Relationship between Concentration of a Force and Energy Density

[NullusSum]

TL;DR

Can energy density model the work for how concentrated a force is, e.g., kinetic energy of object divided by volume of object?

How can we explain the effectiveness of a force in terms of energy and work? For example, a boxer's punch has a similar amount of kinetic energy to a 9x21mm bullet, yet the bullet has far greater penetrating power. From what I understand, this is due to the bullet’s force being applied over a much smaller surface area, resulting in higher pressure (N/m²).
Given this context, could energy density (J/m³), calculated as kinetic energy divided by the volume of the fist or bullet, accurately describe this concept? Or would it be more appropriate to base it on the volume of the impact zone within the target? If so, how should energy distribution within that impact volume be accounted for?

 

[PeroK]

Boxers wear special gloves to cushion the impact and avoid broken bones on the fingers and face.

The force depends on the deformability of the objects that collide.

 

[NullusSum]

Boxers wear special gloves to cushion the impact and avoid broken bones on the fingers and face.

The force depends on the deformability of the objects that collide.

That is true, but if we are comparing a 'gloveless' fist versus a bullet impacting the same rigid surface, would the bullet still not be substantially more effective in terms of localized damage?

 

[A.T.]

Given this context, could energy density (J/m³), calculated as kinetic energy divided by the volume of the fist or bullet, accurately describe this concept?

Compare the damage of an airgun projectile on hard boiled vs. raw egg. It's obvious that the kinetic energy density of the projectile is not the whole story, because the target material determines how much energy is absorbed and how it is distributed.

 

[Lnewqban]

TL;DR Summary: Can energy density model the work for how concentrated a force is, e.g., kinetic energy of object divided by volume of object?

Copied from:
https://en.wikipedia.org/wiki/Energy_density

"In physics, energy density is the quotient between the amount of energy stored in a given system or contained in a given region of space and the volume of the system or region considered.
...
Energy per unit volume has the same physical units as pressure... The energy required to compress a gas to a certain volume may be determined by multiplying the difference between the gas pressure and the external pressure by the change in volume. A pressure gradient describes the potential to perform work on the surroundings by converting internal energy to work until equilibrium is reached."

The penetration and destructive power of any kinetic type of bullet comes from its kinetic energy, which is directly proportional to its mass and the square of its velocity.

A bullet of similar mass and greater volume (less dense material) would receive higher drag resistance to its trajectory from the atmosphere, which would greatly reduce its very important velocity of impact.

Please, see also:
https://en.wikipedia.org/wiki/Projectile#Kinetic_projectiles

https://www.omnicalculator.com/sports/human-punch-force

https://www.omnicalculator.com/physics/bullet-energy

 

[Ibix]

There are a lot of factors. The squishy gloves do two things - they spread the impact over more area and also more time. So your flesh has more time to deform and a larger area to do it with. (As an aside, it also puts more mass on the fist and reduces the risk of injury to hands, so I gather that properly gloved boxers hit harder than bareknuckle and are more likely to get brain damage long term, but less likely to have a match stopped due to bleeding.) So a key factor in the damage a bullet can cause is its speed - flesh simply can't deform fast enough to resist it. Pressure is also important, and what bulletproof vests do is basically spread the impact so you get a large bruise instead of a small hole.

So the difference between a hit from a bullet and a fist with the same energy is both the size of the area and the length of time over which the impact occurs. You have to consider both.

 

[NullusSum]

So the difference between a hit from a bullet and a fist with the same energy is both the size of the area and the length of time over which the impact occurs. You have to consider both.

That makes sense. Thank you for clarifying.

So how would we model mathematically the effectiveness of a force? We have impulse to describe the impact duration and we have pressure to describe the impact area. How would we combine these two concepts into one?

 

[Ibix]

How would we combine these two concepts into one?

You would first need to define what you mean by "effectiveness". Then you would need to gather data on the effectiveness of different strikes. There may be some ethical issues with this.

Bluntly, I think this is the point where the word "art" needs to be stressed in the phrase "martial art". Humans are not simple collections of levers and weights and any simple model of one is wrong and, worse, probably not useful (tip o' the hat to George Box). If you are shooting at something like a tank then you can build identical tanks and shoot them under lab conditions and come up with a reasonably objective definition of a mission kill, and so compare weapon capabilities. With humans it's hard to get ethical approval to punch and shoot them, and with the punching so much depends on whether they see it coming and the wide variety of strengths and abilities to absorb punishment that a model is unlikely to tell you more than "if you hit them harder it usually hurts them more", I'm afraid.

 

[NullusSum]

You would first need to define what you mean by "effectiveness".

What I mean by the effectiveness is how much a force can puncture or the degree of localized damage based on overcoming a material's strength. For example, a sword would be more 'effective' at puncturing a soft target when compared to a blunt object like a club, even though the club may possess more total force. The bullet versus punch example was used to illustrate that both forces can possess a similar amount of kinetic energy, yet the bullet offers significantly more penetration, regardless how easily the target deforms... or at least that is how I understood it.

Then you would need to gather data on the effectiveness of different strikes. There may be some ethical issues with this.

I can assure you my question is not going to lead to any experiments involving living test subjects, haha. It was simply a shower thought I had that developed into a genuine question. I suppose my goal would be to develop a table describing how much pressure different forces, such as sharp, blunt, explosive, etc., exert on different materials using predictive calculations instead of experiments.

 

[Ibix]

What I mean by the effectiveness is how much a force can puncture or the degree of localized damage based on overcoming a material's strength

Yes, but you need things you can measure. And you need those measurements to correspond to whatever your intuitive measure of destructiveness. For something like shooting at a tank this is relatively easy to specify - did the shot penetrate the armour, and did it do enough damage inside to wreck the engine, or armament, or magazine, or kill the crew? It's pretty binary, and the measurements are pretty obvious. Is there a hole in the armour and did stuff go flying around the inside, and how does that vary with armour design? And you can do these experiments fairly easily if you've got the budget.

But what exactly would you measure in the response of a human to a punch? You can measure things like bruising or cuts or broken bones. You can ask if people are seeing double after a strike to the head. But does that capture effectiveness? It captures some of it, but timing and breaking your opponent's posture make a huge difference, and those are more nebulous concepts. This makes it really hard to work out what question you're asking in a way that's precise enough to answer.

The bullet versus punch example was used to illustrate that both forces can possess a similar amount of kinetic energy, yet the bullet offers significantly more penetration, regardless how easily the target deforms...

Well - a bullet may pass through a window leaving a small hole while a punch will shatter the whole pane. Yet a bullet hole will probably kill a person while the punch probably won't. It's really complicated to model this, even with a fairly straightforward and stark bullet-vs-punch example.

I can assure you my question is not going to lead to any experiments involving living test subjects, haha.

I wasn't expecting you to go all Mengele on us - I was more pointing out why it's quite difficult to gather good data in this space. Anything that's respectably scientific is likely to face ethical issues.

All that said, @Dr. Courtney did do some work in the area of blast injuries, and would probably be able to make useful comments. Unfortunately I don't think he posts here anymore.

 

[NullusSum]

This makes it really hard to work out what question you're asking in a way that's precise enough to answer.

I will try to frame my question more precisely and if you are feeling generous, you can point out if my question is still too vague or answer it as is.

What I essentially want to predict is applied force over time, using $F = \frac {m \cdot \Delta v}{\Delta t}$, and how that force distributes over an area, $p = \frac {F}{A}$. I assume this pressure changes depending how much the target deforms during impact.

When I say "localized damage", I am referring to the effects of a pressure impacting a single solid material or object that is a combination of different materials. If we take the tank example, I would only be interested in penetrating its composite armor because any kind of shrapnel or concussive blast that damages more vulnerable targets like the human operators or the engine is less defined and a more complex equation. A hypothetical experiment would involve a solid block of armor, and I want to determine how deep the force penetrates into it. This is similar to bullet penetration tests using stacked bricks or water bottles—except in this case, I’d treat the armor as an idealized continuous medium.

So... in your original comment, you bring up "the length of time over which the impact occurs". The impact duration should depend on how easily the target deforms, right? A bullet hitting rubber would have a longer impact duration due to deformation, whereas hitting tank armor would result in a shorter impact duration due to its rigidity. I want to model how pressure changes based on the impacted material, considering that $\Delta t$ in the force equation influences the force applied. Since pressure depends on both force and contact area, I assume materials that deform more will spread the impact out, reducing peak pressure. Would this approach be a valid way to model penetration depth, or am I overlooking key factors?

Well - a bullet may pass through a window leaving a small hole while a punch will shatter the whole pane.

I [hope I] have conveyed that I am more interested in penetration potential rather than the area of effect. A glass pane might not be the best material for testing because it structurally fails easily when deformed. The bullet’s high speed doesn’t allow it much time to deform, unlike a fist, which applies force over a longer duration.

All that said, @Dr. Courtney did do some work in the area of blast injuries, and would probably be able to make useful comments. Unfortunately I don't think he posts here anymore.

Thank you for sharing. I am not sure if his work describes what I am looking for, but I will see what he has to say regardless.

 

[jrmichler]

What I essentially want to predict is applied force over time, using F=m⋅ΔvΔt, and how that force distributes over an area, p=FA. I assume this pressure changes depending how much the target deforms during impact.

This is the starting point. Then you look at the material properties of your moving object, say a bullet, and the target. Two cases to show how I tackle this problem:

Case 1: A .22 LR Standard Velocity will have muzzle velocity approximately 1000 ft/sec. Shoot that straight into water perpendicular to the water surface. Assume the bullet has a flat face, such as a wadcutter bullet. Since water is a liquid, and the entrance velocity is well below the speed of sound in water, the initial impact stress over the face of the bullet will be $\sigma = 0.5*\rho*V^2$. That comes to about 6700 PSI. The bullet will decelerate rapidly. Also, that stress is more than the yield strength of the lead bullet alloy, so the bullet will start to mushroom. You can check your results by searching how far does a bullet travel in water.

Case 2: Same bullet at the same velocity, except this time it hits a hard thick steel plate perpendicular to the plate. The plate does not deform. We know that the contact stress will be much higher than in Case #1, so we start by assuming that the contact stress will be so much greater than the yield strength of the lead bullet that we can treat the bullet as a cylinder of liquid lead to start the calculations. The velocity pressure is over 70,000 PSI, so the bullet splatters and the steel plate is not deformed.

This is an actual test that I did back in grad school when we wanted to measure the natural frequencies of a load cell. We needed impact duration less than 50 microseconds in order to properly excite frequencies up to 10 kHz. The impact force needed to be high enough to get a useable signal. Blasting caps were fast enough, but not powerful enough. A hammer was powerful enough, but the impact duration was too long.