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LORNA GIBSON: All right.

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And I really wanted to show
you my little hook video

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00:00:30,120 --> 00:00:32,380
and I downloaded it so I
thought we'd start just

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by watching that and then I'll
pick up about modeling phones.

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So this takes like
nine or 10 minutes,

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but I just thought it was cute.

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And I made it and I
want you to see it.

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So let's do that to start.

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[VIDEO PLAYBACK]

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We're here at the Harvard
University Botany Library,

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looking at a first edition of
Robert Hooke's Micrographia,

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00:00:51,070 --> 00:00:51,850
published--

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00:00:51,850 --> 00:00:53,920
How do I get rid
of the bar, Greg?

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Oh, there it is.

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00:00:54,710 --> 00:00:57,855
show the microscopic
structure of materials.

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And it has a number of
remarkable drawings in it.

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Here we see drawings of silk.

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These are two different silks.

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On the top here, we
have a fine-waled silk.

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And in this more details
drawing down here,

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you can see the patterned
weaving of the silk.

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The bottom image here is
a drawing of watered silk.

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And over here, there's another
higher magnification image.

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And you can see the pattern
here is more sharply angled.

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And it appears that
this sharper angle here

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gives the different texture to
the surface finish of the silk.

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So here we see a
drawing of charred wood.

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And one of the things I find
interesting about this drawing

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is how similar it is to modern
electron micrographs which

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we've seen before.

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00:01:42,130 --> 00:01:45,190
And in this drawing, we can
see two of the main features.

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We see these small cells,
which are fibers that provide

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structural support to the tree.

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And we see these
larger cells here,

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which are vessels
which allow fluids

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to go up and down the tree.

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And here we see a drawing of
the surface of a rosemary leaf,

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with the unexpected,
tiny, little bars.

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And this is something
that you can only

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see with the microscope.

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You wouldn't expect to
see those when you just

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feel the surface of
the rosemary leaf.

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So it's kind of interesting
that with the microscope,

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you can see these features that
are invisible to the naked eye.

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One of the main themes
of material science

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is that the property
of materials

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are related to their structure.

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And so being able
to see the structure

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at a microscopic
scale is very helpful.

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And today, we can even see the
structure at the atomic scale.

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Robert Hooke
understood this idea.

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And in the description
of the cork,

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Hooke states, "I
no sooner discerned

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these-- which were the first
microscopical pores I ever

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saw-- but methought that I had
with the discovery of them,

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perfectly hinted to me the true
and intelligible reason for all

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of the phenomena of cork."

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So what he's saying here is
that by looking at the structure

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and looking at the cells
here in the drawing,

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he thinks he can understand
the properties of cork

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or the phenomena of cork.

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What was it about
Robert Hooke that

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allowed him to make this book?

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Why was it him and
not somebody else?

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Well, Robert Hooke had kind
of an interesting history.

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He grew up on the Isle of Wight.

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And as a boy, he
loved making drawings.

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And he got quite skilled
at making drawings.

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The other thing was, he loved
making models of things.

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He made models of ships.

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He made a wooden clock
that was a working

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clock when he was a kid.

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And as a teenager,
he moved to London.

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And he became an apprentice
to Sir Peter Lilley,

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who was a famous
painter of the time.

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So his drawing was good enough
that he would be working

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with a very well-known painter.

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After he did that, he went
to the Westminister School.

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And he studied classics.

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He studied mathematics.

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But he also learned
to use a lathe.

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And this was also very
helpful in him making

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various sorts of apparatus.

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And as a student at
Oxford, he worked

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in the lab of Robert Boyle.

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And his job in that lab was to
develop scientific apparatus.

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00:03:54,060 --> 00:03:56,390
And he did things
like he built pumps

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that allowed Robert Boyle
to do the experiments that

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led to Boyle's Law.

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00:04:01,800 --> 00:04:04,150
When he returned to
London after Oxford,

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he became the Curator
of Experiments

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at the Royal Society.

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00:04:07,430 --> 00:04:10,590
And one of the things he
did was he got a microscope.

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He improved that microscope,
increasing their magnification,

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which was what allowed him to
make the beautiful drawings

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that we see today.

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And here in the
preface of the book,

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we see that he even made a
drawing of his microscope.

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So this thing down here-- this
is Robert Hooke's microscope.

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The development
of new microscopes

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with higher and
higher magnifications

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continues to this day.

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00:04:31,540 --> 00:04:35,090
Scanning electron microscopes
were invented in the 1960s.

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00:04:35,090 --> 00:04:37,600
And today, we have transmission
electron microscopes

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00:04:37,600 --> 00:04:40,220
and atomic force
microscopes with even higher

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magnifications.

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At these higher
magnifications, we

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00:04:43,360 --> 00:04:45,960
can see details that Hooke
was unable to see because

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of the limitations
of the microscope

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that he had-- the
optical microscope.

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But it's interesting
to see today

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the images we see in
a scanning electron

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microscope at a similar
magnification to those

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that Hooke saw in his
optical microscope.

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And it's remarkable to see
how many of the features

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that we see in these much
more fancy microscopes

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that he was able to
capture in his drawings

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with his simple
optical microscope.

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So here we have a
picture of cork.

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We have Hooke's drawings showing
two perpendicular planes.

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We also have this nice, little
drawing of a cork branch here.

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Cork is the bark from
the cork oak tree.

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00:05:21,340 --> 00:05:23,330
And in Hooke's drawings
of the microstructure,

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we can see these cells
here are roughly box-like.

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They're more or
less rectangular.

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And these cells here look
more or less circular.

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So there's these two
different perpendicular

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planes in the cork.

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00:05:34,750 --> 00:05:37,220
And when we look at these
scanning electron micrographs,

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we can see very
similar structure.

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There are some cells
that are roughly boxlike,

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and others that are more or less
hexagonal or roughly rounded.

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One feature that Hooke was
not able to see, though,

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that you do see on the
scanning electron micrographs,

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is the waviness
in the cell walls.

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And that was because the
resolution of his microscope

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was insufficient to see
that level of detail.

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And here in this
illustration on the bottom

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00:06:01,690 --> 00:06:03,740
here is a drawing of sponge.

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00:06:03,740 --> 00:06:06,380
And when we look at the
scanning electron micrograph,

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00:06:06,380 --> 00:06:09,320
we see that the structure is
remarkably similar to what

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Hooke has drawn.

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So here we have Hooke's
drawing of feathers.

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And we can see he's made several
drawings at different length

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scales.

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And if we look at this one
here, we see the barbule.

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And you can see these
little hooked regions there.

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And those hooks lock
into the little feathers

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over on this side over here
of the adjacent barbule.

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And in the higher
magnification picture,

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you can see on one barbule,
there's hooks on one side

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but not on the other.

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And it's this hooking of
the two sections together

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that allows the
feathers to maintain

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a smooth surface for the
wing when the bird is flying.

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And you can see the
same sort of thing

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when you look at the
electron micrograph.

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So you can see the little hooks
on one side of the barbules.

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And you can see how they
interconnect together

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with the next barb.

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One of the most reproduced
images from Hooke's book

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is that of the flea--
this image we see here.

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And you can see why.

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It's a gorgeous image.

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00:07:09,460 --> 00:07:12,330
And it shows details that
people had never seen before.

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People were amazed to see that
the little flea that they might

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have found on their
dog or something

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was actually made up
of this compound body,

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with all these little plates
and little hairs here.

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And you can see these little
tiny claws on the legs,

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and the legs have
all these hairs.

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Nobody had any idea that
this is what a flea actually

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looked like.

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And so it was an
amazing drawing.

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And it was something that
people were just stunned by when

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Hooke's book came out.

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And if we look at a modern
electron micrograph,

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we can see it's
remarkably similar if we

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look at the same magnification.

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So Hooke showed many
of the same details,

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showed some of the
same hairs on the legs,

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showed the same sorts of
plates, showed the claws

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at the ends of the legs.

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And our modern image is probably
from a different species

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of flea.

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We don't know what
species of flea

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that Hooke actually looked at.

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But you can see there's
a tremendous similarity

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between the two images.

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And it's remarkable how
many of the features

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that Hooke was able to
capture in his drawing.

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And here we have the
compound eye of the fly.

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And this, again, was astonishing
to people in Hooke's day.

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And even today, people
look at this image,

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and they're pretty amazed at
the detail in this drawing.

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And again, we can compare
this with a modern electron

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micrograph.

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And again, you can
see the similarities

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between what Hooke
saw and what we

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see in a modern scanning
electron microscope

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at a similar magnification.

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00:08:35,030 --> 00:08:37,600
In the 1980s, atomic
force microscopes

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were invented, which have
a resolution down to tens

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of nanometers.

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00:08:41,710 --> 00:08:44,380
And today, there's transmission
electron microscopes,

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00:08:44,380 --> 00:08:46,820
which allow you to see
the atomic structure.

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So for instance in
a crystal lattice,

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you can see the individual
atoms and the regular crystal

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structure.

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Today, most experimental
studies of materials

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include photographs of
the microscopic structure

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of the material taken through
some sort of microscope.

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And the remarkable thing is
that all of these studies

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really trace back
to this book here

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that we're looking at today--
to Robert Hooke's Micrographia.

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[END PLAYBACK]

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There you have it.

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So I just thought
that was kind of cute.

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You might enjoy that.

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So that was that.

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All right, let's
get out of there.

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Stop.

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So let's go back to the foams.

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00:09:28,000 --> 00:09:29,690
So I think last
time, we got as far

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as talking about the linear
elastic behavior of foams

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and modeling that.

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But we didn't quite
get to looking

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at the compressive
strength of the foam.

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So I think we got
as far as comparing

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the models with
these equations here,

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and these plots of the data.

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And what I wanted to
pick up with today

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was looking at the
compressive strength.

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And we'll look at the fracture
toughness as well in tension.

242
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So we're going to
start with nonlinear

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00:09:54,520 --> 00:09:57,055
elasticity and the
elastic collapse stress.

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So if we have an open-cell
foam, the derivation

245
00:10:15,750 --> 00:10:17,910
for the elastic collapse
stress is really

246
00:10:17,910 --> 00:10:19,370
pretty straightforward.

247
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We say the elastic
collapse occurs

248
00:10:21,840 --> 00:10:23,120
when the cell walls buckles.

249
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So in this schematic here,
you can see the vertical cell

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00:10:25,980 --> 00:10:27,120
walls have buckled.

251
00:10:27,120 --> 00:10:29,490
And so there's going to
be some Euler load that's

252
00:10:29,490 --> 00:10:31,310
related to that buckling.

253
00:10:31,310 --> 00:10:33,920
And that's just the usual
Euler load-- n squared, the n

254
00:10:33,920 --> 00:10:36,780
constraint factor,
pi squared E. This

255
00:10:36,780 --> 00:10:40,110
is going to be E of
the solid, I over l

256
00:10:40,110 --> 00:10:43,300
squared-- the length
of the member.

257
00:10:43,300 --> 00:10:46,310
And then the stress
that corresponds to that

258
00:10:46,310 --> 00:10:49,810
is just going to be proportional
to that buckling load

259
00:10:49,810 --> 00:10:52,440
over the area of the
cell, which is just

260
00:10:52,440 --> 00:10:56,000
l squared, so just P
critical over l squared.

261
00:10:56,000 --> 00:10:58,820
So that just goes as Es.

262
00:10:58,820 --> 00:11:00,760
I is going to go
as t to the fourth,

263
00:11:00,760 --> 00:11:04,737
because we have that
square sectioned member.

264
00:11:04,737 --> 00:11:06,320
And now this is going
to be l squared.

265
00:11:06,320 --> 00:11:07,850
And that's an l squared.

266
00:11:07,850 --> 00:11:09,830
So that's l to the fourth.

267
00:11:09,830 --> 00:11:12,030
And so if I combine
all of that together,

268
00:11:12,030 --> 00:11:14,800
I can say that the elastic
buckling stress is going

269
00:11:14,800 --> 00:11:17,420
to be some constant-- and
I think we're up to C4

270
00:11:17,420 --> 00:11:21,650
now-- times the Young's
modulus of the solid times

271
00:11:21,650 --> 00:11:26,130
the relative density
of the foam squared.

272
00:11:26,130 --> 00:11:33,270
So that's our equation for
the elastic buckling stress.

273
00:11:33,270 --> 00:11:35,900
And if you compare
this with data,

274
00:11:35,900 --> 00:11:39,050
you can make an
estimate of what C4 is.

275
00:11:39,050 --> 00:11:45,190
And we find that
C4 is about 0.05.

276
00:11:45,190 --> 00:11:47,130
And you can also
say that 0.05 really

277
00:11:47,130 --> 00:11:50,220
corresponds to the strain at
which the buckling occurs.

278
00:11:50,220 --> 00:11:53,470
Because the Young's modulus
goes as the constants 1 times

279
00:11:53,470 --> 00:11:55,310
Es times the relative
density squared.

280
00:11:55,310 --> 00:11:58,030
So the strain's just going to
be the stress over the modulus.

281
00:11:58,030 --> 00:11:59,615
So that does correspond
to the strain.

282
00:12:16,820 --> 00:12:19,650
So that's saying that
buckling compressive stress

283
00:12:19,650 --> 00:12:21,145
occurs at a strain of about 5%.

284
00:12:25,540 --> 00:12:28,470
So that's open cells.

285
00:12:28,470 --> 00:12:31,670
And then if we look
at closed cells,

286
00:12:31,670 --> 00:12:33,829
if you recall when we
looked at the moduli

287
00:12:33,829 --> 00:12:35,370
we looked at a couple
of extra terms.

288
00:12:35,370 --> 00:12:38,609
One was associated with face
stretching for the modulus.

289
00:12:38,609 --> 00:12:40,650
And the other was associated
with the compression

290
00:12:40,650 --> 00:12:41,920
of the gas.

291
00:12:41,920 --> 00:12:43,834
For the buckling, the
faces don't really

292
00:12:43,834 --> 00:12:46,250
contribute that much, because
typically the faces are very

293
00:12:46,250 --> 00:12:48,240
thin relative to the struts.

294
00:12:48,240 --> 00:12:49,850
And because they're
so thin, they

295
00:12:49,850 --> 00:12:52,730
buckle at a much lower load, and
they don't contribute too much.

296
00:12:52,730 --> 00:12:55,450
So we're not going to worry
about that contribution.

297
00:12:55,450 --> 00:12:58,360
So I'm just going to say that
the thickness of the face

298
00:12:58,360 --> 00:13:01,960
is often small
compared to the edges.

299
00:13:09,200 --> 00:13:18,060
And that really is from the
surface tension in processing

300
00:13:18,060 --> 00:13:20,525
that draws material away from
the face and into the edges.

301
00:13:45,240 --> 00:13:49,550
There can be some contribution
from the internal pressure.

302
00:13:49,550 --> 00:13:52,220
So if the internal
pressure is greater

303
00:13:52,220 --> 00:13:55,350
than atmospheric pressure,
then the cell walls

304
00:13:55,350 --> 00:13:57,870
are pre-tensioned, and you'd
have to account for that.

305
00:14:07,070 --> 00:14:10,310
So the buckling would have to
overcome that pressure as well.

306
00:14:21,490 --> 00:14:24,370
So then you would have
the buckling stress

307
00:14:24,370 --> 00:14:27,920
would just be what we
have up there-- C4 times

308
00:14:27,920 --> 00:14:31,560
Es times the relative
density squared.

309
00:14:31,560 --> 00:14:38,070
And then we just add on that
factor P0 minus P atmospheric.

310
00:14:38,070 --> 00:14:40,336
The thing with the gas
which tends to affect more

311
00:14:40,336 --> 00:14:41,710
than the buckling
stress, though,

312
00:14:41,710 --> 00:14:43,860
is the post-collapse behavior.

313
00:14:43,860 --> 00:14:46,830
So let me just show you
a couple of things here.

314
00:14:46,830 --> 00:14:50,290
So here's some data for the
elastic collapse stress.

315
00:14:50,290 --> 00:14:52,600
And you can see on
the y-axis, we've

316
00:14:52,600 --> 00:14:56,460
got the stress normalized by the
Young's modulus of the solid.

317
00:14:56,460 --> 00:15:00,140
And on the x-axis, we've
got the relative density.

318
00:15:00,140 --> 00:15:04,660
And that solid line there--
sort of solid, dark line--

319
00:15:04,660 --> 00:15:07,610
is that equation there, which
is the same as this one up here.

320
00:15:07,610 --> 00:15:11,237
And you can see the data
lie fairly close to that.

321
00:15:11,237 --> 00:15:13,320
But what's interesting is
if you look at the-- why

322
00:15:13,320 --> 00:15:14,153
is this not working?

323
00:15:16,832 --> 00:15:19,610
Maybe my batteries finally died.

324
00:15:19,610 --> 00:15:23,600
If we look at the
post-collapse behavior,

325
00:15:23,600 --> 00:15:27,450
you can see if these are
the stress-strain curves,

326
00:15:27,450 --> 00:15:29,110
they're not flat here.

327
00:15:29,110 --> 00:15:31,920
They have some rise to them.

328
00:15:31,920 --> 00:15:33,692
And this is a closed-cell foam.

329
00:15:33,692 --> 00:15:35,400
And you can imagine
as you're compressing

330
00:15:35,400 --> 00:15:37,910
the closed-cell
foam, you're reducing

331
00:15:37,910 --> 00:15:39,200
the volume of the cell.

332
00:15:39,200 --> 00:15:41,408
And as you doing that, you're
increasing the pressure

333
00:15:41,408 --> 00:15:43,060
inside the cell from the gas.

334
00:15:43,060 --> 00:15:44,880
And you can calculate
what that is.

335
00:15:44,880 --> 00:15:46,540
And I'll do that in a second.

336
00:15:46,540 --> 00:15:49,810
And if you subtract off that
gas pressure contribution,

337
00:15:49,810 --> 00:15:52,090
that works out to
this line here.

338
00:15:52,090 --> 00:15:56,320
Then these lines will
be more flat, like this.

339
00:15:56,320 --> 00:15:58,270
And we already really
pretty much worked

340
00:15:58,270 --> 00:15:59,521
out that gas contribution.

341
00:16:03,200 --> 00:16:13,360
So I'll just say for the
post-collapse behavior,

342
00:16:13,360 --> 00:16:15,900
the stress rises due
to the gas compression.

343
00:16:27,120 --> 00:16:29,201
And that's as long as
the faces don't rupture.

344
00:16:33,252 --> 00:16:34,710
So if you have an
elastomeric foam,

345
00:16:34,710 --> 00:16:35,918
typically they don't rupture.

346
00:16:40,940 --> 00:16:43,760
And what we had
worked out before

347
00:16:43,760 --> 00:16:46,185
was that that
pressure-- we called

348
00:16:46,185 --> 00:16:51,360
it P prime-- it was P0
minus P atmospheric-- that

349
00:16:51,360 --> 00:17:00,340
was equal to P0 times
the amount of strain,

350
00:17:00,340 --> 00:17:06,609
epsilon, times 1 minus 2
times the Poisson's ratio

351
00:17:06,609 --> 00:17:12,795
divided by 1 minus
epsilon times 1 minus 2 nu

352
00:17:12,795 --> 00:17:13,920
minus the relative density.

353
00:17:21,140 --> 00:17:24,530
And once you get to
the buckling stress,

354
00:17:24,530 --> 00:17:26,573
then the Poisson's
ratio becomes 0.

355
00:17:36,770 --> 00:17:39,270
So if you take a foam-- so I
brought a little foam in so you

356
00:17:39,270 --> 00:17:41,580
can play around with this one--
so if you take a foam like this

357
00:17:41,580 --> 00:17:44,090
and you compress it, once
you've buckled it like this,

358
00:17:44,090 --> 00:17:45,632
it's not getting
any wider this way.

359
00:17:45,632 --> 00:17:47,340
And part of the reason
for that is you've

360
00:17:47,340 --> 00:17:48,540
got all these pores in here.

361
00:17:48,540 --> 00:17:50,660
And the cells just
collapse into the pores.

362
00:17:50,660 --> 00:17:52,755
They don't really need
to move out sideways.

363
00:17:52,755 --> 00:17:54,380
So you can smush that
yourself, and try

364
00:17:54,380 --> 00:17:57,480
to convince yourself that the
Poisson's ratio is just 0.

365
00:17:57,480 --> 00:17:58,770
Yes, Matt.

366
00:17:58,770 --> 00:18:05,405
AUDIENCE: [INAUDIBLE] I guess
I want to measure [INAUDIBLE]

367
00:18:05,405 --> 00:18:06,787
the gas contribution?

368
00:18:06,787 --> 00:18:08,620
LORNA GIBSON: Yes, so
there is a strain rate

369
00:18:08,620 --> 00:18:09,940
effect with these things.

370
00:18:09,940 --> 00:18:11,648
But I wasn't going to
get into that here.

371
00:18:11,648 --> 00:18:13,900
If you look in the book,
it's described in the book.

372
00:18:13,900 --> 00:18:15,320
So I think there's two things.

373
00:18:15,320 --> 00:18:18,537
One is that the solid itself
can have a rate dependency.

374
00:18:18,537 --> 00:18:20,370
And then there could
be something connected.

375
00:18:20,370 --> 00:18:21,570
AUDIENCE: [INAUDIBLE].

376
00:18:21,570 --> 00:18:24,194
LORNA GIBSON: Yeah, I mean, I'm
not going to go into that here.

377
00:18:24,194 --> 00:18:27,720
But one could look at that.

378
00:18:27,720 --> 00:18:29,790
So let me just write
down one more thing here,

379
00:18:29,790 --> 00:18:32,180
because if we let nu be
0, then this thing here

380
00:18:32,180 --> 00:18:33,035
becomes simpler.

381
00:18:50,180 --> 00:18:53,330
So we could say the
stress post collapse

382
00:18:53,330 --> 00:19:03,190
as a function of strain would
be our buckling stress and then

383
00:19:03,190 --> 00:19:04,566
plus this factor here.

384
00:19:27,970 --> 00:19:32,350
So that curve on the
bottom over here--

385
00:19:32,350 --> 00:19:34,350
if this is the stress-strain
curve-- this little

386
00:19:34,350 --> 00:19:37,210
dashed line here-- that's
the gas contribution.

387
00:19:37,210 --> 00:19:39,257
And that is this term here.

388
00:19:39,257 --> 00:19:41,340
So you can kind of see how
the shape of the curves

389
00:19:41,340 --> 00:19:43,035
reflects that gas contribution.

390
00:19:43,035 --> 00:19:44,410
And when you
subtract it out, you

391
00:19:44,410 --> 00:19:47,245
get pretty much a horizontal
plateau over here.

392
00:19:51,280 --> 00:19:53,230
Are we happy?

393
00:19:53,230 --> 00:19:54,230
Yeah?

394
00:19:54,230 --> 00:19:55,870
AUDIENCE: [INAUDIBLE]?

395
00:19:55,870 --> 00:19:57,620
LORNA GIBSON: This is
for the closed cell.

396
00:19:57,620 --> 00:19:59,120
Because the closed
cell are the ones

397
00:19:59,120 --> 00:20:01,118
that are going to
have the gas pressure.

398
00:20:01,118 --> 00:20:03,576
If it's open cells, the gas
can just move out of the cells.

399
00:20:03,576 --> 00:20:04,550
AUDIENCE: [INAUDIBLE]?

400
00:20:04,550 --> 00:20:07,059
LORNA GIBSON: Oh, sorry,
that was to show you

401
00:20:07,059 --> 00:20:08,350
that the Poisson's ratio was 0.

402
00:20:08,350 --> 00:20:09,725
And that's true
for both of them.

403
00:20:33,320 --> 00:20:36,250
So then we can look at the
plastic collapse stress.

404
00:20:36,250 --> 00:20:38,710
Say we had a metal foam.

405
00:20:38,710 --> 00:20:40,800
And we do a calculation
a little bit

406
00:20:40,800 --> 00:20:43,080
like the one we did for
the honeycombs, too.

407
00:20:43,080 --> 00:20:46,130
So we say the failure occurs
when the applied moment equals

408
00:20:46,130 --> 00:20:47,210
the plastic moment.

409
00:20:55,630 --> 00:20:57,950
And the applied
moment is proportional

410
00:20:57,950 --> 00:21:00,736
to the applied stress
times the length cubed.

411
00:21:04,640 --> 00:21:06,940
So I'm going to call that
applied stress-- our strength

412
00:21:06,940 --> 00:21:09,920
sigma star plastic
times the length cubed.

413
00:21:09,920 --> 00:21:13,760
So if you think of, say, the
little schematic up here,

414
00:21:13,760 --> 00:21:16,230
the force is going to
go with stress times

415
00:21:16,230 --> 00:21:17,439
the length squared.

416
00:21:17,439 --> 00:21:19,480
And the moment's going to
force times the length.

417
00:21:19,480 --> 00:21:26,100
So it's the stress
times the length cubed.

418
00:21:26,100 --> 00:21:33,010
And then the plastic moment
goes as the yield strength

419
00:21:33,010 --> 00:21:34,300
times the thickness cubed.

420
00:21:37,410 --> 00:21:41,950
And then if I just
combine those,

421
00:21:41,950 --> 00:21:45,760
I get that the plastic
collapse stress in compression

422
00:21:45,760 --> 00:21:49,550
is another constant-- I'm going
to call it C5-- times the yield

423
00:21:49,550 --> 00:21:55,921
strength times the relative
density to the 3/2 power.

424
00:22:00,680 --> 00:22:03,610
And if we look at data,
we find that the constant

425
00:22:03,610 --> 00:22:11,080
is about equal to 0.3.

426
00:22:11,080 --> 00:22:16,290
And if I go to the
next slide, here's

427
00:22:16,290 --> 00:22:22,480
a plot of the yield strength or
the plastic collapse strength

428
00:22:22,480 --> 00:22:24,840
of the foam divided by the
yield strength of the solid,

429
00:22:24,840 --> 00:22:26,900
plotted against the
relative density.

430
00:22:26,900 --> 00:22:30,410
And that dark, bold line
is this equation here.

431
00:22:30,410 --> 00:22:33,187
And you can see the data lie
pretty well on that line.

432
00:22:33,187 --> 00:22:35,520
There's one data set that's
a little bit above the line.

433
00:22:35,520 --> 00:22:38,470
But you can see the slope
of that data set is still

434
00:22:38,470 --> 00:22:39,116
about 3/2.

435
00:23:21,560 --> 00:23:23,240
OK, and the same as
in the honeycombs,

436
00:23:23,240 --> 00:23:25,640
we could say that we
can get elastic collapse

437
00:23:25,640 --> 00:23:28,070
before the plastic collapse
if we were at a low density.

438
00:23:28,070 --> 00:23:30,100
You can get the same
thing in the foams.

439
00:23:30,100 --> 00:23:33,510
And you calculate out what the
critical relative density is

440
00:23:33,510 --> 00:23:36,280
for that the same kind of way.

441
00:23:36,280 --> 00:23:42,690
So we can say we can get
elastic collapse precedes

442
00:23:42,690 --> 00:23:51,450
the plastic collapse
if the elastic buckling

443
00:23:51,450 --> 00:23:57,190
stress is less than the
plastic collapse stress.

444
00:23:57,190 --> 00:24:02,772
So all we do is make those
two things equal to figure out

445
00:24:02,772 --> 00:24:04,230
the critical relative
density where

446
00:24:04,230 --> 00:24:06,104
you get the transition
from one to the other.

447
00:24:23,690 --> 00:24:28,000
So the relative density has to
be less than 36 times the yield

448
00:24:28,000 --> 00:24:30,910
strength of the solid
over the Young's modulus

449
00:24:30,910 --> 00:24:38,070
of the solid squared in order
to get buckling before yielding.

450
00:24:38,070 --> 00:24:39,570
And let's see, where
can I put that?

451
00:24:44,410 --> 00:24:47,390
So for rigid
polymers, that ratio

452
00:24:47,390 --> 00:24:51,140
of the strength of the solid
over the modulus of the solid

453
00:24:51,140 --> 00:24:52,528
is about one over 30.

454
00:24:55,040 --> 00:25:00,420
And so the critical relative
density for the transition

455
00:25:00,420 --> 00:25:03,130
is about 0.04.

456
00:25:03,130 --> 00:25:05,130
So you'd have to have a
pretty low-density foam,

457
00:25:05,130 --> 00:25:06,850
but it's possible.

458
00:25:06,850 --> 00:25:18,300
And for metals, that
ratio is about 1/1,000.

459
00:25:18,300 --> 00:25:23,960
And then the critical transition
density is less than 10

460
00:25:23,960 --> 00:25:25,437
to the minus 5.

461
00:25:25,437 --> 00:25:27,645
So essentially, it never
happens for the metal foams.

462
00:25:47,250 --> 00:25:49,100
And then for the
closed-cell foams,

463
00:25:49,100 --> 00:25:51,860
we could include the
terms for face stretching

464
00:25:51,860 --> 00:25:53,080
and for the gas.

465
00:25:53,080 --> 00:25:56,690
But in practice, the faces don't
really contribute very much.

466
00:25:56,690 --> 00:25:59,900
And typically for foams
like say metal foams

467
00:25:59,900 --> 00:26:03,616
or a rigid polymer that had a
yield point, the faces rupture.

468
00:26:03,616 --> 00:26:05,240
And then if the faces
rupture, then you

469
00:26:05,240 --> 00:26:07,986
don't get the gas
compression term, either.

470
00:26:07,986 --> 00:26:09,610
So I'll just write
the full thing down.

471
00:26:09,610 --> 00:26:12,475
But typically, you
don't need to use it.

472
00:26:59,600 --> 00:27:02,110
So the first term would
be from the edges bending.

473
00:27:06,439 --> 00:27:08,730
And the second term would be
from the faces stretching.

474
00:27:13,430 --> 00:27:16,160
And this would be from the gas.

475
00:27:16,160 --> 00:27:18,410
But in practice, the first
term is really the only one

476
00:27:18,410 --> 00:27:20,120
that is significant.

477
00:27:52,210 --> 00:27:54,699
So for closed-cell
foam, this equation

478
00:27:54,699 --> 00:27:56,240
works pretty well,
too-- the same one

479
00:27:56,240 --> 00:27:57,537
as for the open-cell foams.

480
00:28:49,850 --> 00:28:53,850
OK, so if we had, say, a
ceramic foam that was brittle,

481
00:28:53,850 --> 00:28:55,790
there'd be a brittle
crushing strength.

482
00:28:55,790 --> 00:28:57,790
And then we get failure
when the applied moment

483
00:28:57,790 --> 00:29:01,630
M is equal to the
fracture moment Mf.

484
00:29:01,630 --> 00:29:05,810
And this works very similar
to the plastic yield strength.

485
00:29:05,810 --> 00:29:08,090
So we find the
applied moment goes

486
00:29:08,090 --> 00:29:13,310
as the global stress
times the length cubed.

487
00:29:13,310 --> 00:29:18,230
And the fracture moment goes
into the cell wall strength

488
00:29:18,230 --> 00:29:20,946
times the cell wall
thickness cubed.

489
00:29:24,030 --> 00:29:26,460
So the brittle
crushing strength goes

490
00:29:26,460 --> 00:29:30,580
as another constant-- let's
call it C6-- times the wall

491
00:29:30,580 --> 00:29:35,580
strength times the relative
density to the 3/2 again.

492
00:29:39,650 --> 00:29:47,330
And C6 is about equal to 0.2.

493
00:29:47,330 --> 00:29:49,620
And typically, ceramic
foams have open cells.

494
00:29:49,620 --> 00:29:53,920
So I'm just going to leave it at
the open-celled formula there.

495
00:29:53,920 --> 00:29:56,850
So there's one last thing
for the compressive behavior,

496
00:29:56,850 --> 00:29:59,210
and that's the
densification strain.

497
00:29:59,210 --> 00:30:01,080
And we just have an
empirical relationship

498
00:30:01,080 --> 00:30:02,740
for the densification strain.

499
00:30:18,610 --> 00:30:22,450
So if you compress the foam and
you get to very large strains,

500
00:30:22,450 --> 00:30:24,770
then the cell walls
start to touch,

501
00:30:24,770 --> 00:30:26,825
and the stress starts
to rise steeply.

502
00:30:26,825 --> 00:30:28,700
And there's some strain
at which that occurs.

503
00:30:28,700 --> 00:30:30,970
And we call that the
densification strain.

504
00:30:30,970 --> 00:30:32,970
And in the limit, the
modulus at that point

505
00:30:32,970 --> 00:30:34,830
would go to the
modulus of the solid.

506
00:30:34,830 --> 00:30:37,150
If you could completely
squeeze all the pores out,

507
00:30:37,150 --> 00:30:41,269
the stiffness of that would go
to the modulus of the solid.

508
00:30:41,269 --> 00:30:43,810
And you might expect that that
densification strain is just 1

509
00:30:43,810 --> 00:30:45,930
minus the relative
density, but it actually

510
00:30:45,930 --> 00:30:47,680
occurs at a slightly
smaller strain.

511
00:30:56,200 --> 00:30:59,110
So in a large compressive
stress, or strain,

512
00:30:59,110 --> 00:31:02,205
I guess we could say,
cell walls touch,

513
00:31:02,205 --> 00:31:03,830
and we start to get
this densification.

514
00:31:32,030 --> 00:31:33,500
So the modulus in
the limit would

515
00:31:33,500 --> 00:31:36,900
go to the modulus of the solid.

516
00:31:36,900 --> 00:31:41,800
And you might expect
that the densification

517
00:31:41,800 --> 00:31:47,570
strain was just equal to 1
minus the relative density.

518
00:31:47,570 --> 00:31:50,100
But it occurs at a little
bit less than that.

519
00:31:50,100 --> 00:31:55,780
So empirically, we find that
it's just 1 minus 1.4 times

520
00:31:55,780 --> 00:31:58,850
the relative density.

521
00:31:58,850 --> 00:32:00,640
And then I have this
plot here, which

522
00:32:00,640 --> 00:32:03,565
is really just fitting a line
to that data for densification

523
00:32:03,565 --> 00:32:04,065
strain.

524
00:32:09,300 --> 00:32:12,060
So those equations describe
the compressive stress

525
00:32:12,060 --> 00:32:13,940
or the compressive
behavior of the foam.

526
00:32:13,940 --> 00:32:17,670
So we've got the moduli, we've
got the three compressive

527
00:32:17,670 --> 00:32:20,210
strengths, and we've got
the densification strain.

528
00:32:22,840 --> 00:32:25,420
So what we're going to do
later on in the course is

529
00:32:25,420 --> 00:32:28,010
we'll use those
models to look at how

530
00:32:28,010 --> 00:32:30,220
we can use foams and
things like sandwich panels

531
00:32:30,220 --> 00:32:32,102
and looking at
energy absorption.

532
00:32:32,102 --> 00:32:34,060
And we'll also look at
these equations in terms

533
00:32:34,060 --> 00:32:37,080
of some biomedical materials--
looking at trabecular bone,

534
00:32:37,080 --> 00:32:39,979
and looking at tissue
engineering scaffolds.

535
00:32:39,979 --> 00:32:42,020
So there's one last property
I wanted to go over,

536
00:32:42,020 --> 00:32:43,610
and that's the
fracture toughness.

537
00:32:43,610 --> 00:32:45,360
So if we were pulling
the foam in tension,

538
00:32:45,360 --> 00:32:46,894
and we had a crack
in the foam, we'd

539
00:32:46,894 --> 00:32:48,810
want to know what the
fracture toughness would

540
00:32:48,810 --> 00:32:50,170
be for a brittle foam.

541
00:32:50,170 --> 00:32:52,550
And this follows the
same sort of argument

542
00:32:52,550 --> 00:32:53,752
as we had for the honeycomb.

543
00:32:53,752 --> 00:32:55,460
So all of these
equations really are just

544
00:32:55,460 --> 00:32:58,411
following the same
kinds of arguments.

545
00:32:58,411 --> 00:33:00,410
But you can kind of see
how having the honeycomb

546
00:33:00,410 --> 00:33:05,810
calculations makes it
easier to do the foam ones.

547
00:33:05,810 --> 00:33:09,340
So we'll do the fracture
toughness calculation,

548
00:33:09,340 --> 00:33:12,970
and then I want to talk a little
bit about material selection

549
00:33:12,970 --> 00:33:15,830
and selection charts for foams.

550
00:33:15,830 --> 00:33:17,210
So that's less equation-y.

551
00:33:47,070 --> 00:33:49,700
OK, and we're just going
to look at open cells here.

552
00:33:52,640 --> 00:33:54,660
So imagine we have a
crack of length 2a.

553
00:33:59,860 --> 00:34:02,890
And we have some
remote stress applied,

554
00:34:02,890 --> 00:34:09,510
so remote tensile stress, so
I'm going to call that sigma

555
00:34:09,510 --> 00:34:11,830
infinity-- the far-away stress.

556
00:34:11,830 --> 00:34:16,090
And then we have a local
stress on the cell walls.

557
00:34:16,090 --> 00:34:19,730
I'm going to call
that signal local.

558
00:34:19,730 --> 00:34:21,552
So I have a little
schematic that kind of

559
00:34:21,552 --> 00:34:22,510
shows what we're doing.

560
00:34:22,510 --> 00:34:24,260
So we're pulling on it.

561
00:34:24,260 --> 00:34:25,139
There's some crack.

562
00:34:25,139 --> 00:34:28,580
The crack length is large
compared to the cell size.

563
00:34:28,580 --> 00:34:32,389
And we want to know what
the fracture toughness is.

564
00:34:32,389 --> 00:34:36,110
So we can say from fracture
mechanics the local stress is

565
00:34:36,110 --> 00:34:38,540
going to be equal to
some constant times

566
00:34:38,540 --> 00:34:43,389
the faraway stress times
the square root of pi a

567
00:34:43,389 --> 00:34:46,380
over the square root of 2 pi r.

568
00:34:46,380 --> 00:34:49,329
And that's at a distance r
from the head of the crack tip.

569
00:35:03,130 --> 00:35:05,190
And if we look at our
little schematic here,

570
00:35:05,190 --> 00:35:08,580
we could say it's hard to say
exactly where the crack tip is,

571
00:35:08,580 --> 00:35:10,100
but it would be
somewhere in here.

572
00:35:10,100 --> 00:35:12,160
And we'd say this
next unbroken cell

573
00:35:12,160 --> 00:35:14,810
wall is a distance r
ahead of the crack tip.

574
00:35:14,810 --> 00:35:16,580
And that r is going
to be related to l.

575
00:35:16,580 --> 00:35:19,560
It's going to be
some function of l.

576
00:35:19,560 --> 00:35:34,000
So I can say the next unbroken
wall ahead of the crack tip

577
00:35:34,000 --> 00:35:39,280
at some distance r is
going to be related to l.

578
00:35:39,280 --> 00:35:45,950
And that's subject
to a force, which

579
00:35:45,950 --> 00:35:48,125
is going to be the local
stress times l squared.

580
00:35:53,240 --> 00:35:56,500
So that force is going to go as
local stress times l squared.

581
00:35:56,500 --> 00:35:59,140
And the local stress-- I can
substitute this thing here

582
00:35:59,140 --> 00:36:00,910
in-- that's going
to be proportional

583
00:36:00,910 --> 00:36:03,160
to the faraway stress.

584
00:36:03,160 --> 00:36:05,874
And I'm going to
get rid of the pi's.

585
00:36:05,874 --> 00:36:07,290
And I'm going to
substitute for r.

586
00:36:07,290 --> 00:36:08,690
I'm going to put in l.

587
00:36:08,690 --> 00:36:10,940
So it's going to be proportional
to the faraway stress

588
00:36:10,940 --> 00:36:15,907
times the root of a over l
and times l squared there.

589
00:36:20,862 --> 00:36:22,320
And then we're
going to say, again,

590
00:36:22,320 --> 00:36:24,430
the edges are going to fail
when the applied moment equals

591
00:36:24,430 --> 00:36:25,300
the fracture moment.

592
00:36:46,420 --> 00:36:48,590
And the fracture
moment is going to go

593
00:36:48,590 --> 00:36:53,640
as the modulus of rupture of
the cell walls times t cubed.

594
00:36:53,640 --> 00:36:57,480
And the applied moment is
going to go as f times l.

595
00:36:57,480 --> 00:36:59,900
And I've got f from
up there, so that

596
00:36:59,900 --> 00:37:03,910
goes as the faraway
stress, sigma infinite,

597
00:37:03,910 --> 00:37:06,200
times the root of a over l.

598
00:37:06,200 --> 00:37:08,340
And now I've got l
cubed, because there's

599
00:37:08,340 --> 00:37:10,740
an l squared there and
there's an l down here.

600
00:37:15,210 --> 00:37:17,960
And then if I just
equate those, then this

601
00:37:17,960 --> 00:37:23,110
is going to go as sigma fs
times t cubed, like that.

602
00:37:23,110 --> 00:37:26,780
So then I can say
the fracture strength

603
00:37:26,780 --> 00:37:29,090
is equal to my faraway stress.

604
00:37:29,090 --> 00:37:32,560
That's going to go as my
modulus of rupture times

605
00:37:32,560 --> 00:37:40,396
the root of l for a
times t over l cubed.

606
00:37:40,396 --> 00:37:41,770
And then my fracture
toughness is

607
00:37:41,770 --> 00:37:45,150
going to be this tensile
stress times the root of pi a.

608
00:37:55,360 --> 00:37:57,610
So there's going to be some
other constant here, which

609
00:37:57,610 --> 00:37:59,405
I'm going to call that C8.

610
00:37:59,405 --> 00:38:03,070
We've got the modulus
of rupture of the solid.

611
00:38:03,070 --> 00:38:05,030
I've got the square
root of l, and I'm

612
00:38:05,030 --> 00:38:07,317
going to multiply it by pi
so it's like other fraction

613
00:38:07,317 --> 00:38:08,525
mechanics kinds of equations.

614
00:38:13,054 --> 00:38:15,220
And then we multiply that
times the relative density

615
00:38:15,220 --> 00:38:16,484
to the 3/2 power.

616
00:38:19,210 --> 00:38:23,320
And here, if we look at data,
we find that that constant

617
00:38:23,320 --> 00:38:28,840
is about equal to 0.65.

618
00:38:28,840 --> 00:38:30,760
And here's another
one of these plots.

619
00:38:30,760 --> 00:38:34,240
So here I've normalized the
fracture toughness of the foams

620
00:38:34,240 --> 00:38:35,920
by the modulus of
rupture of the cell

621
00:38:35,920 --> 00:38:37,770
walls times the root of pi l.

622
00:38:37,770 --> 00:38:40,334
So I've taken the cell
size into account here,

623
00:38:40,334 --> 00:38:42,250
and I've plotted against
the relative density.

624
00:38:42,250 --> 00:38:43,720
And that equation
there is the same

625
00:38:43,720 --> 00:38:45,900
as this equation I've
got down on the board.

626
00:38:48,610 --> 00:38:50,360
And this is the only
one of the properties

627
00:38:50,360 --> 00:38:53,570
that we've looked at that
depends on the cell size.

628
00:38:53,570 --> 00:38:55,350
There's a cell size
dependence here.

629
00:39:07,020 --> 00:39:11,755
All right, so I think that's
all the modeling of the foams.

630
00:39:17,570 --> 00:39:19,910
Are we good?

631
00:39:19,910 --> 00:39:21,240
I gave you a lot of equations.

632
00:39:21,240 --> 00:39:21,850
We're good?

633
00:39:21,850 --> 00:39:22,350
All right.

634
00:40:13,950 --> 00:40:16,410
So I want to talk about
how we might design foams

635
00:40:16,410 --> 00:40:17,625
to improve their properties.

636
00:40:17,625 --> 00:40:19,000
And then I want
to talk about how

637
00:40:19,000 --> 00:40:21,890
we might select foams
for certain applications

638
00:40:21,890 --> 00:40:24,100
and look at selection charts.

639
00:40:24,100 --> 00:40:26,350
So when we've been talking
about the foams, especially

640
00:40:26,350 --> 00:40:28,080
the open-cell foams,
we've been saying

641
00:40:28,080 --> 00:40:32,990
their deformation is largely
by bending of the cell edges.

642
00:40:32,990 --> 00:40:34,720
And if we could do
something to increase

643
00:40:34,720 --> 00:40:37,710
the stiffness of the edges
or the strength of the edges,

644
00:40:37,710 --> 00:40:41,180
then that would increase the
overall properties of the foam.

645
00:40:41,180 --> 00:40:44,150
And there's a couple of ways
to think about doing that.

646
00:40:44,150 --> 00:40:46,600
So the foam
properties-- if the foam

647
00:40:46,600 --> 00:40:48,270
is controlled by
bending of the edges,

648
00:40:48,270 --> 00:40:50,190
and the edges have
some flexural rigidity,

649
00:40:50,190 --> 00:40:53,790
EI, if we could increase
that EI of the edges,

650
00:40:53,790 --> 00:40:56,730
we would increase the
properties of the foam.

651
00:40:56,730 --> 00:40:59,980
And one way to do that is
by making the edges hollow.

652
00:40:59,980 --> 00:41:03,540
So if we had hollow
edges, and you had a tube,

653
00:41:03,540 --> 00:41:05,270
then that would increase the EI.

654
00:41:05,270 --> 00:41:08,870
And we can work out how much
it's going to increase them.

655
00:41:08,870 --> 00:41:16,520
And I have a little example
here of-- a natural example

656
00:41:16,520 --> 00:41:20,150
of hollow foam struts.

657
00:41:23,010 --> 00:41:24,677
So this is a grass.

658
00:41:24,677 --> 00:41:26,260
I don't know what
kind of grass it is.

659
00:41:26,260 --> 00:41:27,379
I just saw this grass.

660
00:41:27,379 --> 00:41:28,920
And we picked some
different grasses,

661
00:41:28,920 --> 00:41:31,440
and we took some SEM pictures.

662
00:41:31,440 --> 00:41:35,270
And it has a really kind of
common structure for grasses.

663
00:41:35,270 --> 00:41:37,730
It's very common for
grass stems to have

664
00:41:37,730 --> 00:41:42,130
sort of a solid outer part and
then a foam-like inner part.

665
00:41:42,130 --> 00:41:44,310
It's so common that
botanists have a name for it.

666
00:41:44,310 --> 00:41:46,060
They call it the
core-rind structure.

667
00:41:46,060 --> 00:41:48,440
And if you take
one of these grass

668
00:41:48,440 --> 00:41:51,780
stems, and you look at the sort
of foamy bit in the middle,

669
00:41:51,780 --> 00:41:54,426
and you do a SEM
picture of that,

670
00:41:54,426 --> 00:41:56,550
you can see that the little
cell walls are actually

671
00:41:56,550 --> 00:41:57,740
little hollow tubes.

672
00:41:57,740 --> 00:42:00,930
So one of these things--
it's a little hollow tube.

673
00:42:00,930 --> 00:42:02,610
So what I wanted
to do is work out

674
00:42:02,610 --> 00:42:05,374
how much the modulus
of the foam would

675
00:42:05,374 --> 00:42:07,040
increase if you could
make all the edges

676
00:42:07,040 --> 00:42:09,406
into little hollow tubes.

677
00:42:09,406 --> 00:42:10,780
So we're going to
start by saying

678
00:42:10,780 --> 00:42:13,825
the foam behavior is
dominated by cell bending,

679
00:42:13,825 --> 00:42:14,860
so edge bending.

680
00:42:32,170 --> 00:42:36,120
And the foam properties can
be increased by increasing

681
00:42:36,120 --> 00:42:37,620
the EI of the cell wall.

682
00:42:52,256 --> 00:42:54,130
So there's a couple of
ways we could do that.

683
00:42:54,130 --> 00:42:57,242
So the first one is
looking at hollow walls.

684
00:43:03,940 --> 00:43:06,280
So imagine I have a
thin-walled tube-- just

685
00:43:06,280 --> 00:43:07,867
a circular, thin-walled tube.

686
00:43:15,120 --> 00:43:17,130
There's my little wall there.

687
00:43:17,130 --> 00:43:23,460
It has some radius little
r, and a wall thickness t.

688
00:43:23,460 --> 00:43:25,510
And then imagine I have
the same amount of mass,

689
00:43:25,510 --> 00:43:29,340
but now I have a solid
circular section.

690
00:43:29,340 --> 00:43:35,520
And I'm going to say the
radius of that is big R.

691
00:43:35,520 --> 00:43:42,590
So for our thin-walled
tube, the moment of inertia

692
00:43:42,590 --> 00:43:48,540
is pi r cubed times the
thickness, t, if it's thin.

693
00:43:48,540 --> 00:43:57,554
And for our solid
circular section,

694
00:43:57,554 --> 00:44:04,180
I is going to be pi big
R to the 4th over 4.

695
00:44:04,180 --> 00:44:06,600
And if I say I want to set
this up so that the masses are

696
00:44:06,600 --> 00:44:09,740
equal, then the areas
of the cross-sections

697
00:44:09,740 --> 00:44:12,541
have to be equal-- say it's
from the same material.

698
00:44:15,212 --> 00:44:16,670
So the masses are
going to be equal

699
00:44:16,670 --> 00:44:24,280
if pi R squared is
equal to 2 pi r t.

700
00:44:24,280 --> 00:44:26,610
So I'm going to
solve here for R.

701
00:44:26,610 --> 00:44:28,780
So the pi's are
going to cancel out.

702
00:44:28,780 --> 00:44:33,210
So the masses are equal if R is
equal to the square root of 2

703
00:44:33,210 --> 00:44:34,637
times r times t.

704
00:44:38,284 --> 00:44:39,700
And then what we're
going to do is

705
00:44:39,700 --> 00:44:42,080
see how the big is
the moment of inertia

706
00:44:42,080 --> 00:44:45,910
of the tube relative
to the solid.

707
00:44:45,910 --> 00:44:50,040
And the tube is pi
little r cubed t.

708
00:44:50,040 --> 00:44:56,355
And the solid was pi R
to the 4th, divided by 4.

709
00:44:56,355 --> 00:44:57,730
And I'm going to
get rid of the R

710
00:44:57,730 --> 00:44:59,811
here, and get rid
of the pi's there.

711
00:45:03,870 --> 00:45:10,270
So R to the 4th is going
to be 4r squared t squared.

712
00:45:10,270 --> 00:45:11,860
So the 4s are going to go.

713
00:45:11,860 --> 00:45:16,850
And this boils down to r over t.

714
00:45:16,850 --> 00:45:21,340
So if I had a thin-walled
tube, the moment of inertia

715
00:45:21,340 --> 00:45:25,830
is going to be r over t bigger
than if I had the same mass

716
00:45:25,830 --> 00:45:27,472
in a solid circular section.

717
00:45:27,472 --> 00:45:28,930
So you can see for
the little plant

718
00:45:28,930 --> 00:45:30,460
here, by making a
thin-walled tube,

719
00:45:30,460 --> 00:45:32,310
you're increasing the
stiffness of the foam

720
00:45:32,310 --> 00:45:34,060
with the same
amount of material.

721
00:45:34,060 --> 00:45:37,120
That's the idea.

722
00:45:37,120 --> 00:45:39,887
And you can do a similar kind of
analysis for other properties.

723
00:45:56,360 --> 00:45:59,440
So that's if we
have hollow tubes.

724
00:45:59,440 --> 00:46:01,840
So another option is we
could have cell walls that

725
00:46:01,840 --> 00:46:03,390
are sandwich structures.

726
00:46:03,390 --> 00:46:05,440
So imagine if the
cell walls themselves

727
00:46:05,440 --> 00:46:07,120
were little, tiny
sandwich structures.

728
00:46:43,440 --> 00:46:48,400
So when you have a
sandwich beam, what

729
00:46:48,400 --> 00:46:50,960
you have is too
stiff, strong faces

730
00:46:50,960 --> 00:46:53,180
that are separated by
some sort of porous core,

731
00:46:53,180 --> 00:46:56,330
like a honeycomb or
a foam or balsa wood.

732
00:46:56,330 --> 00:46:59,220
And the idea with the
sandwich structure--

733
00:46:59,220 --> 00:47:06,080
if I draw a little sketch of
the sandwich, here's my faces.

734
00:47:06,080 --> 00:47:07,730
So imagine those are solid.

735
00:47:07,730 --> 00:47:09,800
So they might be
aluminum sheets,

736
00:47:09,800 --> 00:47:12,567
or they might be fiber
reinforced composites.

737
00:47:12,567 --> 00:47:14,400
And then we have some
sort of cellular thing

738
00:47:14,400 --> 00:47:19,080
here as the core.

739
00:47:19,080 --> 00:47:23,750
And the idea is, that's
analogous to an I-beam.

740
00:47:23,750 --> 00:47:28,690
So in the sandwich
beam, we have two,

741
00:47:28,690 --> 00:47:37,600
stiff, strong faces separated
by a lightweight core.

742
00:47:43,120 --> 00:47:48,770
So the core is typically
a honeycomb, or a foam,

743
00:47:48,770 --> 00:47:49,980
or balsa wood.

744
00:47:54,400 --> 00:47:56,320
And the idea is, you
increase the moment

745
00:47:56,320 --> 00:47:59,337
of inertia of the cross-section
with little increase in weight.

746
00:48:13,550 --> 00:48:16,550
And if you think of
an I-beam, an I-beam

747
00:48:16,550 --> 00:48:19,060
has a large moment of inertia,
because you're separating

748
00:48:19,060 --> 00:48:20,840
the flanges by the web.

749
00:48:20,840 --> 00:48:22,900
And the sandwich beam
works in the same way.

750
00:48:22,900 --> 00:48:25,359
You're separating the
faces by the core.

751
00:48:25,359 --> 00:48:26,900
But the core doesn't
weigh very much,

752
00:48:26,900 --> 00:48:29,410
because it's a cellular thing.

753
00:48:29,410 --> 00:48:33,780
So the faces of the sandwich are
like the flanges in the I-beam.

754
00:48:40,170 --> 00:48:41,810
And then the core
is like the web.

755
00:48:48,440 --> 00:48:51,465
So the idea is to make something
called a micro-sandwich foam.

756
00:48:57,530 --> 00:49:00,660
So what you want to do is make
the cell walls into sandwiches.

757
00:49:00,660 --> 00:49:03,760
And one way to do that is
to disperse a large volume

758
00:49:03,760 --> 00:49:06,680
fraction of thin-walled
spheres into the foam.

759
00:49:23,210 --> 00:49:27,130
And you have to get the
geometry right to make it work.

760
00:49:27,130 --> 00:49:32,070
So let me draw a little kind
of sketch here of how it works.

761
00:49:32,070 --> 00:49:33,818
So here's our
thin-walled spheres.

762
00:49:41,000 --> 00:49:44,880
And then you're going to
distribute those in a foam.

763
00:49:44,880 --> 00:49:47,135
Here's another sphere over here.

764
00:49:47,135 --> 00:49:48,445
The spheres are not perfect.

765
00:49:51,570 --> 00:49:53,195
Let's say there's
another one in here.

766
00:49:58,170 --> 00:50:00,934
And then the idea is this stuff
in here would be the foam.

767
00:50:05,680 --> 00:50:07,206
So these guys are
hollow spheres.

768
00:50:13,250 --> 00:50:17,100
And say the spheres
have a diameter D.

769
00:50:17,100 --> 00:50:21,570
And say they have a wall
thickness here of t.

770
00:50:21,570 --> 00:50:24,930
And say that the separation
of the spheres I'm

771
00:50:24,930 --> 00:50:27,290
going to call c.

772
00:50:27,290 --> 00:50:28,870
You can see that there.

773
00:50:28,870 --> 00:50:32,710
And then the cell size of
the foam I'm going to call e.

774
00:50:37,450 --> 00:50:41,060
So there's a bunch of parameters
you have to kind of play with

775
00:50:41,060 --> 00:50:42,044
to get this to work.

776
00:50:46,100 --> 00:50:50,080
So you have to have thin-walled
spheres so the faces are thin.

777
00:50:50,080 --> 00:50:52,640
The sandwich panels work
best when the faces are thin.

778
00:50:52,640 --> 00:50:56,570
So you need the thickness of the
sphere to be much less than D.

779
00:50:56,570 --> 00:50:59,770
You need the faces to be
stiff relative to the foam.

780
00:50:59,770 --> 00:51:02,800
So you need the modulus
of the sphere material

781
00:51:02,800 --> 00:51:05,110
to be greater than the
modulus of the foam.

782
00:51:07,810 --> 00:51:10,910
And you need the volume
fraction of the spheres

783
00:51:10,910 --> 00:51:13,550
to be relatively high to
get the spheres close enough

784
00:51:13,550 --> 00:51:15,777
together for this to work.

785
00:51:15,777 --> 00:51:17,360
So you want that
volume fraction to be

786
00:51:17,360 --> 00:51:20,690
something like 50% to 60%.

787
00:51:20,690 --> 00:51:23,850
And for the foam, you
need to have the foam cell

788
00:51:23,850 --> 00:51:27,075
size less than the separation
between the spheres.

789
00:51:27,075 --> 00:51:28,950
You need to have a number
of-- you can't just

790
00:51:28,950 --> 00:51:30,090
have one pore in here.

791
00:51:30,090 --> 00:51:31,660
That's not really like a foam.

792
00:51:31,660 --> 00:51:34,180
It won't behave like
a foam as a continuum.

793
00:51:34,180 --> 00:51:36,700
So you need to have a number
of different cell sizes

794
00:51:36,700 --> 00:51:40,010
in between each sphere.

795
00:51:40,010 --> 00:51:41,830
And so you need the
cell size of the foam

796
00:51:41,830 --> 00:51:44,670
to be a lot less than the
separation of the spheres

797
00:51:44,670 --> 00:51:45,810
there, c.

798
00:51:45,810 --> 00:51:47,950
But if you can
control this geometry,

799
00:51:47,950 --> 00:51:49,750
you can get the sandwich effect.

800
00:51:49,750 --> 00:51:51,992
And you can get improved
properties by doing that.

801
00:51:51,992 --> 00:51:53,450
So there's ways
you can play around

802
00:51:53,450 --> 00:51:58,297
with the structure of the foams
to improve their properties.

803
00:51:58,297 --> 00:51:59,880
So that was one thing
I wanted to say.

804
00:52:32,250 --> 00:52:34,730
Another way to
improve the properties

805
00:52:34,730 --> 00:52:37,570
of a foam-like
material is to use

806
00:52:37,570 --> 00:52:39,420
one of those lattice materials.

807
00:52:39,420 --> 00:52:41,570
So we've been talking
about ways to improve

808
00:52:41,570 --> 00:52:42,580
the bending stiffness.

809
00:52:42,580 --> 00:52:43,650
But if you could get
rid of the bending

810
00:52:43,650 --> 00:52:46,410
altogether and have axial
deformation in the cell walls,

811
00:52:46,410 --> 00:52:47,950
that would be much stiffer.

812
00:52:47,950 --> 00:52:49,330
And you can get
axial deformation

813
00:52:49,330 --> 00:52:52,520
by having those 3D
truss kind of materials.

814
00:52:52,520 --> 00:52:54,880
So I have a picture of this.

815
00:52:54,880 --> 00:52:57,430
There we go, so there's one
of those 3D truss materials.

816
00:52:57,430 --> 00:52:59,890
So another alternative is to
sort of get rid of the bending

817
00:52:59,890 --> 00:53:02,642
altogether, and to try to
make a truss-type material.

818
00:53:41,297 --> 00:53:42,880
So there's various
ways to make these.

819
00:53:42,880 --> 00:53:45,510
I think that we talked about
a few of them earlier on.

820
00:53:45,510 --> 00:53:49,640
And you can analyze them
as truss-type structures.

821
00:53:49,640 --> 00:53:51,380
And I can just
run through a sort

822
00:53:51,380 --> 00:53:54,300
of little dimensional
argument to get the modulus.

823
00:53:54,300 --> 00:53:59,410
So the modulus is going to go
as the stress over the strain.

824
00:53:59,410 --> 00:54:03,630
The stress is going to go as
a force over a length squared.

825
00:54:03,630 --> 00:54:06,350
The strain's going to go
as a deformation over l.

826
00:54:06,350 --> 00:54:09,370
So this is just like what
we had before for the foams.

827
00:54:09,370 --> 00:54:11,640
But in this case,
the deformation

828
00:54:11,640 --> 00:54:13,400
is going to go with
the force times

829
00:54:13,400 --> 00:54:18,210
the length over the area of
the cross-section divided

830
00:54:18,210 --> 00:54:22,840
by Es, because we're pulling
it or pushing it axially.

831
00:54:22,840 --> 00:54:30,640
So that goes as Fl
over t squared Es.

832
00:54:30,640 --> 00:54:32,740
And if I just put that
back in the equation

833
00:54:32,740 --> 00:54:37,180
here for the modulus, I get
that we've got F over l.

834
00:54:37,180 --> 00:54:43,910
And I've got delta here,
so that's F l t squared Es.

835
00:54:43,910 --> 00:54:45,930
And you just get
the modulus goes

836
00:54:45,930 --> 00:54:51,940
as the modulus of the solid
times t over l squared.

837
00:54:51,940 --> 00:54:57,020
And that goes as the
modulus of the solid times

838
00:54:57,020 --> 00:54:58,770
the relative density.

839
00:54:58,770 --> 00:55:00,920
So for the open-celled
foams, the modulus

840
00:55:00,920 --> 00:55:02,690
went as the relative
density squared.

841
00:55:02,690 --> 00:55:06,330
So if it was 10% solid,
the modulus would be 0.01.

842
00:55:06,330 --> 00:55:09,140
And this is saying if it's
10% solid, the modulus is 0.1.

843
00:55:09,140 --> 00:55:11,480
So it's much bigger.

844
00:55:11,480 --> 00:55:13,970
So this is all sort
of well and good.

845
00:55:13,970 --> 00:55:18,740
The only difficulty is that
when you look at the modulus,

846
00:55:18,740 --> 00:55:20,020
you can do reasonably well.

847
00:55:20,020 --> 00:55:21,820
But when you look at the
strength, some of the members

848
00:55:21,820 --> 00:55:23,570
are going to be
inevitably in compression.

849
00:55:23,570 --> 00:55:25,224
When you have these
truss materials,

850
00:55:25,224 --> 00:55:26,890
some members are going
to be in tension.

851
00:55:26,890 --> 00:55:28,723
Some members are going
to be in compression.

852
00:55:28,723 --> 00:55:30,790
And the compression
members tend to buckle.

853
00:55:30,790 --> 00:55:32,530
And once the compression
members buckle,

854
00:55:32,530 --> 00:55:35,260
then you're back to the same
kind of strength relationship

855
00:55:35,260 --> 00:55:36,690
that you have for the foam.

856
00:55:36,690 --> 00:55:39,280
So that's one of the
difficulties of this.

857
00:55:41,800 --> 00:55:48,450
So let me say that the
strength-- so if the strength

858
00:55:48,450 --> 00:55:50,770
was controlled by
uni-axial yield,

859
00:55:50,770 --> 00:55:52,934
it would go linearly
with relative density.

860
00:55:52,934 --> 00:55:55,100
But if it goes with buckling,
it goes as the square.

861
00:56:17,830 --> 00:56:20,678
So I'll just say the
compression members can buckle.

862
00:56:35,568 --> 00:56:40,680
And say you had a metal lattice.

863
00:56:40,680 --> 00:56:43,890
Then there's some interaction
between the plastic behavior

864
00:56:43,890 --> 00:56:44,670
and the buckling.

865
00:56:44,670 --> 00:56:47,620
And you use what's called
the tangent modulus instead

866
00:56:47,620 --> 00:56:49,680
of just the Young's modulus.

867
00:56:49,680 --> 00:56:52,040
And the tangent
modulus is lower.

868
00:56:52,040 --> 00:56:54,400
And there's also what's
called knock-down factors that

869
00:56:54,400 --> 00:56:55,433
can be large, too.

870
00:56:58,810 --> 00:57:01,183
So the knock-down
factor can be like 50%.

871
00:57:07,170 --> 00:57:09,790
So the measured strength
can be half of what

872
00:57:09,790 --> 00:57:12,820
you thought it was going to be.

873
00:57:12,820 --> 00:57:14,861
This should be a
squared over here.

874
00:57:14,861 --> 00:57:15,360
Sorry.

875
00:57:18,450 --> 00:57:21,600
So even though the stiffness
of these 3D trusses

876
00:57:21,600 --> 00:57:25,150
can be quite good, the strength
often isn't quite as good

877
00:57:25,150 --> 00:57:26,270
as one might hope.

878
00:57:26,270 --> 00:57:30,825
So that's one of the
issues with them.

879
00:57:30,825 --> 00:57:31,325
All right.

880
00:57:43,474 --> 00:57:45,140
So do you see the
idea, though, with all

881
00:57:45,140 --> 00:57:48,040
these different
micro structures,

882
00:57:48,040 --> 00:57:49,970
is that you can control
the structure in a way

883
00:57:49,970 --> 00:57:51,770
to try to increase
the bending stiffness

884
00:57:51,770 --> 00:57:53,228
or get rid of the
bending stiffness

885
00:57:53,228 --> 00:57:54,620
and increase the
axial stiffness?

886
00:57:54,620 --> 00:57:56,870
So there's things you can
do to play around with that.

887
00:58:20,110 --> 00:58:22,860
And I wanted to talk a bit
today about material selection

888
00:58:22,860 --> 00:58:23,880
charts for foams.

889
00:58:23,880 --> 00:58:26,460
So when we talked about woods,
we started talking about this.

890
00:58:26,460 --> 00:58:29,320
Remember, I derived a
little performance index.

891
00:58:29,320 --> 00:58:31,370
We said if we had
a material and we

892
00:58:31,370 --> 00:58:32,880
wanted to have a
given stiffness,

893
00:58:32,880 --> 00:58:34,338
and we wanted to
minimize the mass,

894
00:58:34,338 --> 00:58:37,110
we had that performance index
that was E to the 1/2 over rho.

895
00:58:37,110 --> 00:58:39,910
And we had a chart of
modulus versus density.

896
00:58:39,910 --> 00:58:41,530
And we saw that wood
was really good.

897
00:58:41,530 --> 00:58:44,310
You can do that for other
sorts of properties, not just

898
00:58:44,310 --> 00:58:44,810
modulus.

899
00:58:44,810 --> 00:58:49,140
So you can make-- depending on
what the mechanical requirement

900
00:58:49,140 --> 00:58:52,050
is, you can work out
different performance indices.

901
00:58:52,050 --> 00:58:54,500
So I want to go into that
in a little bit more detail.

902
00:58:54,500 --> 00:58:56,880
So the question is,
how do we select

903
00:58:56,880 --> 00:59:01,032
the best material for some
mechanical requirement?

904
00:59:33,522 --> 00:59:35,730
So in the wood section, we
looked at the minimum mass

905
00:59:35,730 --> 00:59:38,216
of a beam of a given stiffness.

906
00:59:42,240 --> 00:59:45,300
And we saw that the performance
index was E to the 1/2

907
00:59:45,300 --> 00:59:46,094
over rho.

908
00:59:48,767 --> 00:59:50,850
So let me do another one
of these little examples,

909
00:59:50,850 --> 00:59:52,558
and then I'll show
you some more of them.

910
00:59:52,558 --> 00:59:55,620
So another example would
be what material-- minimize

911
00:59:55,620 --> 00:59:58,130
the mass of a beam of a given
strength or a given failure

912
00:59:58,130 --> 00:59:58,630
load.

913
01:00:22,020 --> 01:00:26,560
So we'll call the
failure load Pf.

914
01:00:26,560 --> 01:00:29,450
And we can see the
maximum stress in the beam

915
01:00:29,450 --> 01:00:32,370
is going to be the moment in
the beam times the distance

916
01:00:32,370 --> 01:00:34,600
from the neutral
axis y, and divided

917
01:00:34,600 --> 01:00:36,170
by the moment of inertia.

918
01:00:36,170 --> 01:00:40,274
So here, M is the maximum
moment in the beam.

919
01:00:45,800 --> 01:00:48,535
And y is the maximum distance
from the neutral axis.

920
01:00:57,680 --> 01:01:00,560
And I is the moment of inertia.

921
01:01:00,560 --> 01:01:05,800
And I'm going to say
i goes as t to the 4.

922
01:01:05,800 --> 01:01:08,830
And I'm going to define a
failure stress of the material

923
01:01:08,830 --> 01:01:09,610
sigma f.

924
01:01:19,440 --> 01:01:24,250
So sigma max is going
to go as my failure

925
01:01:24,250 --> 01:01:25,490
load times the length.

926
01:01:25,490 --> 01:01:27,080
That would be the moment.

927
01:01:27,080 --> 01:01:30,750
The distance from the neutral
axis is going to go as t.

928
01:01:30,750 --> 01:01:33,576
And the moment of inertia is
going to go as t to the 4th.

929
01:01:36,470 --> 01:01:39,740
And that's going to be the
failure strength there.

930
01:01:39,740 --> 01:01:41,626
So I can solve this for t.

931
01:01:41,626 --> 01:01:43,750
And then I'm going to write
the mass in terms of t,

932
01:01:43,750 --> 01:01:45,640
and put that in there.

933
01:01:45,640 --> 01:01:55,410
So here t goes as Pf
l divided by sigma f.

934
01:01:55,410 --> 01:01:57,530
And that's going to
be to the 1/3 power.

935
01:02:03,720 --> 01:02:05,107
I guess I can scoot over here.

936
01:02:27,080 --> 01:02:32,470
Then we can say that the mass
M goes as the density of times

937
01:02:32,470 --> 01:02:35,520
t squared times l.

938
01:02:35,520 --> 01:02:48,950
So the mass M is going to go
as rho times l times t squared.

939
01:02:48,950 --> 01:02:53,240
So that whole thing
goes to the 2/3 power.

940
01:02:53,240 --> 01:02:55,450
So if we look at the
material properties,

941
01:02:55,450 --> 01:03:01,010
the mass goes as the density
times the failure stress

942
01:03:01,010 --> 01:03:02,642
raised to the 2/3 power.

943
01:03:02,642 --> 01:03:04,100
So if we want to
minimize the mass,

944
01:03:04,100 --> 01:03:07,450
we want to minimize rho
over sigma f to the 2/3,

945
01:03:07,450 --> 01:03:12,020
or we want to maximize
sigma F to the 2/3 over rho.

946
01:03:19,150 --> 01:03:21,456
So that's the performance
index for that case.

947
01:03:25,460 --> 01:03:27,950
So we can obtain these
performance indices

948
01:03:27,950 --> 01:03:29,960
for different loading
configurations

949
01:03:29,960 --> 01:03:32,330
and different
mechanical requirements.

950
01:03:32,330 --> 01:03:34,530
And I don't want to go
through a whole lot of them,

951
01:03:34,530 --> 01:03:37,840
but I'm going to put
this up with the notes.

952
01:03:37,840 --> 01:03:40,135
So this is from Mike Ashby's
book on Material Selection

953
01:03:40,135 --> 01:03:41,720
in Mechanical Design.

954
01:03:41,720 --> 01:03:44,680
And this is a whole series
of these performance indices

955
01:03:44,680 --> 01:03:47,560
for different
situations, for things

956
01:03:47,560 --> 01:03:49,760
loaded in torsion, for
columns and buckling,

957
01:03:49,760 --> 01:03:51,640
for panels and bending.

958
01:03:51,640 --> 01:03:53,210
So these ones are
all for stiffness.

959
01:03:53,210 --> 01:03:56,220
And they all involve a modulus
raised to some power divided

960
01:03:56,220 --> 01:03:57,530
by a density.

961
01:03:57,530 --> 01:04:01,800
So a tie in tension, c over
rho, the beam in bending

962
01:04:01,800 --> 01:04:03,580
is E to the 1/2 over rho.

963
01:04:03,580 --> 01:04:06,370
A plate in bending
is E to 1/3 over rho.

964
01:04:06,370 --> 01:04:08,420
So you don't need
to memorize those.

965
01:04:08,420 --> 01:04:11,850
But you can see you can derive
these for different situations.

966
01:04:11,850 --> 01:04:14,710
And here's another one for
strength-limited design.

967
01:04:14,710 --> 01:04:19,450
So the shaft is, depending on
what the specifications are,

968
01:04:19,450 --> 01:04:22,780
it's the strength raised
to the 2/3 power over rho.

969
01:04:22,780 --> 01:04:25,740
The beam loaded in bending--
the top one there-- sigma f

970
01:04:25,740 --> 01:04:26,790
to the 2/3 over rho.

971
01:04:26,790 --> 01:04:27,940
That's what we just did.

972
01:04:27,940 --> 01:04:30,600
So there's all these different
kind of performance indices.

973
01:04:30,600 --> 01:04:32,670
So depending on what
your situation is,

974
01:04:32,670 --> 01:04:34,770
you would pick one
of these indices.

975
01:04:34,770 --> 01:04:37,440
And then what you can do is
use these material selection

976
01:04:37,440 --> 01:04:40,580
charts, which plot one
property against another

977
01:04:40,580 --> 01:04:42,247
on log-log scales.

978
01:04:42,247 --> 01:04:44,080
And because all of these
performance indices

979
01:04:44,080 --> 01:04:46,160
involve a power,
they always end up

980
01:04:46,160 --> 01:04:48,940
being a straight line
on your log-log plot.

981
01:04:48,940 --> 01:04:50,630
And here this one,
I think, is the same

982
01:04:50,630 --> 01:04:52,850
as what I showed
you for the wood.

983
01:04:52,850 --> 01:04:57,080
This one's the modulus here
plotted against density.

984
01:04:57,080 --> 01:04:58,260
So foams are down here.

985
01:04:58,260 --> 01:05:00,487
And other engineering
materials are over here.

986
01:05:03,229 --> 01:05:05,520
And these guidelines here
are the different performance

987
01:05:05,520 --> 01:05:05,810
indices.

988
01:05:05,810 --> 01:05:06,955
So this one's E over rho.

989
01:05:06,955 --> 01:05:08,670
This one's E to
the 1/2 over rho.

990
01:05:08,670 --> 01:05:10,530
This one's E to
the 1/3 over rho.

991
01:05:10,530 --> 01:05:12,850
And for this case here,
as you move the lines up

992
01:05:12,850 --> 01:05:15,930
to the top left-hand
corner, E is getting bigger.

993
01:05:15,930 --> 01:05:17,090
Rho is getting smaller.

994
01:05:17,090 --> 01:05:19,100
And so the actual value
of the performance index

995
01:05:19,100 --> 01:05:20,200
is getting bigger.

996
01:05:20,200 --> 01:05:22,630
So you can use this
to select a material.

997
01:05:22,630 --> 01:05:24,950
So we've made these
charts for foams as well.

998
01:05:27,530 --> 01:05:30,017
So here's a couple
of charts for foams.

999
01:05:30,017 --> 01:05:32,350
And I think what I'm going
to do is just go through them

1000
01:05:32,350 --> 01:05:32,854
quickly.

1001
01:05:32,854 --> 01:05:34,520
And there aren't
really that many notes,

1002
01:05:34,520 --> 01:05:36,120
so I'll just put the
notes on the website.

1003
01:05:36,120 --> 01:05:38,170
And you can come and
write all the notes down.

1004
01:05:38,170 --> 01:05:40,330
And then we can
finish this today.

1005
01:05:40,330 --> 01:05:42,980
So this one here is the
Young's modulus versus density.

1006
01:05:42,980 --> 01:05:45,380
And these are all sorts
of different foams.

1007
01:05:45,380 --> 01:05:47,970
So the low modulus ones
tend to be flexible.

1008
01:05:47,970 --> 01:05:50,950
The higher modulus ones
tend to be more rigid.

1009
01:05:50,950 --> 01:05:55,420
And you could use this to
select foams, if you wanted.

1010
01:05:55,420 --> 01:05:57,730
You can also see what
the range of values is.

1011
01:05:57,730 --> 01:05:59,780
So the values of
the modulus here

1012
01:05:59,780 --> 01:06:03,930
goes from a little
less than a 100--

1013
01:06:03,930 --> 01:06:06,030
because this is two
orders of magnitude here,

1014
01:06:06,030 --> 01:06:09,266
I think, each one of these--
down to about 10 to the minus 4

1015
01:06:09,266 --> 01:06:10,580
or a little less than that.

1016
01:06:10,580 --> 01:06:11,580
So there's a huge range.

1017
01:06:11,580 --> 01:06:13,538
There's almost a range
of a factor of a million

1018
01:06:13,538 --> 01:06:15,339
in those moduli.

1019
01:06:15,339 --> 01:06:16,880
And the same with
the strengths here.

1020
01:06:16,880 --> 01:06:19,980
The strengths go from
10 to the minus 3

1021
01:06:19,980 --> 01:06:22,880
mega-pascals up to about
maybe 30 mega-pascals,

1022
01:06:22,880 --> 01:06:24,570
something like that.

1023
01:06:24,570 --> 01:06:27,420
And you can see for the
modulus and the strength,

1024
01:06:27,420 --> 01:06:29,050
things like the
metal foams are good.

1025
01:06:29,050 --> 01:06:29,987
The balsa's good.

1026
01:06:29,987 --> 01:06:31,910
Here's the balsa up here.

1027
01:06:31,910 --> 01:06:33,370
Metal foam's up there.

1028
01:06:33,370 --> 01:06:38,940
So you can kind of see
the range of properties

1029
01:06:38,940 --> 01:06:41,040
that you could get.

1030
01:06:41,040 --> 01:06:45,740
And then you could also--
need a drink, hang on.

1031
01:06:50,860 --> 01:06:53,840
You can also plot the
specific property.

1032
01:06:53,840 --> 01:06:55,460
So here's the
compressive strength

1033
01:06:55,460 --> 01:06:58,580
divided by the density plotted
against the Young's modulus

1034
01:06:58,580 --> 01:07:00,006
divided by the density.

1035
01:07:00,006 --> 01:07:01,630
And here you want to
be up at this end.

1036
01:07:01,630 --> 01:07:04,260
So you would have a high
strength and a high stiffness.

1037
01:07:04,260 --> 01:07:07,790
So the balsa and the metal
foams are good up here.

1038
01:07:07,790 --> 01:07:10,695
This next plot-- this is
the compressive stress

1039
01:07:10,695 --> 01:07:13,100
at 25% strain.

1040
01:07:13,100 --> 01:07:15,530
And this is the
densification strain.

1041
01:07:15,530 --> 01:07:21,020
And if you think of having
your stress-strain curve

1042
01:07:21,020 --> 01:07:24,560
looks like this,
something like that,

1043
01:07:24,560 --> 01:07:27,780
so you could say
that's a strain of 0.25

1044
01:07:27,780 --> 01:07:30,920
and that's the stress
that corresponds to that.

1045
01:07:30,920 --> 01:07:34,520
So that stress times the
densification strain,

1046
01:07:34,520 --> 01:07:37,730
which is out here
someplace, is an estimate

1047
01:07:37,730 --> 01:07:41,230
of the energy underneath
the stress-strain curve.

1048
01:07:41,230 --> 01:07:45,240
So you can think of this
right-hand plot here--

1049
01:07:45,240 --> 01:07:48,920
those dashed lines-- these lines
like this and this and this--

1050
01:07:48,920 --> 01:07:51,630
each one of those corresponds
to how much energy you

1051
01:07:51,630 --> 01:07:54,100
would absorb under the
stress-strain curve.

1052
01:07:54,100 --> 01:07:57,840
So points that lie
on here would have

1053
01:07:57,840 --> 01:08:01,500
an energy of 0.001
megajoules per cubic meter.

1054
01:08:01,500 --> 01:08:04,130
And over here, we're at
10 joules per cubic meter.

1055
01:08:04,130 --> 01:08:07,600
So again, the balsa and the
metal foams are good over here.

1056
01:08:07,600 --> 01:08:10,660
So you can use these plots
to try to identify foams

1057
01:08:10,660 --> 01:08:12,700
for particular applications.

1058
01:08:12,700 --> 01:08:14,890
And I think there's
a couple more.

1059
01:08:14,890 --> 01:08:16,740
It doesn't have to be
mechanical properties.

1060
01:08:16,740 --> 01:08:19,699
Here is thermal conductivity
versus compressive strength.

1061
01:08:19,699 --> 01:08:21,740
So you can imagine if you
wanted some insulation,

1062
01:08:21,740 --> 01:08:24,130
you wanted to have a certain
thermal conductivity value,

1063
01:08:24,130 --> 01:08:26,550
you probably also need at
least some minimal compressive

1064
01:08:26,550 --> 01:08:27,729
strength.

1065
01:08:27,729 --> 01:08:30,130
You could also have something
like a maximum service

1066
01:08:30,130 --> 01:08:32,074
temperature, that
maybe the foam is going

1067
01:08:32,074 --> 01:08:33,240
to melt at some temperature.

1068
01:08:33,240 --> 01:08:35,670
You can't go beyond that.

1069
01:08:35,670 --> 01:08:37,850
So there's some property there.

1070
01:08:37,850 --> 01:08:40,420
And I think there's
one more here.

1071
01:08:40,420 --> 01:08:42,090
You can look at things
like the density

1072
01:08:42,090 --> 01:08:43,729
in terms of the
buoyancy of a foam,

1073
01:08:43,729 --> 01:08:45,840
if you have some
buoyancy application.

1074
01:08:45,840 --> 01:08:48,560
And you can look at cell
size on this one here.

1075
01:08:48,560 --> 01:08:50,510
And cell size can be
important for things

1076
01:08:50,510 --> 01:08:52,250
like filtration and catalysis.

1077
01:08:52,250 --> 01:08:55,510
So the amount of surface area
goes as 1 over the cell size--

1078
01:08:55,510 --> 01:08:57,680
the surface area
per unit volume.

1079
01:08:57,680 --> 01:09:00,010
And so the cell size can be
important for those sorts

1080
01:09:00,010 --> 01:09:01,132
of applications.

1081
01:09:01,132 --> 01:09:03,340
So the idea is, you can make
these material selection

1082
01:09:03,340 --> 01:09:04,130
charts for foam.

1083
01:09:04,130 --> 01:09:06,287
And you can put data on there.

1084
01:09:06,287 --> 01:09:07,370
And you can compare foams.

1085
01:09:07,370 --> 01:09:09,745
And you can use these
performance indices.

1086
01:09:09,745 --> 01:09:11,120
So I'm going to
leave it at that.

1087
01:09:11,120 --> 01:09:12,760
There is a little
bit more notes.

1088
01:09:12,760 --> 01:09:14,460
But I'll just put
them on the website,

1089
01:09:14,460 --> 01:09:17,050
and you can get them from there.

1090
01:09:17,050 --> 01:09:19,409
So I think we're good for today.