Latest / Mechanical Engineering Made Simple / How Stress Waves Rupture Solid Steel

Transcript

How Stress Waves Rupture Solid Steel · Mechanical Engineering Made Simple

0:00
Welcome back to mechanical engineering made simple.
0:02
You know, if two solid steel spheres collided anything faster
0:06
than a casual walking pace, like literally just one foot per
0:10
second, they permanently dent each other.
0:12
Yeah, the physics of impact are incredibly unforgiving.
0:15
Right. It's so violent that the math
0:18
governing our static, predictable world just
0:21
completely unravels in a matter of milliseconds.
0:24
We spend well most of our time engineering for equilibrium.
0:27
Exactly. We want the bridge to stand
0:29
still. We want the machine casing to
0:31
hold its shape. But today, our mission for this
0:34
deep dive into the sources is to explore what happens during that
0:37
microscopic window of pure chaos when 2 bodies actually slammed
0:41
together. It's a fascinating shift in
0:43
perspective. It really is.
0:44
So whether you are designing heavy machinery or, I don't
0:47
know, investigating A structural failure or just trying to figure
0:51
out why your phone screen shattered when it fell off the
0:53
table, you have to understand the invisible violence of an
0:56
impact. And I like that we're building
0:58
this up as an escalation today because an impact isn't just a
1:01
force being applied, you know, it's a profound rapid disruption
1:04
of a system. We'll start with the perfect
1:07
mathematical illusion of a clean bounce, and then gradually turn
1:10
up the speed until the equations literally breakdown and the
1:13
material shatters. That's the perfect road map,
1:15
because when you generate massive impulse type forces at a
1:19
tiny point of contact for just a fraction of a second, you send a
1:23
shockwave of complexity through the material.
1:26
Which is wild to think about. Right.
1:28
And our understanding of how to handle that complexity has
1:31
evolved. We've gone from treating objects
1:34
like indestructible billiard balls to recognizing them as
1:39
highly volatile, vibrating webs of energy.
1:43
Let's start at that baseline though, the perfect billiard
1:45
ball scenario. Before we can understand how
1:48
massive steel structures bend and vibrate, we have to look at
1:51
the classic direct central impact between 2 rigid spheres.
1:56
Yes, the sources immediately point us back to Isaac Newton
1:59
for this one. Good old Newton.
2:00
Always right. He made this fundamental
2:03
experimental observation. He noticed that when 2 bodies
2:06
collide, their relative velocity after they bounce off each other
2:10
is in a constant ratio to their relative velocity before they
2:12
hit. They're just traveling in the
2:14
opposite. Direction.
2:15
Exactly. And he called that constant
2:17
ratio the coefficient of restitution.
2:20
Which is basically the mathematical embodiment of a
2:23
material's bounce. That's a great way to put it,
2:25
and the brilliance of Newton's observation, especially when you
2:28
pair it with the law of conservation of linear momentum,
2:31
is that it lets us treat the actual collision like a locked
2:35
black. Box OK, this is the blankest
2:38
level insight from the sources that I find so fascinating.
2:41
Let me do some spoken math here to explain it.
2:43
Go for it. The core equation for Newton's
2:45
observation is this. You have U prime minus X prime
2:50
equals negative E multiplied by the quantity U -, X.
2:54
Right where U&X are velocities before the impact.
2:57
Yeah, and the primes are the velocities after.
2:59
And then you bring in the conservation of linear momentum,
3:01
which is, let me get this right, M sub U * U prime plus M sub X *
3:06
X prime equals M sub U * U plus M sub X * X.
3:10
So basically, if we know the masses of the two spheres, and
3:14
we know exactly how fast they were moving before they hit each
3:16
other, we can map out the velocities after the crash
3:20
without ever having to calculate the messy details.
3:23
Without knowing anything about the force during the split
3:25
second they touch. Yeah, you just check the entry
3:28
and exit receipts. We don't actually need to know
3:30
what happens inside the black. Box.
3:32
It is the ultimate cheat code in classical mechanics because I
3:36
mean trying to map out a specific reference to the force
3:40
of contact as a function of time during a high speed collision.
3:43
I'm guessing it's a nightmare. Oh, it requires horrific
3:47
calculus. Heavy time integrations.
3:49
The forces are spiking, the geometry is warping.
3:52
It's a mess. But by relying on energy and
3:54
momentum, you bypass the microscopic violence entirely.
3:58
Completely deducible from what went in.
4:00
Which perfectly explains the classic ballistic pendulum
4:03
experiment. If you want to know the muzzle
4:05
velocity of a bullet, you don't try to mathematically model the
4:09
bullet literally tearing its way through wood fibers in real
4:12
time, no. That would be practically
4:14
impossible. Right, you fire the bullet into
4:17
a heavy block of wood hanging from a string, Let it in bed and
4:21
just measure the final velocity of the swinging block.
4:23
You work backward from the resulting swing.
4:26
Using the conservation of momentum.
4:28
You find out how fast that bullet was traveling when it
4:31
left the barrel. It's beautifully elegant.
4:34
It is, but as with all simplifications in engineering,
4:38
it carries severe limitations. The ballistic pendulum only
4:42
works because the bullet embeds itself and we can conveniently
4:45
ignore the internal vibrations of the wood block.
4:48
But what happens when things actually bounce?
4:51
Like what is physically happening at the exact
4:53
microscopic point of contact between two supposedly solid
4:57
spheres? Well, that's the catch.
4:58
Nothing in the real world is perfectly rigid at the
5:01
microscopic level. Even the hardest steel spheres
5:03
squish when they hit each other. They deform.
5:06
Yeah, and that squish forces us to abandon the simple rigid body
5:10
illusion. This is where we have to
5:11
introduce the Hertz theory of impact.
5:14
Heinrich Hertz. Right.
5:16
Hertz decided to look at the relation between the total
5:18
pressure being applied and the displacements at the point of
5:21
contact. He wanted to measure what he
5:24
called the relative approach. The relative approach of the
5:26
centers of the spheres. Basically how much they compress
5:30
into one another. Exactly.
5:31
The sources mentioned that Hertz found this relative approach
5:34
varies as the 2/3 power of the total pressure.
5:38
Let's pause on that for a second.
5:40
Why the 2/3 power? It's a very specific number.
5:43
Yeah. What is the physical mechanism
5:44
behind that specific non linear number?
5:47
Well, it comes down to the geometry of a sphere.
5:50
Think about when 2 perfectly flat plates press together.
5:54
The contact area stays the same. Right, so the pressure grows
5:56
linearly. But when a sphere presses into a
5:59
flat surface or into another sphere, it starts with an
6:02
infinitely small point of contact.
6:04
Oh, I see. And as it squishes deeper, that
6:07
circular flattened contact area expands word nonlinearly because
6:12
the area distributing the force is constantly growing wider the
6:15
deeper you push. The relationship between the
6:17
pressure and the compression follows that 2/3 power rule.
6:20
OK, geometrically that makes perfect sense, but I have to
6:23
push back on the core premise of Hertz's theory here.
6:28
Oh. What caught your eye?
6:30
I read through this section in the sources twice because I was
6:34
so confused. Hertz derive these formulas for
6:37
the force between the spheres and the radius of that circular
6:41
contact area based on bodies being pressed together under
6:45
purely statical condition. Statical.
6:47
Yes, but we are talking about a high speed dynamic collision.
6:51
Why are we modeling it using math designed for things sitting
6:55
at rest being slowly squeezed together in a vice?
6:58
I know it sounds completely counterintuitive, right?
7:00
You would assume the speed of the impact changes the
7:02
fundamental material properties exactly, but Hertz's brilliant
7:05
workaround functions perfectly under one very specific crucial
7:09
condition. The duration of the contact has
7:12
to be long in comparison with the period of the fundamental
7:15
mode of vibration for either sphere.
7:17
Wait, let me make sure I'm translating that correctly.
7:19
If hitting the sphere is going to make it ring like a bell,
7:21
that internal ringing vibration has to cycle much faster than
7:25
the actual time the two objects spin physically touching each
7:28
other. That captures the physics
7:29
perfectly. By making that assumption,
7:32
engineers can treat the spheres as perfectly rigid bodies
7:35
everywhere except for a tiny localized elastic spring exactly
7:39
at the point of contact. Yeah, and to prove this actually
7:43
works in the real world, the math relies on a dimensionless
7:46
ratio, usually denoted by the letter R.
7:49
Which is the ratio of the maximum kinetic energy of
7:52
vibration to the total kinetic energy of the spheres just
7:55
before they collide. Right.
7:57
And under almost all ordinary conditions, that ratio R is an
8:01
incredibly small quantity that tells us that the collision
8:04
simply doesn't excite much internal vibration in the
8:07
spheres. So because the vibration is
8:09
practically non existent, the energy of the colliding spheres
8:12
remains translational. Exactly.
8:14
They compress that microscopic spring, the spring pushes back,
8:16
and they bounce away. The statical assumption holds up
8:19
beautifully in a dynamic event because the energy didn't get
8:22
lost to internal ringing. So the math survives the squish.
8:26
That is fascinating. But that brings us to the next
8:29
escalation. Real engineering rarely involves
8:31
perfectly isolated spheres bouncing off each other in a
8:34
vacuum, you know. Oh, definitely not.
8:36
We are dealing with machine parts slamming into casings.
8:39
We are dealing with objects striking extended elastic
8:42
structures like plates or massive beams.
8:45
Right. And when a sphere hits a massive
8:47
steel boundary, you can't ignore the vibrations anymore because
8:50
the boundary can't just bounce away.
8:51
The game changes entirely when a solid sphere hits an elastic
8:55
plate. The plate acts as a vibrating
8:58
body. Yes, the kinetic and potential
8:59
energies of the impact don't just translate into motion, they
9:03
ripple out through the plate in the form of flexural waves.
9:06
Flexural waves Instead of the gentle Pebble in a pond analogy,
9:10
trying to visualize a flexural wave in steel feels more like
9:13
cracking a whip made out of a solid I-beam.
9:16
That is a terrifying but very accurate visual.
9:19
You are sending a massive undulation of stress traveling
9:22
through a solid medium. And the energy spreads over a
9:25
circular area growing outward from the impact point.
9:29
Which introduces a highly destructive variable into our
9:32
structural designs. Because if you want that plate
9:35
or beam to remain entirely elastic, meaning it absorbs the
9:39
flexural wave, bounces back completely, and doesn't suffer
9:42
any permanent plastic deformation, the rules are
9:46
shockingly strict. This is the one foot per second
9:49
rule I mentioned at the very beginning.
9:50
Yes, for a collision involving steel striking steel under
9:54
standard conditions, the impact velocity usually must be less
9:59
than one foot per second to avoid yielding.
10:01
That is barely a crawl. It really highlights just how
10:04
unbelievably concentrated the forces of impact are at that
10:07
initial point of contact. This all happens before the
10:11
flexural waves even have a chance to carry the load away.
10:14
Anything faster than a gentle tap and you are creating
10:17
localized plastic deformation. You permanently dent the steel.
10:20
And the engineering gets even more complex when we look at
10:23
transverse impacts on damped, elastically supported beams.
10:26
Let's definitely explore that damping issue, because normally
10:29
damping is the structural engineer's best friend.
10:32
It drains the energy out of a vibrating system and stops the
10:35
shaking. Right, you would naturally
10:36
assume heavy damping would save a machine from impact damage.
10:40
You really would, but the sources show that the analysis
10:42
of different types of damping reveals a pretty disturbing
10:46
reality. It does.
10:47
There is external damping, which is like the air resistance
10:50
around the beam or the shock absorbers it's mounted to, and
10:54
then there is internal viscous damping, which relies on Stokes
10:58
law. That's the resistance generated
11:00
within the material itself, right as the internal molecules
11:03
essentially rub together and dissipate the vibrational energy
11:06
as microscopic heat. Spot on, but the sources point
11:09
out that external damping actually loses its effectiveness
11:12
incredibly fast at the higher modes of vibration.
11:16
So the high frequency ripples racing through the steel just
11:18
completely ignore the external mounts.
11:20
Basically, yes, internal viscous damping does a better job with
11:23
those higher modes. If the frequency is high enough,
11:26
the internal damping can actually make the vibration a
11:29
periodic, meaning the wave just dies out without oscillating
11:32
back and forth. OK, so that sounds good.
11:34
It does, but here is the most crucial and frankly terrifying
11:38
finding for anyone designing machinery.
11:41
Ordinary damping affords almost zero protection against the
11:44
sharp initial rises and strain caused by the impact.
11:47
Wait, really? If the damping can't protect the
11:50
structure from that initial violent spike in strain when the
11:54
wave first hits, isn't that a massive red flag for fatigue
11:58
failure? It is the primary mechanism for
12:00
long term machine failure. You cannot treat a structure
12:03
subjected to impacts as a simple single degree of freedom system.
12:07
Because it's too complex. Right, a machine part
12:09
experiencing repeated strikes is subjected to a chaotic web of
12:13
high frequency flexural waves. Even if you're damping kills the
12:16
vibration quickly after the strike, that massive undamped
12:20
strain spike in the 1st millisecond is doing microscopic
12:22
damage. So over thousands of cycles,
12:25
those initial spikes slowly tear the material apart.
12:27
Exactly. Even if the impact is visually
12:29
small and the vibration seems to vanish instantly, the true
12:33
damage is happening faster than the damping can physically
12:35
react. Wow.
12:37
And that is all while we are still mostly in the elastic
12:39
range. Which brings us to the final
12:41
escalation for this deep dive. The scary part?
12:44
Yeah. What happens when we completely
12:46
blow past that one foot per second limit?
12:50
What happens when the velocity is so high and the force is so
12:53
great that those elastic flexural waves simply cannot
12:57
carry the energy away fast enough?
12:59
The material is overwhelmed, it stops vibrating elastically, and
13:02
it enters the plastic phase. Meaning it permanently yields
13:05
into forms. Yes, we start dealing with the
13:08
concept of the plastic hinge, where large structural
13:11
deformations occur under transverse impact.
13:14
But the most fascinating aspect of this escalation is how the
13:17
physics of the stress waves completely changes once you
13:20
cross that yield point. The speed limit of the material
13:22
shifts because in an elastic material the stress and strain
13:26
are proportional, governed by Young's modulus, and the elastic
13:30
stress waves travel at a known relatively high speed.
13:33
But the sources note that when the yield point is exceeded, the
13:35
slope of the stress strain curve changes drastically.
13:39
Well, the wave speed in a solid is directly tied to the
13:42
stiffness of the material in the elastic zone.
13:45
The material is stiff, the curve is steep so the wave travels
13:48
incredibly fast, but the moment you hit the yield point, the
13:52
material starts giving way. The slope of that stress strain
13:55
curve flattens out. Exactly.
13:57
The stiffness plummets and because of that, the velocity of
14:00
the plastic stress waves is vastly slower than the elastic
14:04
waves. So as the strain gets
14:05
indefinitely increased, the velocity of that plastic wave
14:08
actually approaches 0. The wave literally stalls out
14:12
and this leads to a phenomenon known known as the critical
14:15
velocity of impact. The critical velocity.
14:17
Right. The velocity of an incoming
14:19
object, say a speeding car or a runaway piece of industrial
14:23
tooling, is completely independent of the material's
14:26
internal wave speed. You can easily reach a point
14:28
where the physical impact velocity is higher than the
14:31
velocity of the plastic stress waves trying to propagate
14:33
through the steel. Think about what that actually
14:35
means. The material at the point of
14:38
contact is getting obliterated, but the internal warning system,
14:43
the stress wave trying to travel down the beam to say hey, we're
14:47
taking a massive load, everyone share the burden.
14:50
Literally can't move fast enough to deliver the message.
14:53
The continuity of the material is violated because the stress
14:56
cannot be communicated away from the impact zone.
14:59
It builds up instantaneously at the point of contact and the
15:02
material simply snaps. You get a catastrophic
15:05
continuity failure before the rest of the structure even knows
15:07
it's been hit. That is exactly what happens.
15:09
It's a terrifying physical reality, so.
15:11
How on earth do engineers design for this?
15:14
You can't track a stress wave that isn't moving, and all the
15:17
complex mathematical time integrations breakdown
15:20
completely when the material yields.
15:22
You have to abandon the micro level wave tracking.
15:24
You shift to a macro level approach known as the energy
15:26
method. The energy method.
15:28
Yeah, instead of trying to calculate the force at a
15:30
specific millisecond, we evaluate the dynamic resistance
15:34
of the material as a total budget of energy.
15:37
We look at the total energy absorbed by the material through
15:40
plastic deformation before it finally ruptures.
15:42
Oh I see. It's essentially A crumple zones
15:44
calorie count. You figure out exactly how much
15:47
kinetic energy the structure can digest before it dies.
15:51
That's a brilliant way to phrase it.
15:53
It relies entirely on the ductility of material, its
15:56
physical ability to stretch, warp and deform to soak up that
16:00
energy. The sources provide a fantastic
16:03
classic example using a tension specimen to explain.
16:05
This Yes. Picture a standard medium steel
16:08
bar 10 inches long and half an inch in diameter.
16:11
OK, I'm visualizing a heavy solid metal rod.
16:15
Laboratory testing shows that this specific 10 inch bar can
16:18
absorb a total of 1000 foot pounds of energy through plastic
16:21
deformation before it next down in the center and rips apart.
16:24
So 1000 foot pounds is his total energy budget.
16:27
Right now imagine a heavy 200 LB weights striking that bar with
16:31
an initial velocity of 80 feet per second.
16:34
If we run the kinetic energy calculation on that, remembering
16:38
that velocity is squared, which means speed kills far faster
16:42
than mass, that £200 weight moving at 80 feet per second
16:46
represents roughly 20,000 foot pounds of energy.
16:49
And the bar only has a budget of 1000 foot pounds.
16:52
So that 20,000 foot pound impact doesn't just bend the bar, it
16:55
ruptures it instantly. Total catastrophic failure.
16:58
But if we take a 50 LB weight and let it strike the same bar
17:02
at 30 feet per second. The mass is 1/4 of the original,
17:05
but more importantly, the velocity is drastically lower.
17:08
The kinetic energy of that strike is only about 700 foot
17:11
pounds. Exactly the the bar has a 1000
17:14
foot pound budget and the impact only hands it a bill for 700.
17:18
So the bar absorbs the energy. It deforms plastically, it
17:23
permanently stretches out, and it is ruined as a precise
17:26
mechanical component. But it holds.
17:29
It doesn't snap. The energy method allows
17:31
engineers to size protective structures based on these
17:33
survivability budgets. But there is a massive caveat in
17:37
the sources that we have to talk about before we wrap up the
17:39
notch effect. Oh yes, the notch effect.
17:43
Because this entire energy budget, this idea that a 10 inch
17:46
bar can safely digest 1000 foot pounds of energy, assumes that
17:51
the entire 10 inch length of the bar is participating in the
17:54
stretch. It assumes uniform yielding.
17:57
And that assumption is the critical vulnerability of the
17:59
entire energy method. The theory demands material
18:02
perfection. Because if there is a single
18:04
notch in that steel bar, or say a poorly executed butt Weld
18:10
halfway down, the energy doesn't distribute nicely through the
18:13
crystalline structure of the metal.
18:15
It acts like a bottleneck. Yeah, it's like a chain being
18:17
only as strong as its weakest link.
18:20
The impact Energy hyper focuses directly on that notch.
18:23
The bulk of the material in the specimen might actually remain
18:26
totally fine. The vast majority of the 10 inch
18:28
bar stays below the yield point, meaning it stays completely
18:31
elastic. But that tiny localized area
18:34
right at the notch or right at the edge of the bad Weld is
18:37
suddenly forced to take on 100% of the plastic deformation
18:41
required to absorb the impact. And a fraction of an inch of
18:44
steel simply does not have the physical volume to absorb 700
18:49
foot pounds of energy, so it reaches its maximum stretch
18:52
limit instantly and the bar snaps prematurely.
18:55
The full energy absorption capacity of the overall
18:57
structure never gets a chance to develop.
18:59
This is why material uniformity and rigorous quality control are
19:03
absolutely paramount in any impact resistant design.
19:07
A single scratch left by a careless machinist or a single
19:10
bad pass with a welding torch completely invalidates your 1000
19:14
foot pound energy budget. It turns what should have been a
19:16
survivable elastic bounce into a catastrophic lethal failure.
19:21
All from a microscopic flaw. It is wild to consider that a
19:24
tiny scratch on a massive steel beam can fundamentally alter how
19:29
that entire structure handles a split second collision.
19:33
We spend our days surrounded by objects in a state of quiet
19:36
equilibrium, but as we push the limits, building faster
19:39
vehicles, denser cities, and more aggressive machinery, the
19:42
invisible mathematics of a split second collision dictate the
19:46
safety of our physical world. Next time you drop your phone or
19:50
hear the deafening crack of a pile driver at a construction
19:52
site, think about those elastic flexural waves racing through
19:56
the materials and the structural energy budgets fighting against
19:59
incredible forces to keep the object whole.
20:02
It really makes you appreciate the engineering behind everyday
20:05
objects. Absolutely master the critical
20:07
few or the trivial many will master you.
20:09
If this episode gave you a solid take away drop.
20:11
A comment takes you a few seconds, but it tells me this
20:14
workbench is worth keeping alive.