How to reason about the force, pressure or impulse to break something?

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Flumble
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How to reason about the force, pressure or impulse to break something?

Example scenario: I was testing the thickness of the ice behind my house today and it cracked after stomping the heel of my shoe a few times. I could see the ice was less than 4cm thick, so not safe to skate on anyway, but I'm wondering how stomping a heel compares to walking/skating on it.

Now ice, porcelain, glass and such are materials with a special property that they like to form big cracks rather than just disconnect like wooden beams or spaghetti, so to relate to the ice example, let's *ahum* crack the behaviour of cracking materials.

As far as I can tell, the ice and my heel have a small crumple zone(?), so any impulse would result in a large force over a small area, so the ice would get a large shear stress, which I guess is the main cause of ice cracks.
Is this kind of correct? And how does this compare to skating (during which you hopefully don't stomp the ice too much, but you do exert your body weight+weird movement on a narrow piece of metal)?

Zamfir
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Re: How to reason about the force, pressure or impulse to break something?

A guy called JE Gordon wrote a pair of amusing, cheap and insightful books about this. They're called "The New Science of Strong Materials: Or Why You Don't Fall Through the Floor", focused on fracture mechanics of materials, and "Structures, or why things don't fall down" that takes a wider look at structures.

A fracture requires energy to form, and so does permanent deformation of a structure. On the other hand, a loaded structure contains stored "spring" energy, and a fracture releases part of that energy. Also, materials contain plenty of microscopic holes, a bit like nucleation sites for steam bubbles..

If a structure is loaded enough, the growth of a crack would release more energy (from unloading the spring) than the formation of the crack requires. Once a structure reaches such a state, it's unstable, and you'll get exponentially fast crack propagation through it. That's breaking.

A shock will have a similar effect: its energy is temporarily absorbed into the springiness of the structure, which can make it unstable. As you can see with a shattering cup, this might lead to crack growth all over the structure at once, while a statically loaded structure will typically fail at one decisive fracture. Basically, the shock puts in energy faster than the crack formation can take out, so there's excess energy everywhere.

Now ice, porcelain, glass and such are materials with a special property that they like to form big cracks

This is called "brittleness". The typical cause is that the material has almost no capability for permanent deformation (think of a deformable metal cup as the opposite). Deformation absorbs a lot of energy, which is then not available for crack growth.

Also, such materials often take on permanent internal stresses during formation. So it has a built-in reservoir of energy, and only needs a push over the edge to break.
than just disconnect like wooden beams or spaghetti

Wood is difficult, it's an anisotropic composite material. It's fracture behaviour is an advanced topic... Cooked spaghetti lacks so much stiffness that it cann't absorb much energy at all under static load, it doesn't contain internal stresses either, and it avoids concentrated loads in any specific place. But note that you canthrow soft spaghetti to a pulp - that's a similar overall energy overload as the shattering of brittle object.

jewish_scientist
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Re: How to reason about the force, pressure or impulse to break something?

The proper way to fire a shotgun is to have the stock pressed right against your shoulder when you fire. If there is a gap between the two, the recoil of the shotgun can hurt your shoulder. Everything is identical in these two situations except the distance between the shotgun and you, so why are the results different?

speising
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Re: How to reason about the force, pressure or impulse to break something?

jewish_scientist wrote:The proper way to fire a shotgun is to have the stock pressed right against your shoulder when you fire. If there is a gap between the two, the recoil of the shotgun can hurt your shoulder. Everything is identical in these two situations except the distance between the shotgun and you, so why are the results different?

If you fire a shotgun, it accellerates over some period of time. When it's pressed against your shoulder, your body receives the energy over that period, when there's a gap, the accellerated gun slams into your shoulder and delivers all of the energy over a much shorter timespan.

Soupspoon
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Re: How to reason about the force, pressure or impulse to break something?

Ditto the advantage of a seatbelt, even/especially with an airbag. The car, upon impact, is designed so the front may immediately come to rest with the solid stationary thing you collided (or to now start moving more like the moving thing you struck) but the compartment in which you sit decelerates more gradually due to the crumpling. If you're strapped in, you'll similarly more gradually decelerate. But if you're not strapped in you tend to slow less slowly than the robust compartment, as you slide (forward/sideways/whatever) off the seat and then you hit the wheel/airbag/passenger-shelf that has slowed more, while you were happily unretarded.

Wearing the seat belt starts your second-or-so-derivitives of position changing right at the start so that on the way to attaining your final stationary position the forces don't mostly act in a shorter division of time than they ideally should do.

Flumble
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Re: How to reason about the force, pressure or impulse to break something?

Thank you, Zamfir, for explaining the framework of loads and shockwaves. A lot of vague thoughts are now solidified and combined to a coherent system in my head.

So if, by kicking the ice, the main component to breaking it is a shockwave, does the energy of a shockwave (specifically the energy that just about cracks the ice, assuming there is such a point) relate to its load-bearing capacity?
Or would that heavily depend on other properties of the ice, like crystal size and contaminants in the water?

Zamfir wrote:Cooked spaghetti lacks so much stiffness that it cann't absorb much energy at all under static load, it doesn't contain internal stresses either, and it avoids concentrated loads in any specific place. But note that you canthrow soft spaghetti to a pulp - that's a similar overall energy overload as the shattering of brittle object.

My bad, I meant raw spaghetti. But I fail to see now how raw spaghetti could be a counterexample.

Zamfir
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Re: How to reason about the force, pressure or impulse to break something?

http://www.tms.org/pubs/journals/JOM/9902/Schulson-9902.html

This contains more than you probably want to know about ice strength...

jewish_scientist
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Re: How to reason about the force, pressure or impulse to break something?

speising wrote:If you fire a shotgun, it accellerates over some period of time. When it's pressed against your shoulder, your body receives the energy over that period, when there's a gap, the accellerated gun slams into your shoulder and delivers all of the energy over a much shorter timespan.

When the stock is against your shoulder, the energy from the explosion travels through the stock at the speed of sound as a shock wave.. When the stock is not against your shoulder, the energy from the explosion accelerates the stock to a speed much less than the speed of sound.

Soupspoon wrote:Ditto the advantage of a seatbelt, even/especially with an airbag.

One of the most important functions of the seat belts and airbags is to distribute the forces over a larger area, resulting in less pressure, and to areas better able to cope with the compressive forces. That is why seat belts go across the rib cage, which can decrease its volume, and not the skull, which is designed to be perfectly rigid (I am ignoring the jaw). In the shotgun situation, the area where the force is applied is constant, and the same body parts are effected.

speising
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Re: How to reason about the force, pressure or impulse to break something?

jewish_scientist wrote:When the stock is against your shoulder, the energy from the explosion travels through the stock at the speed of sound as a shock wave.. When the stock is not against your shoulder, the energy from the explosion accelerates the stock to a speed much less than the speed of sound.

It's not about a shock wave. It's simple Newtonian action/reaction: the bullet gets accellerated (for the whole time it travels down the barrel), so something has to be accellerated in the opposite direction. This something is either the gun, or the gun plus your body.

morriswalters
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Re: How to reason about the force, pressure or impulse to break something?

I thought the idea was to make the car and the driver act as a single unit. So that when the car decelerates by absorbing the energy as it collapses, the body inside slows with it. The placement of the seat belt is through the center of mass so ideally the body won't rotate or fold. Airbags protect you from the cars interior, more or less while also decelerating you. I assumed something similar for a shotgun.

Xanthir
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Re: How to reason about the force, pressure or impulse to break something?

Yes, that's basically what Soupspoon said.
(defun fibs (n &optional (a 1) (b 1)) (take n (unfold '+ a b)))

morriswalters
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Re: How to reason about the force, pressure or impulse to break something?

Yep.

Zamfir
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Re: How to reason about the force, pressure or impulse to break something?

The seatbelt and the shotgun stock are not exact analogues.

In the case of the shotgun: the necessary transfer here is only momentum, not energy. Some (mass * velocity) has to equal the momentum of the bullet. If that'a big mass, the velocity can e low. Since kinetic energy is (1/2 mass * velocity^squared), a fixed amount of transferred momentum can result in different amounts of transferred energy. High mass => less energy transfer. Same logic as a high-bypass jet engine.

If the shotgun plus you (and perhaps the earth) form a single rigid body, the energy transfer is minimal. If you don't shoulder the gun, there's initially only the shotgun's mass for the momentum transfer, and it will absorb much more energy (leaving less for the bullet). Then it slams into you, and that energy is transferred to blunt trauma. That energy comes from a slightly slower bullet.

In the case of the seatbelt: your bodies kinetic energy has to go somewhere, no way around it. In the old days, the seatbelt material itself absorbed a little bit of energy, but most of the energy went into you body anyway. Just distributed over a wider area, and especially not into your head.

With airbags, the working of the seatbelt has changed. They now have "load limiters", a mechanism that allows the seatbelt to unroll during an impact, absorbing part of energy in the process. Which has an obvious downside: it allows your upper body to pivot, and therefore your head is decelerated least of all body part. That's why old-time seatbelts kept you fixed in the chair - head trauma is concern number one.