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PROFESSOR: All right, then, I guess we may as well start.
So what I wanted to talk about today
was natural sandwich panels and sandwich beams.
So there's lots of examples of sandwich structures in nature,
and we've been looking at the engineering sandwich
structures.
And we've seen that you can get a lightweight structure
by having this sandwich construction.
And so there are several examples
I was going to talk about today.
And I think because this isn't really on the test,
I'm not going to write a lot on the board.
So there's some notes.
I'll just put them on the website,
and you can look at that if you want.
Because we have kind of a shorter time today.
I'll just try and talk and explain what's what.
Hey, Bruno.
How are you?
So this is the first example.
So many leaves of Monocotyledon plants
have a sandwich structure.
And this is an iris plant and iris leaves.
And for those of you in 3032, I think
you know that these are glass flowers.
So the Harvard Museum of Natural History
has a glass flower collection that was made in the 1800s.
And there was a botany professor there
who made these as sort of a lecture demonstration vehicle.
And so he would bring then to class,
and he would show different things
about the plants with the glass flowers.
But now they're just in the museum,
and they're very realistic.
So I just wanted to show you those.
So let's see, it's not working.
Turn it on.
There we go.
So if we look at a cross-section of an iris leaf,
it looks like the diagram on the left.
So here's the iris.
And you can see there's these kind of solid fibers,
and those solid fibers are called schlerchyma.
And they only exist at the top and the bottom of the leaf.
So I went out this morning.
And if you look outside of the Stata building,
there's that little kind of river-y thing,
and there's some iris leaves growing there.
So I went and got some iris leaves.
And you can tell we had a horrible winter because usually
when I give this lecture in the spring,
the leaves are like twice as big.
But this year, they're just little, short, wimpy ones.
But I'm going to pass it around.
And if you just like move your thumb over the top,
you can feel little ridges, little bumps.
And those little ridges that you can feel
are these little schlerchyma fibers.
So you kind of see they kind of stick up a little.
And so when you move your thumb over it, you can feel that.
And then you can see that the middle of the iris leaf
has this kind of foamy-type structure here,
and that's called parenchyma cells.
So you can think of the leaf as very much like one
of the sandwiches.
This is like a fiber-reinforced composite at the top
and at the bottom.
And then this is kind of like a foam core in between,
separating the fiber-reinforced faces.
And so the iris leaf behaves mechanically
like a sandwich beam.
So I'm going to talk a little bit about how we can actually
demonstrate that using the equations that we developed
in class.
This is another example.
This is, I guess, what Americans call the cat tail,
but Canadians and English people call it a bull rush.
And you can see this is a slightly different
construction, but there's the same sort of idea.
So instead of having a foamy core as in the iris leaf,
you've got these kind of webs here
that go in between the top and the bottom,
and that forms like a series of I-beams almost.
And you can think of that also like a sandwich
panel or a sandwich beam.
So you've got two stiff top and bottom pieces,
and then you've got these kind of webs that separate them,
kind of like a honeycomb core would be.
So that's another example of a leaf that has
the sandwich-type structure.
And this is very common in these Monocotyledon leaves.
So if you think of a cat tail or you think of an iris,
they tend to be kind of narrow at the base,
maybe an inch or two wide at the base,
and they can be quite tall.
The iris leaves can get two or three feet tall.
The cat tails can get five or six feet tall.
And they stand up more or less straight.
They bend over a little, but they stand up more or less
straight.
And this sandwich structure is one
of the things that lets them stand up straight
at a fairly low weight.
And from the plant's point of view,
there's a sort of metabolic cost associated
with making more material.
So it we can minimize the amount of material,
it's a better thing for the plant.
These are some other examples of grasses that are
sandwich-type constructions.
This is from some papers by Julian Vincent.
And the black little circles here are the schlerchyma,
are those sort of dense fibers.
Then you can see in both of these cases,
the dense fibers are on the outside,
and the parenchyma cells, which is the white,
are on the inside.
And so this is sort of another set of micrographs of the iris.
So this is just showing the outside,
and these are the ribs viewed from the outside.
And this is the core, just sort of viewed
along the length of it.
And so you can idealize the structure
as being like a sandwich that's got sort of fibers on the top
and on the bottom.
So the top and the bottom are like a fiber composite.
And the middle part, with the parenchyma cells,
is kind of like a foam.
And so we did a little project on iris leaves,
and we wanted to see if you could
show that they behave mechanically,
like a sandwich beam.
So you remember that we had that equation for the deflection
of the sandwich beam.
There were two terms.
There was a bending term, and then there was a shearing term.
And so we took some little sandwich beams.
We cut little kind of rectangular beams.
We hung little weights.
We measured how much they deflected,
and we wanted to see if we could use this equation
to predict their stiffness and how much they deflected.
So to do that, we needed to know a bunch of things.
We needed to know some of the geometrical parameters.
So we needed to know what volume fraction of the face
is those solid ribs, how thick's the core, how thick's the face?
And so we measured a bunch of these geometrical parameters.
We tested it like a cantilever so we knew what B1 and B2 were
for the cantilever.
We knew how long the beam was, so we know what l is.
We knew what loads we applied, so we knew what P was.
But we needed to make some estimate
of what the face modulus was and what the core shear
modulus was, too.
And so we made some estimates of that.
So this table here just shows some
of the dimensions of the leaf.
The leaf tapers, and this is at the thin end,
so here's the face thickness.
Here's the sort of length of this.
Some square cells in the face.
This is the core thickness here.
This is the dimensions of the core cells.
This is the diameter of the ribs, the spacing of the ribs,
the volume fraction of solids in the ribs.
And we did that at different lengths
along the different positions along the length of the rib,
or length of the leaf.
So we had the geometrical parameters,
but we needed to get this E of the face and G of the core.
And to do that, we looked at the literature.
And people had done tests on the fiber parts of leaves.
They'd done little tensile tests,
and they'd measured modulii between about two
and 20 gigapascals.
And then we did some tension tests on the iris leaf.
And in tension, those ribs are going
to take most of the stress.
And if you know the volume fraction of the ribs,
you can back out what the stiffness of the ribs
must have been.
If you know the stiffness of the ribs,
you can figure out the stiffness of the face.
So we calculated that, and then we looked at the literature.
And people have done tests on parenchyma cells
and different types of tissue on things like apples and potatoes
and carrots.
And these are the values for the Young's modulus they get.
They're between about 1, and the highest one was 14 megapascals.
But most of these values for the Young's modulus
are around about four.
And the shear modulus is roughly about half
of the Young's modulus.
So we said the shear modulus was around two.
So we have these values we could plug it in and then calculate
what the stiffness would be for the iris leaf.
And so this was a little analysis we did.
So this was the measured beam stiffness up here.
We had four beams, and they were different stiffnesses.
They all had the same length.
They all the same face thickness.
The core thickness varied.
They all had the same width.
We cut them to have the same width so we could
calculate a flexural rigidity.
That's the EI equivalent.
We could calculate the bending deflection term, the shear
deflection term.
And this is the calculated beam stiffness.
And then this is the ratio of the calculated
over the measured.
So it's not exactly right.
Obviously, there's some difference here.
But it's in the same order of magnitude.
It's in the same ballpark.
And one of the complications that we didn't really
try to take into account was that the leaf isn't
a nice rectangular structure.
The leaf has this kind of curved cross-section to it.
And we made a bit of an approximation to that,
but it wasn't that close, really.
We could have probably done better on that.
But I think the idea that the iris behaves like a sandwich is
a reasonable one.
So that was the iris leaf.
And then I wanted to show you some other structures in nature
that are sandwiches.
So this is a seakelp, help like a seaweed thing,
in New Zealand.
This is the largest intertidal seaweed.
The fronds, the sort of long pieces of it,
are up to 12 meters long.
So that's almost 40 feet.
So 40 feet is probably like from one side of this room
to the other side of the room.
It's quite long.
And you can see, if you look at this section here,
this is all like a honeycomb-type section here.
And the honeycomb is like a honeycomb in a sandwich,
and the top and the bottom faces are
like the face of the sandwich.
So this would be like the face here.
That would be the honeycomb core.
And that would be the other face on the other side over there.
And those honeycomb-like cores, apparently,
have some gas-filled pockets that then provide buoyancy
to keep the whole thing floating.
So it photosynthesizes.
So one of the things about these leaves
is that they have multiple functions.
It's not just that they have to have a certain stiffness
so they don't fall over.
The plant wants to photosynthesize,
so you want to maximize the surface area as well,
and you want to have exposure to the sunlight.
So there's a number of things that the plant's trying
to do in having this structure.
So that seakelp is one example.
These are skulls from birds.
And so this is a pigeon here.
This is a magpie.
If you come from the West you see magpies out West.
You see them in Europe as well.
And this is a long-eared owl.
This long-eared owl's around here.
And I brought in a couple of bird skulls as well.
And you can see that all of those birds skulls
are sandwich structures.
The one for the pigeon has sort of a foam-like core here.
And you can see that the two faces
aren't sort of concentric for the pigeon skull.
They sort of not following each other.
But here, this would be, say, on the top shell
of the magpie, where the two, the inner and outer face,
are sort of concentric.
Then you get these kind of little ribs of trabecular bone
in between them, and then the same with a long-eared owl.
You get these little ribs in between them.
And so you can see that there's a sandwich structure there.
And obviously, birds want to be light.
They have to be light to fly, to take off,
and so they want to be light.
So I've got two skulls here.
And I'll pass them around.
Please be careful because they're kind of delicate.
This one is from a screech owl, and you see screech owls
around here.
This was a screech owl that had an intersection with a car.
Yeah, so the skull fractured, but you can see the sandwich
right there.
You see the two little bits?
So you can see the inner plate and the outer plate
and the foam, the trabecular bone.
So that's the screech owl.
And this is a red tail hawk.
So you can't really see the shell and the sandwich
structure here.
But I want to pass it around just
so you can see how light it is.
So it's amazingly light.
So a red tail hawk is probably about this big,
something like that.
And this is one of the things that makes them very light.
So those are the bird skulls.
Oh, yes, so now I have to tell you about the owl.
So I think the people in 3032 have heard this before.
But the other people haven't.
So one of the things about the owl
is if you look at the whole skull,
if you look at this picture here, one of the things
is that this bone here is not symmetrical with that bone
there.
Normally, when you think of a body,
you think of the bones being symmetrical.
But those bones are not symmetrical,
and those bones are near where the ear is.
And it turns out on owls, at least on some owls,
the ears are at different heights on their heads.
And people think that one of the things that
allows the owls to do is it allows their hearing
to sort of pinpoint where something is.
And owls can catch little creatures at night,
but they can also catch little creatures underneath the snow.
So they can catch things that they can't even see.
And they have a number of adaptations
to improve their hearing, but this is one of them.
So here's a little owl Allison Curtis is a Canadian friend who
lives in northern Ontario, and this is looking out
of her living room window.
And that's a barred owl.
And you can see the barred owl has
caught this little vole here.
And you can see in the background
it's winter in Canada.
and there's snow all over the place.
So this owl has probably caught that little vole
underneath the snow.
And then it's come to eat it.
And this is another picture of-- you
can see this is where an owl landed in the snow.
It's wings hit the snow, trying to catch something underneath.
And this is another kind of beautiful print
of the owl's wings hitting the snow in the winter time.
So did I show you the fox video?
Should I show you the fox video?
You saw it, right?
I think I showed it last time in 3032.
But you guys haven't seen it.
Let me show you the fox video because foxes
do the same kind of thing.
Their ears are the same as ours.
They're in the same position.
But they have this-- let me see.
Where's the sound thing?
We don't really need the sound for this,
but there's BBC sound.
So we get this music, even though the fox
can't hear the music.
Here we go, fox no drive.
Check this out.
Is it going to come up?
Is that going to play?
OK
[VIDEO PLAYBACK]
-It listens for the tiny sounds of its prey moving about below.
PROFESSOR: So you see how it cocks its head,
and it does this with its head?
It's putting its ears at different heights
when it does that.
So check this out.
And look carefully, you can see the little animal
it's got in its mouth when it comes out.
There's a little tail.
So part of the reason dogs and foxes and coyotes
do that thing, I think, is because they
put their ears at different heights,
and it helps them pinpoint where something is.
[END PLAYBACK]
You know I love these Nature videos, right?
So that's the fox video.
Let me see if I can stop that.
So that's one of the interesting things about owls.
Let me go back to my little PowerPoints.
So here's another example of a creature that
has a sandwich-type structures.
So here's the sandwich here.
Here is the ever so charming looking cuttlefish.
And the cuttlefish is not actually a fish.
It's a mollusk.
So it's related to things like octopus, things like that,
and squids.
It's a cephalopod.
And you can't see it so well in this picture,
but I'm going to show you something else and you see it.
It's got like little tentacles.
These things here are actually separate little tentacles.
And because it's not a fish, it doesn't have like fins
that can kind of swim with.
And it's got this thing called the cuttlefish bone.
And this is a cuttlefish bone here.
And that bone has the sandwich structure here.
And it's not actually a bone.
It's really a shell.
It's a calcium carbonate thing, not a calcium phosphate thing.
But the cuttlefish can control how much air
goes into those little pockets.
And it can control its buoyancy by controlling how much air
goes into those little pockets.
And I brought with me a cuttlefish bone.
Have you ever owned like, I don't know,
like a parrot or a pet bird?
Apparently, pet birds love to sharpen their beaks
on this cuttlefish bone.
So if you go to a pet store, you can buy this stuff.
So you won't be able to see the little sandwich
structure because it's a very small length scale.
But you can kind of see there's a sort
of different material on the inside
than there is on the outside of that.
So do people know the other thing
that cuttlefish are famous for, besides the bone?
Change colors.
Can I show you a video of cuttlefish changing color?
Yeah, of course.
So let me get rid of this again.
Go back to this.
Let's see, somewhere-- where's the cuttlefish?
Here we go.
Did I do it?
Is it thinking?
Here we go.
Where's the cuttlefish?
So this is another one of these Science Friday
videos from National Public Radio with Flora Lichtman.
[VIDEO PLAYBACK]
-OK, let's play a game.
[GAME SHOW MUSIC PLAYING]
[APPLAUSE]
PROFESSOR: See it?
-Biologist Sarah Zielinski took these shots.
And if you needed a helping hand to find the cuttlefish,
don't feel bad.
-I've certainly taken photos in the past then come back
to look at them and gone, I'm sure there was a cuttlefish
in there somewhere!
-These cephalopods are master camouflagers.
But while they're hiding their body,
they're revealing something about their mind,
or at least their visual system.
-In very simple terms, they can tell us
what they can see by the body patterns they
produce on their skin.
-They produce these body patterns
by expanding or contracting chromatophores,
these little ink sacks on their skin.
And they use different displays for different reasons,
like for male-to-male combat.
-Two males will turn into each other
and pass these kind of waves of dark chromatophores
over a really bright sort of iridescent stripey body pattern
and somehow solve these combats.
Eventually, one male gives up and goes away.
-And then there's this unsolved mystery.
It changes color when it grabs a snack.
-That doesn't make perfect sense because it seems
to make it very conspicuous.
So one theory is that it's just a happy signal
of how excited it is to have caught
something, some response that it doesn't have any control over.
-But most of the time they seem to be
using their chromatophores more intentionally,
primarily to blend in.
-Because otherwise they're more likely to be eaten,
so it's very important they don't make mistakes
about ambiguous visual information.
-And ambiguous visual information
is specifically what Zielinski's interested in.
So here's the experimental setup.
Print out laminated patterns, like this checkerboard,
and stick them in a tank.
-And we place the animals in the tank.
And we record the body patterns that they produce.
-You're seeing them on squares, but they do the same thing
on top of circles.
They produce--
- --the disruptive pattern, where you get these blocky
components of high-contrast components.
-But when you put a cuttlefish over squiggles, it produces--
- --a sort of mottley pattern, where you get these little
groups of dark spots showing across the body.
-So what happens when you put a cuttlefish on something
in between, when you put them on incomplete circles?
When we see something like this, our visual system
likes to fill in the blanks, something we do constantly,
Zielinski says.
-The reason why cartoons and sketches work
is because we can recognize objects
based on their edges alone.
-And we can identify objects even if they're broken up or--
- --have an object that is occluded by another object.
That's no problem for us.
We can still work out what the object is most of the time.
And I was interested to know whether cuttlefish
can solve similar problems.
-And Zielinski and colleagues report this week that
cuttlefish do seem to--
- --fill in those gaps and interpret those little segments
as a whole circle.
-Or anyway, the broken circles prompted the same camo pattern
as full circles.
So if you're wondering, uh, I see these as circles, too.
What's the big deal?
The weird thing here is that there's no reason why
cuttlefish, which are--
- --invertebrates, and they're in the same group as slugs
and snails.
- --should see the world the way we do.
-Yes, it's like they're alien, but we also
seem to have so much in common with them.
-So the next step?
-Because we can't share the perceptive experience
of a cuttlefish, it's hard to know
exactly what it is that they're doing to fill in that missing
information.
And I want to try to get a better grasp on that
and also see whether they actually respond
to true illusory contours.
-So you're going to show optical illusions to cuttlefish?
-(LAUGHING) That's what I'm hoping to do, yes.
[END PLAYBACK]
PROFESSOR: So let's go back to sandwiches.
I think I have-- do I have one more?
There we go.
So horseshoe crab shells, so different sorts
of arthropods, the shells are sandwiched too.
This is from Mark Myers' work.
So we're looking at the cross-section
of a horseshoe crab shell.
So again, it's the same idea-- the animal wants
to minimize the amount of material
or minimize the weight, and this is a way of doing that.
And I went to the Galapagos about a year ago.
And there was a place where they had these giant Galapagos
tortoise shells.
And one of them was broken, and you
could see there was a sandwich structure in the Galapagos
tortoise shells.
These Galapagos tortoises, their shell is like this big.
They're gigantic.
They're huge.
So those are my examples of sandwich panels and beams
and shells and whatnot in nature.
So the idea is that nature too wants
to minimize weight and minimize the amount of material,
and the sandwich structure is a way of doing that.
So I have one more thing I wanted to talk about today.
So this isn't quite sandwich structures,
but it's looking at another kind of natural structure
that is designed to reduce the weight of plant stems,
in this case, palm stems.
And there's a couple of interesting things about this.
So when you look at palms, like let's pretend
we're not in Boston.
We're in California, where they have palms.
And we're in LA, and they don't have winter.
And if you look at the palms growing, when the palm's short,
it's about this big in diameter.
And as it gets taller and taller,
the diameter doesn't really change.
It gets taller and taller and taller,
but the diameter doesn't change, at least in some species.
Whereas if you think of a tree, a tree
starts out with a little skinny diameter.
And as the tree gets taller, the diameter gets bigger.
And it sort of tapers and does that whole thing.
So palms don't do that.
And palms are not trees.
They're a botanically different thing from trees.
So here's a coconut palm.
And so the question is, as the stem gets taller and taller,
how does it resist the bending loads that
get bigger and bigger?
So probably, the main load on these sorts of things
is from the wind.
And often these plants are in areas
where they have hurricanes.
And you see them in hurricanes, you
see the pictures of the palm stem blowing way over.
And so how do they resist the larger internal stresses
as they get taller and taller, if the diameter doesn't
get bigger and bigger?
And the way they do that is that they deposit additional layers
of cell wall as the plant ages.
So if you think of a tree, when a tree grows,
it just deposits more and more cells.
And the cells have roughly the same thickness.
So there's ones that are deposited in the spring have
thinner walls.
Then the summer and the fall have thicker walls.
But more or less, it's similar.
Whereas the palm, it deposit cells,
and then as the trunk of the palm
gets taller, as the stem gets taller,
it deposits more layers on the cell wall.
So this is an example in an SCM.
You can see here this is a young cell,
and it's got-- this one that's not marked is a primary cell
wall, and then this is the first layer of the secondary cell
wall.
And then this is an older palm.
And you can see here it's got more layers,
and so the cell wall itself has gotten thicker.
So that means that the density of the tissue
changes as the palm ages.
And it does so in a very kind of clever way.
If you think of the palm as being like a cantilever that's
vertical and it's bending in the wind, when we have a cantilever
beam or any kind of beam, the stresses
are going to be biggest on the periphery, right?
They're going to be biggest on the outside.
And if you think of the palm as having
a circular cross-section, that outer periphery
is going to see the biggest stresses.
So it would make the most sense if that was the densest tissue.
And that's exactly what the palm does.
So there was a nice study done by Paul Rich quite
a number of years ago.
And he studied palms in Central America
and looked at the density and measured
the mechanical properties.
And I'm going to talk about his stuff today.
So the white is the low density.
The gray's the medium, and the black's the high.
So you can see the low density's on the middle
of the young stem, and just at the very base
and then the periphery is the dense tissue.
But as the stem gets taller and gets older,
then stuff that was low density is now high density.
And only the very middle here is the low density.
And that some stuff that was low density
has turned to middle density.
And some stuff that was low density
has turned to high density.
So it's done this by adding more and more layers to the cell
wall, making the cell wall thicker
and making the cells themselves denser.
So this is looking just at a single palm.
So each one of these lines is a single palm.
And this is looking at how the density changes
from the periphery to the center of the palm.
So if you cut the palm down and, say,
we take a little sample radially from the middle to the outside
or from the outside to the middle,
he then measured the density.
And it's probably easiest to think about the dry ones
because that's kind of what you would compare wood to.
So the dry densities varied from about one gram per CC,
that's about 1,000 kilograms per cubic meter,
down to almost zero in this particular species
here, probably like 50 or something like that.
And if you compare this with woods,
this little arrow here is the density of most common woods.
So if you looked at pine and spruce an oak and maple
and ash and hickory, they would all be in that little range
there.
So a single palm stem can have a bigger range
of densities than many different species of wood.
So it has this kind of profile of the density.
And the thing I was interested in
is seeing how mechanically efficient
that was to put the denser material at the outside.
So I looked at the stiffness of the palm,
and I also looked at the strength.
So I just replotted that data on this slightly different axes
here.
So this is the radial position relative to the outer radius,
and this is the density.
And I subtracted off the minimum and then took the range.
And for this species here, the minimum density
was almost zero.
So this expression simplifies to something like that.
And just because it's mathematically simpler,
that's what we're going to look at.
So the density goes roughly as the radius squared.
And Paul Rich also did a lot of mechanical tests on the palm,
and he took out little beams of different densities.
And he measured the stiffness and the strength of the beams.
So he measured the modulus of elasticity here versus density.
And he measured the modulus of rupture here.
And these are all along the grain.
And he found that the Young's modulus varied with the density
to the 2.5 power, and the strength
varied as the density squared.
And if the--
[BUZZING SOUND]
Oh, hello.
[LAUGHTER]
So these were just sorts of empirical findings
that he made.
If you have prismatic cells and you deform them axially,
and the cell wall was the same in the different specimens,
then the solid modulus would be a constant.
And you would expect that the modulus of the beam
would go just linearly with the density, sort
of like a honeycomb loaded [? at a ?] plane.
But what he measured was that the modulus and the strength
varied with some power of the density.
And the reason for that really was
that the cell walls of the denser material
had more layers.
And in the additional layers, the cellulose microfibular
angle was probably different, so that the different layers
had different stiffnesses.
And if you have layers of differences,
then you're going to get this power relationship.
So what I then did was I took his data,
and I tried to see how efficient that would be in bending.
So he had found that the density varied with the radius raised
to some power.
This power n was 2, but I wanted to do it just
for a general case, so I said I was just n.
And he said that he found that the modulus varied
with the density raised to some other power m.
And for him, m was 2 and 1/2.
And so I could write just another equation
saying that the modulus goes as the radius to the mn power.
And then you could do a little calculation
where you work out with the equivalent flexural rigidity
is.
So you have to integrate up.
You kind of say you have a little band
at a certain radius.
That radius has a certain modulus.
And you can figure out the moment of inertia
that goes with that particular radius.
And then if you integrate it up over the whole thing,
you can say that the flexural rigidity
for the gradient density is some constant times
pi times the outer radius to the fourth power divided
by those two powers mn plus 4.
So m was the power here for the modulus.
And n was the power there for the density.
And then you could compare that with having the same mass
just uniformly distributed over the whole cross-section.
And then if you take the ratio of the flexural rigidity
for the density gradient versus the flexural rigidity
for the uniform density, you can show
that it's this equation here.
And then if you plug in these measured values
for those exponents for n and m, you
find that the flexural rigidity with the gradient density
relative to the uniform density is a factor of 2 and 1/2.
So the stem is 2 and 1/2 times stiffer
by having that density profile.
So there's a huge sort of mechanical advantage
to doing that.
And just sort of physically, if you
know the stresses our biggest on the outside,
it would make sense to put the denser material on the outside.
And then the other thing I looked at
was the strength of the palm.
So imagine this is our very schematic palm here,
and then there's a circular cross section.
So I wanted to compare the bending stress distribution
with the bending strength distribution.
So the stress goes as the modulus times the strain,
just Hooke's law.
And here we're assuming that plane sections remain plane,
like that's the standard assumption of bending.
So if you assume plane sections remain plane,
then the strain goes with the curvature times the distance y
from the neutral axis, the distance from the middle.
So this distance here would be the dis-- [? same ?]
at loaded with a loaded p here.
That distance would be y there.
And then I can plug in some things here.
So instead of E, I'm going to plug-in my relationship
with the radius to that mn power.
And here's my curvature, and instead of y,
if I say that some radius, I'm going
to say y is our r cos theta.
And so I'm going to say that the stress goes--
[SNEEZE]
Bless you.
Goes as radius raised to some power mn plus 1.
And again, for the species I know what n and m are,
so the stress goes as the radius to the sixth power.
And then I can also compare with what Paul Rich had
found for the strength.
He found that the strength-- so sigma star is the strength--
was proportional to the density raised to some power q,
and that power was 2 in the measurements that he made.
And so I can say that the strength
goes as the radius to this power nq, so to the fourth power.
And then if I plot the stress distribution and the strength
distribution-- so imagine, this is through the cross-section
here.
So this is the diameter of the stem.
And this is the neutral axis here in the middle.
The strength goes as that solid line there.
It goes as the fourth power.
And the stress goes as that dashed line there,
as the sixth power.
So they're not exactly on top of each other,
but they're very close to being on top of each other.
So basically what the palm has done
is it's arranged the material in such a way
that the strength matches the stresses that
are applied to it.
So if I just had a constant density,
my stress profile would look like that.
And if I had a constant density, the strength profile
would kind of like that.
So the strength here would be a constant,
and this would be the stress here.
So the stuff in the middle, it's much stronger
than it needs to be.
Whereas the palm has arranged things so that it's got
just the right amount of strength for the stress,
as a function of the radial position.
So it's kind of a clever thing.
So that's kind of a beautiful thing.
And I think that is it.
I think that's-- yeah, that's the end of it.
So all these images came from this other book that we wrote.
And if you wanted to get the sources,
you could get them from there.
So that all I wanted to talk about today
was some examples of sort of efficient mechanical design
in nature and the sandwich panel structures as one,
and these radial density gradients is another.
We have a project on bamboo right now,
and the bamboo also has a radial density gradient,
and it's the same thing.
The densest material's on the outside,
and the least dense is on the inside.
So I think I'm going to stop there for today.
So what I was going to do on Monday
is talk a little bit about bio-mimicking.
And that won't take the whole class at all.
And I thought we could spend the rest of the class
on Monday just doing a review.
So the test's on Wednesday.
So if you want to bring questions,
that would be a beautiful thing.
I can't really can I review the whole last six weeks
or something in an hour and a half or something.
So if you want to bring questions, I'll be here
and we can just go over questions.
Does that sound good?