 0.6 inches plus the compression factor for the propellant.  An internal ellipsoidal shoulder angle of 48 to 54 degrees at the neck-to-shoulder shoulder junction was provided, focusing the primer shock wave 0.04 to 0.10 inches from the bullet base, to minimize heat loss to the bullet.  This maximizes adiabatic heating of the propellant that would normally be the last to ignite and burn before the bullet reaches the muzzle.

For example the 22/40 SMc cartridge has a volumetric capacity of 44 grains of water at the neck shoulder junction, midway in capacity between the 22-250 and 220 Swift but about 0.3 inches shorter.  It was loaded with 42 grains of Hodgdon H-335 propellant and a Nosler 40 gr bullet.  Velocities averaged 4655 fps, chamber pressures were moderate and groups were under 0.5 inches at 100 yd.  Further testing with 46.5 grains of Hodgdon H 414 propellant and a 55-grain Sierra bullet gave average velocities of 4171.5 fps, again at moderate chamber pressure (below 62000 psi).  This cartridge has a D to d ratio of 2.071 and a shear line length of 0.349 inches.  The shear line is short as is the propellant plug following the bullet, therefore the peak pressures are low and efficiency is high.  As with other SMc cartridges, the shear line pressure increase is well defined on piezoelectric chamber pressure measurements.

An example of pressure transducer measurements for the 6.5mm/60 SMc and 6.5mm-284 Winchester with the same bullet and propellant load are presented as Enclosure 1.  This data was taken at the Norma Precision AG ammunition plant in Ämotfors Sweden, August 7 thru 10, 2003.  The annotations show the differences between the two cartridges, the 6mm/60 SMc having about 2 gr less case capacity.

Claims:
  1. Use of shear line areas to determine peak pressure and
    pressure as a function of time throughout cartridge burn.
  2. Use of propellant compression data at first bullet movement to determine
    shear line length.  Definition of the texture term “T” and its effect
    on chamber pressure in the burn area equation.
  3. Use of the dO and dI terms to determine maximum
    pressure as a function of burn time.
  4. Use of the lO and lI terms to determine burn areas.
  5. Recognition of the burn area equation and how it changes as a function of time.
  6. How lO is reduced over time by Br2.
  7. How lI is reduced over time by Br2 and powder plug/bullet movement into the barrel.
  8. How dO increases with time by Br2
  9. How dI reduces with time by Br2.