Federal Premium 12-gauge Barnes Expander ¾-ounce slug (P152XT1) vs. 4 layers of 16-ounce denim
Average Expanded Diameter: 0.7898 inch
Retained Mass: 325.4 grains
Impact Velocity: 1,707 feet per second
Test Firearm: Remington 11-87; 21'' rifled barrel
Barrier: 4 layers of 16-ounce cotton denim
Range: 35 feet
Test Medium: H2O @ 53° F
Front:
Side:
Rear:
Predictive Analysis:
Q-model
DoP: 20.974 inches
Wound Mass: 5.060 ounces
Wound Volume: 8.417 cubic inches
mTHOR
DoP: 23.391 inches
Wound Mass: 5.643 ounces
Wound Volume: 9.387 cubic inches
DoP = maximum equivalent depth of penetration in calibrated ordnance gelatin (or soft tissue)
Wound Mass = total weight of tissue damaged/destroyed within the entire wound channel
As much as solid copper 'monolithics' have become 'all the rage' in rifle and self-defense handgun ammunition for their improved performance in those platforms, these designs also demonstrate a similar increase shotgun performance. Unlike a lot of lead deer slugs that typically fragment and lose mass along the shot line, copper monolithics tend to retain most, if not all, of their initial mass ensuring that penetration is adequate in the extreme. In this case, the predicted penetration depth, an average of the two models (Q-model and mTHOR), is 22.1825 inches with an average predicted wound mass of ~5.35 ounces or about ⅓ of a pound worth of soft tissue damage matching―and in many cases, exceeding―the performance of many rifle calibers.
Using the equation for calculating residual projectile velocity in a semi-infinite target (page 19 of Quantitative Ammunition Selection) to model projectile residual velocity by dividing the entire predicted penetration depth into 100 discrete elements (each element being about 5.328 mm thick) to obtain ∆V/∆x, the actual power (in joules per second also called watts) can be determined by dividing the kinetic energy, EK, of the round by the total transit time of the bullet (power is expressed as energy divided by time) through the test medium to the point at which its velocity becomes zero. In order to determine transit time of the bullet, the average retarding force exerted upon the bullet is calculated by using the Work-Energy Theorem, F = ∆EK/∆x, by dividing the change in kinetic energy, ∆EK, over each separate incremental change in distance, ∆x, which allows for the computation of the deceleration of the bullet by dividing the retarding force (F) by the mass (m) of the bullet. Once transit time is computed, dividing EK (in joules) at impact by time yields the power of the bullet in joules per second (watts).
In this particular test case, the power of the Federal Premium 12-gauge Barnes Expander ¾-ounce slug is 587.4255 kW, instantaneous force at impact is 19.794172 kN (or 4,449.907 lb·f), pressure at impact is 20,416.503 psi and total transit time until the projectile comes to rest (V = 0) is 4.858 milliseconds.
Of course, the spreadsheet does all of this for me (as seen below), because I am a typical lazy retired cop and not a glutton for mathematical punishment.
The red curve shows velocity decay, the blue curve shows EK decay and the black curve shows deceleration decay; all curves are with respect to instantaneous projectile position.
So, with all other terminal behavior factors remaining the same, power naturally increases proportionally due to the increased braking forces that result from the larger expanded diameter and greater ELOSS of the expanding slug. That power, which is the rate of energy expenditure (or loss) over a given distance, ELOSS/∆x, is what gives rise to and produces the radial displacement of tissues that come into direct contact with the projectile and creates the stresses that damage tissue.