Structural hierarchy at any rate is
ubiquitous in the biological world and

when we talk about structural hierarchy
what we mean in particular is

structural organization and

many disparate like scales and so
recent review literature for instance.

This network shows us for instance that if
we keep zooming in on bone we can see many

levels of organization ranging from
the macro scale all the way to eventually

down to the molecular scale but we need
to pick on bone we could alternatively

consider the leaves of plants we could
consider the shells of course stations

we could consider the feathers
of birds there are thank you and

so in view of the many
different occasions in

which structural hierarchy has emerged and
the many different evolutionary lineages

in which it's emerged it stands to
reason that there are broader mic.

Mechanics of materials properties to
be appreciated now of course we're not

the first people to consider structural
hierarchy some of the traditional

benefits that have been reported
are improved economy of material and

EDITION we can realize certain
counterintuitive mechanical

responses would not be observed for
conventional solids and

the the use of structural hierarchy
has been noted as as a possible

means of resolving the traditionally
Integra mystical strength and

toughness now the pessimist might
note that as we become hierarchical

we also have far more information
that must be specified and

order to assemble material properly and
so with this additional information

we also have many more opportunities for
errors and we might worry that

with the accrual of these random errors
there would be on the liability of

catastrophic failure at least a failure
to attain our target properties.

But this is not the case and so now we'd
like to ask what is it about hierarchal

materials that makes
them a robust strategy so

we'd like to start to get some mechanistic
understanding of the interplay of

structure at various lengths scales and
to do this we turn to the.

In the ball in spring lattice Now we had
a master course on this this morning back

I'll be talking once again about
a modified form of the dilute triangular

lattice so we suppose that we have
a set of nodes connected by a harmonic

bonds and there will be arranged
in a triangular lattice.

For which we might remove
some portion of bonds and so

will characterize it by the portion P.

that are retained Now there's a century
and a half old here a stick due to Maxwell

that allows us to anticipate whether or
not such a structure will be stiff so

we note that in D.D. mentions we have and
bonds will start with N.

times D.

spatial degrees of freedom and every time
we add a bond between two of those nodes

we are going to remove one of those two
degrees of freedom and by Adams and

energetic cost to varying their separation
and so we expect to need order of N.

times D.

binds to have at least
a marginally rigid structure and.

So from there we reasoned
that average of two D.

bonds per node is going to be
necessary for a stiff frame but

now I would like to ask well so
in particular for the triangular lattice

this structure has been considered now for
over thirty years and

going back to ninety five there
is a simple scaling law that says

that if we retain more
than two thirds of bonds

we should be at least marginally stiff and
above that threshold of two thirds.

The stiffness is expected to scale
linearly with the excess spine portion

of the actual bind portion minors
that critical bond portion so what

happens when multiple lengths scales are
involved is there one true length scale or

does each length scale and
somehow contribute so.

To begin answering this question
we consider a hierarchical

dilute train your lattice so
as before we have this clear to

clearly defined lattice structure on
the large scale but now each bind has been

replaced by an even finer scale lattice
structure and were able to independently

control the latter the activity
on both the large and the small

length scales without influencing
the kind of activity on either.

Scale by by plucking.

By plucking on the larger the small scale
we can we can leave the other scale.

And change and so
this provides this a means of

seeing if if there is indeed
one true length scale or

if there is a more subtle interplay
between the two lengths scales and so

we can further iterate this process but
we'd like to gain tractable

mathematical model because as we keep
adding more and more levels of hierarchy

the the information that must be
specified once again becomes and that and

the task of simulating such a structure
becomes computationally prohibitive so

to begin with I'll consider
a single large scale violent and

I presume they did inherit.

An effective stiffness from that
small scale lattice structure and so

I suppose once again the effective large
scale Beim stiffness is proportional to

small scale by portion
minus critical small scale

by portion but I'm not going to assume
that's two thirds this time because as.

Notes there are bound to be quite
significant finite size effects so.

Kind of combining that.

Small scale where the nation
with an effective

with a large scale find portion
I am going to conjecture that

the overall stiffness should be
in effect an excess large portion

times in excess small scale
portion time some constant and

taking this a step further using the
Fisher's counting one too many were done

we're going to postulate a general
scaling rule something like this and

so we've worked it to check
this with simulations for

two dimensions and
our work in three dimensions is ongoing so

our procedure is much like what jamming
described it this morning actually we're

going to displace the top of the lattice
they were going to hold the top and

the bottom like ordinates fixed
well that all other degrees of

freedom in the lattice relax log
the residual strain energy and

then repeat to build up a curve and then
from the derivative of that curve we can

infer the uni axial times out stickiness
of one of these structures so we do

that for eighty one different combinations
of large and small scale bun portion and

so on the left hand side you see
the simulation data on the right hand

side is a heat map where
we've the free parameters and

that scaling model I proposed earlier
now on the large scale we actually see

it really should be labeled critical.

We see a critical bond portion that
slightly less than two thirds and

a small scale critical bond portion
that's markedly more than two

thirds now this actually kind of
reminds me of our behavior seen.

When there's a moderates
bending stiffness and so

I it appears to us that
perhaps the large scale.

Binds have some effect of bending
stiffness there's actually a lovely paper

of N.P.R. We coauthored by John and know
that appeared two thousand and thirteen.

That explored different regimes of bending
to stretching ratios so if we plot the.

Scaling model versus simulation data
we get a fairly good agreement between

the two so we think that we're
mostly capturing the salient

details with this seemingly simple model
so now that we have a bit of a mechanistic

understanding we feel like we're in
a position to address the question that I

posed at the start why is it that
hierarchical structures are indeed robust.

So we consider some target stiffness so
we suppose that

we attempt to and
to achieve that target stiffness with the.

Point in buying portion space but
we don't actually hit the mark perfectly

there is a little noise and so we could
have some rain in the air and the and

the true bottom portion on
each length scale and so

we consider Gaussian random noise for
for simplicity so.

We have kind of a money Karla's
down process where we're all

add some gaps in random noise
small large scale vine portion.

Over ten thousand trials we see that we
get a much tighter distribution for a two

level network than a single level network
Moreover at one level you're very so

susceptible to descending the low
threshold for marginal stiffness for

a marginally stiff structure we've
worked out a little bit of analytics so

that act in a slightly different
regime we see a scaling.

Of this form work a.

Target stiffness and is the number of
levels of hierarchy and so we see that

you eventually get diminishing returns so
there is an optimal level of.

Optimal number of levels of structural
hierarchy beyond which you can have to

measure the good thing.

So with that I'll be happy to discuss
that analytic approach later but

I'd like to thank you for

your attention and I'll invite our
invite any questions at this time.

Thank you.