OK.
So good morning everyone First I'd
like to thank the organizers for
putting together this wonderful symposium
I'm really enjoying my time here so today
I'm going to talk about the fracturing and
the two interesting model systems we
recently started at that to show very Are
your old behaviors when they fracture so
we all know that a fracturing can be a
major problem in under an earring this is
what happened to us now the oil
tanker during the World War two And
this is what happened the last abounding
Florida to a pedestrian bridge so
when you look at the pictures of these
Scott has traffic events one thing you can
notice is that they are long straight to
cracks that lead up to the brittle failure
of the material and the engineers
certainly know very clearly about of
the importance of these cracks and in
the news it was reported out to two days
before the bridge collapse and reported
to seeing cracks so why cracks are so
important this was explained by of
British under near Griffith in the one
thousand nine hundred is so the basically
the physical picture is like this
if you have a sun pole with a cracking
it and you put it under external load
the stress added the corrective will
be much stronger than the background
the stress and the magnifying factor
is proportional to the square
root of the loans off this crack and
then you can see it out of the areas or
instability for the crack to
propagate because these children
I think I probably did not turn
me there you know this better.
It's working so same.
Try the other one you know you.
OK story hello OK that's better.
So it is OK yeah great so yeah thank you.
If the external stress exceeds
a critical value I told
make a bond out of the correct tape was
started to break and then after the break
the crack will be longer and there will
be more stress added the new tip so
the crack will just to propagate not
stop and do the whole thing will break
break catastrophically OK so this
Christmas Siri has been a guideline for
engineers to estimate a critical stress
of some polls and design structures and
then the question we would like to address
today in this talk is Conway design noble
structures that violated this paradigm
of stress focusing crack propagation and
the catastrophic failure so let me
emphasize that we are going to do those by
looking out of the structure and the
building blocks that we are going to make
up of the material is still brittle and
we are going to use geometry and
the structure to make a tool
to to achieve the fixed out
to material cost of oil catastrophe
failure OK so here is the outline
of my talk the first a modern system I'm
going to talk about our fiber networks and
this is done in collaboration with my
graduate student a lawyer John and
the who is one of our organizer and
my colleague one sender at Michigan and
the second although system I'm going
to talk about our top logical next or
lettuces and this is done with my student
Hello your drum OK So let's start with
the first one when we first gave you a
brief overview of fiber networks and their
mechanics I know many of you are probably
are if are already familiar with fiber
networks so I'll be brief here and just
to go through that you central points and
if there is anything not clear please feel
free to stop me and ask so fiber networks
are a group of theoretical models where
you have a disordered network of fibers
that are tied together by crossing so it's
a very simple structure but it occurs.
The essential structure of many different
types of materials such as natural
materials the subtle skeleton which
is a fiber network of biopolymers and
the money man made the materials such as
felt the paper and the carbon nanotube
paper so all these different systems are
very different to learn scales and they
have different occult chemical composition
but they share the same essential
structural fiber networks and the we would
like to understand how do they fracture.
I think of the areas some chemical
bonding between the two to
to make it into a solid.
OK So also there is entanglement
a dot to give rise.
OK so.
To to in order to think about
of the fracturing Let's first of
spend a minute or two of the mechanics
of these fiber networks so
because the component of the fiber
networks the fibers you can model them as
long elastic rods and then you can
easily see that it's much harder to
structure the rods done to bundle them so
the bending stiffness is much smaller than
the stretching stiffness here I put into
powers of the length of the rather to make
it a measure of right and then you can
think about of the interesting limit where
you turn off bending stiffness of a fiber
network and this just a map into a central
force network which means that out of
the hundred twenty of this network only
cares about of the distance between the
Connected to cross-linking points and it
doesn't care about of the bending between
the neighboring fiber segments OK so
that's what a central force mean and isn't
mechanics of central force not a works
have already been studied for a long
time since the time of the next roll so
Max will go down this equation for
the verge of mechanical
you stability in a central force network
basically you have you call a number of
degrees of freedom in
the constraints if you have too many
degrees of freedom then you will have
zero modes which are waves through did.
Form those that work without
causing an elastic under and
if you have too many constraints you
will have to you will have stress and
then for larger central force network you
could write of this equation in terms of
content is polite is sight so
the degrees of freedom is D.
for point a like particle
we see the motions and
of the constraint is called initial number
divided about tool because the ordination
number is the number of arms
coming out of from each side and
do we need to divide it by tool because
each found to give you one constraint and
it is shared by the two neighboring sides
OK So this is a very simple and this gave
rise to the well known you question four
as a statistic which is equal to two D.
that you could to see in many different
branches of the matter of physics OK so
this was the idea behind this paper
we wrote several years ago on linear
You asked Is it A or fiber networks and
the we find of this first diagram
using the diluted to try and
go like his model so this axis is
the connectivity of the now to work and
in terms of the called initial number and
this axis is bending vs stretching
stiffness of each of the fibers so
in the limit of couple you could have zero
or where in the central force network
unlimited and of the network only
becomes stable above the central force
as a starting point is equal to twenty and
A below that it's floppy and
now if you have funding stiffness
the floppy not to work will can become
rigid with the extra constraints from
bending but that the network is very
floppy because now all of the shear
modelers comes from the bending stiffness
of the fibers it's much still weaker
than the stretching stiffness and then
about this point of the network is already
stable with the stretching stiffness of
all of the fibers so the sheer modelers
will be proportional to the stretching
stiffness and in the middle of the central
force as a starting point it gave rise to
this critical region him where the sheer
modelers depend on both and bending and
stretching are strongly cup.
Board OK And below this Laura critical
connectivity evil way of spending Steven
is the network is still flopping OK So
this is for a linear us to city and
the now if you look at the drama tree
of many fiber networks most of the time
you have cross-linking points that
only connected to fibers together and
did that it tells you that to
the number is always less than two D.
because if you look at the cross we
can points inside of the network
it's four coordinated but
of the ones on the Bunbury.
There are less than four cardinal because
the only earns don't a contributor to
elasticity so I have a ridge the networks
are you're really in the bending dominant
regime and so this is about the linear you
asked is it and then you can think about
what if we keep stretching over to Langley
near you asked is it a regime OK So
what happens to our metric ole when
you structure to more and more
in the beginning there are a lot of these
softer bending modes in the network and
the sheer modelers is very small but
if you keep hearing the network
are stretching the network the fibers are
becoming aligned in the stream direction
and there is a crossover from bending
dominant to stretching dominant under
the elastic model like increase a lot so
with studded of this carefully using
the diluted a triangular latest model and
the we find it is three dimensional.
So this plane couple versus P.
is the oldest The last visit her fifth
diagram where I have just to show
you in the previous slide and
now we have this new axis of strain so
basically if you start off from the
funding dominant to redeem in linear you
last is a theory which corresponded
to a very diluted fiber network and
now you increase the strain
the network will cross over
into a non-linear elasticity region and
then become stretching dominant and what
you can see here is started as a static
pointer controls not only the linear but
also the nonlinear you asked is it
any of these five or not who are.
And this also helps explain the strange
stiffening behavior in many of our polymer
jails where the sheer modelers can
increase by more the order of magnitude as
you strain the network OK so what if
we're straight even more such that out of
the funds in the network started to
break and it does start to fracture so
to study that we designed
the following simulation protocol so
basically where use that diluted
a triangular latest model again
now each band in the network is it
is present to with probability P.
and with set of thresholds for the bundle
to break which is a small number Lunda and
for the first step with side to
the bending stiffness to be zero so
this is a simple force now to
work later a way to test it and
that works with finding stiffness and
the result is essentially the same if
binding stiffness is not too large and
then we have one hundred twenty and that
is simply the sum of the United manager of
every bond in the network ever
existing bunny in the network and
it's worth noting that at this is a for
now only one hundred twenty or
four arbitrarily large the formation
there's still applies OK and
the way put a small crack in the center of
that network it's not a visible in this
case because the network ignore it is so
diluted So the crack is now to obvious and
the way way out of the strain is down
to where increase the strain in the Y.
direction so overstretched the network in
the way direction in very small strain
steps and then after a very average strain
step way through elastic relaxation which
means we minimised one hundred twenty and
with respect to two arbitrary with
respect to two and then one of deformation
of the of the sides in the lattice and
then we ask if there is a new band that is
structured a beyond of the threshold and
if yes we remove the bound and
require liberated them not to work so
where do they cycle and there are no
more bounds beyond of the threshold of
the weight increase the strain again OK So
this is the product of those an illusion.
And then let me show you one
example of what we find so
this is the single limited to of people to
one so we are looking out of the undiluted
a triangular light is and the on
the face diagram is a pass like this.
And if we do this.
You know.
To Win seven the make
sure the movie is played.
OK It's all right if rain Yes So when you
increase the strain we use both color and
the thickness of the bonded
tool represented the stress
as you can guess the stress will build
up to the top of the crack and the van
fracturing will start from the top of the
crack and go through the whole system and
here is a stress strain relations
of the of the whole network OK So
this is a classical case of
the Griffins crack nucleation So
the crack will propagate to where
the instability is so this is for
a dense network and if we look at
a network below I stood hesitating so
which is means following a path like
this on the face diagram so it's very
different now because in the beginning
the network does not have stability.
It's some to force network below the ice
of starting point and then you start to
play in the movie so in the beginning
you add a you increase strain and
then there is no stress at all you're
just preparing the man to work and
it's only after a critical strain
they are started to be for
strains emerging in
the network as you can see.
OK So then after the four strains
emerge bonds in the four strain will
quickly go beyond of the threshold and
it is starting to break but
there's a new force trains will emerge
in the network and of the network and
hearses interesting steady state where
force trains emerge on the brake and
emerge on the brick and it's a pretty
long gradual fracturing process and
it's interesting to see that now there
is no crack propagation the crack
is irrelevant to the whole network of
brakes in a pretty homogeneous way and
the one thing the braking event is not to
close in space in the with the next upon
the braking events it's kind of have
a divergent the correlation runs
I'm going to show you next OK so
when the movie is kind of long that's
we'll have to play the whole thing so
basically the whole network could use to
break through this kind of a steady state
and there is no stress
focusing to crack tips and
there is no crack propagation So
how do we understand this yes.
It's yes it is related to population but
it's going to be different I'll show
you scaling relation this way and
Elias Yeah so so how do we understand
this more competent who lay so
one quantity we can look out is very
convenient connoted out is the total
number of broken bonds when the whole
network is broken and the way counted
the number of broken bones as a function
of peace which is a probability for
each ponder to be there and
the point is to serve them here OK and
what you can see is down to
the number of broken bones peak and
to the as a static a point and then you
can guess where there is as a starting
point if you see it is again
the boundary between these two types of
the ring of behaviors the crack nucleation
behavior for does not a works and
of the kind of distributed a damage for
the on the lot and that works.
So to see that let's try to see him.
We can collapse those curves for
different system size and
the to do that we're going to use the
usual way of doing fanaticized scaling so
basically the idea is that we're supposed
to areas of correlation LUNs that
the diverged to the central force as a
starting point with exponent a needle and
therefore an opposite over a ball you
can write it into their scaling form
which has a point which has a factor.
That is basically the infinite or
size scaling of this continent and
then there is another factor which
is a finite size scaling function
that is controlled by is the ratio between
the system size and of the correlation so
this is a Ural way of doing
fine outside scaling and
that you quibble underlay You can also
write to this equation you can also pull
out of the system size in the front and
rewrite of the scaling function
in either you quibble it away like this
this turns out to be more convenient or
for analyzing our data for
the number of broken bones OK.
So now let's try to collapse
the data using this relation so
here is what's happening for about P.C.
for networks above as of statistics so
we can collapse all of
the data using two exponents
why exponent is new the correlation
month for grid a typical ation and
then another exponent we need to use is
a fractal dimension in the front which
characterize the damage region and so we
can collapse all of the data pretty well
by risk ailing both the horizontal
axis and of the vertical axis so
we collapse the data back to what it
does this tell us physically right.
What is the meaning of this correlation so
this has been very nicely explained
in a recent paper on P.S.A. ice where
the authors studied a different type
of networks of ours as a statistic and
this picture pretty much speaks for itself
physically when you have these alter
the cracks are no longer sharp straight a
line instead it becomes like a meandering.
Lever of a fine with and then the
correlation learns basically characterize
the first Innes of this river and
the other exponent a kind of
tells you how many bones are broken inside
of this bend made his own religion OK So
this is basically what's happening
in the in fourth else networks above
as a state is it a physical a as
usual out of the fracturing of
the network still looks like a crack but
as you is only in there is a length scale
that tells you it's not her sharp
crack there is less gold below Roach
the you you see that you
fact of the disorder OK And
now if we look out of the region below
P C which is truly the fiber network or
region it's becoming very different
from the scenario for about P.C. So
what's happening is that a now
if we try to collapse the data
what we see is that you don't need to
do anything to the horizontal axis
you only need to risk all the vertical
axis and the data already collapse so
what did this mean remember when we look
at the data for people record on P.C.
where you had to risquÃ© all the horizontal
axis using the system size now we
don't need to do that so what this tells
us is that the system size doesn't matter
the correlation learns always goes to
infinity so all system size becomes
the same for networks feel O.P.C.
So the network always have divergent
to learn scale for the fracturing for
the non-linear fracturing process so
this suggests of this diagram like
a there is so this axis is P.
The probability for
having each abundance and
in the middle there is the central force
as a static a pointer to serve for
the tree and the lower lattice and
about the central force as
of sceptical point thus as you
make the system size larger and
larger it afloat to crack a nucleus of
fixed point so basically four and they've.
Into there is correlation LUNs and
that it diverges as you
approach the as a static a point A from
above and the four and they point to above
the isostatic a point you can always make
the system size bigger and then the crack
will become thing there and a thing there
with respect to the system size and
this agrees with the scenario discussed
in this paper by from Jim a Cessna school
basically as you increase the system size
the fracturing behavior cross over from
damage population to crack a nuclear
nation OK However the regime
would be lowkey see the fiber networks is
very different we always have divergent to
learn scale one from the breaking event to
come be arbitrarily far from the next of
on the breaking event it's dominated by
the physics of the force chains instead of
craft nucleation So they are there
is always extensive damage and
the network always takes a lot of energy
to break OK So we also look at how
to undersize distribution of these fiber
networks it turns out of that that they
also have very interesting behavior so
this is the distribution of avalanches so
the ice is the size of the avalanche which
is the number of pounds breaking in one
strain a stab so when you break one pound
it countries are either bonded to break
into the same strain so that is why I
will launch and also as a function of P.
and the system size and in the literature
the undersides distribution
can often be written in a scaling formula
to this so you have a power law part that
it describes the distribution when the
avalanche when you're looking at a smaller
avalanche is an event a large element
of size there is a total function that
are defined under a system and the what we
find is dieted we can't use the scaling
relation we have already derived to using
the number of broken bones to get to the.
Two to get as a part of.
Avalanche size and the using data
we can collapse the data for
our system size for
the Avalanche size distribution and
the one thing we're worth noticing here is
down to the Avalanche exponent here for.
Parle part has an exponent of one
point five which is different from
most a brutal failure in in order to
systems which is two point five so
but it turns out of that it's similar to
what people observed before I wonder is it
in granular flow so why it is why
it is we're talking about this
in the paper we wrote and the way made out
of our toy model which we call the slack
fabric bundle model to explain the
difference between these two exponents so
I don't have time to go into the detail
of that to today if you're interested or
you can look at our paper OK So this is so
much about it the first part of the fiber
networks where you have a divergent to
learn scale that makes the system not to
follow a graph of scenario for
fracturing OK so.
Are there any questions.
If no way I will guess.
Yes So here we are all talking
about a brittle failure so so
there is no plasticity.
OK thank you for
the interesting comment OK so
now let's talk about it a second apart
on top a logical next all out has
a switch also exhibit a very
interesting fracturing behavior so
let me first of give you a brief
introduction of top logical mechanics so
let's start from the index theorem and
where we're talk about a state of
stress which is very important in
understanding our result so for
any elastic and I to work you can always
define these two matrices called of the
equilibrium matrix which maps the Persian
every pound to the total force on every
side and there's a compatibility matrix
which maps the displacement of oversight
to the extension of bounded So
this is about force and
that this is about of the displacement and
both of these two matrices are determined
by the geometry of the network and
if you try to write down you will see
that it's basically just the science and
the cost ions of the angles of the bounded
that and has the matrix and you
can use a no show that they are transpose
of each other then let's think about
of the not space of these two literal says
the last days of the Q matrix give
you the states of self stress So
what does that mean if you have
a way to put tension on the bones so
here grad in the green corresponding
to positive on a negative and
if you have a way to put attention
on the buns resulting No the total
force on any of the site is called a state
of self stress so this is a way for
the network to carry stress leaving
all of the sites in force balance and
then if you look back to the last days of
the seen the tricks that gave you all of
the zero modes because they are ways to
displace the sides without extending or
compressing any of the bond and then
because these two matrices are related to
two You try their bad by a transpose they
must be they must have the same rank and
then you can apply rank knowledge of.
Serum to the two matrices and
the value of Dr of this actual color
down in that serum that tells you
the number off zero modes minus the number
of states of self stress must be
equal to the difference between the number
of degrees of freedom minus the number of
constraints in your network and therefore
the special case of next to a lattice
is which by definition as you call number
of degrees of freedom and the constraints
you must have the same number of zero
modes understates of self stress.
OK so this was the starting point for
this seminal paper by candle Bensky
on problematical mechanics of.
Of Mexico like this basically they find
it hard to buy it changing the geometry
of the unit cell of a max or
like this you can make the non-trivial
top logical structure to emerge from
the structure of the network and
that is the card of the top
logical polarization and
then the physical significance of the top
logical polarization is that they dictated
the exponential organization of zero
modes are states of self stress so
in particular in this figure if you
put a domains of maps or likenesses
of different geometry so such that if they
have different top logical polarization
together then all the domain was there
can be exponentially localized and
zero modes and exponentially localized
states of self stress so you're really
in stuff the matter with pay a lot of
attention to the zero modes and it turns
out of that that the state of self strikes
are also very important so why is that.
You can think about of the sense of self
stress they give you all the possible ways
to carry stress while leaving all of the
light his sights in force balance OK so
that you can imagine if you put
a lattice under external load
then the stress pattern can only come from
a linear combination of the all of
the states of self stress in the linear.
Elasticities region and you can write
it into a question like this so
here is the coefficients of each of
the states of self stress comes from the.
Comes from the inner product of
the external load where the states of self
stress and
then in this paper from trans over to
the school they tested to putting Max or
lattice where the state of self stressed
domain war under external pressure and
do they find out the structure was back
all the buckling of the structure and
their external pressure is exponential or
localized I did a storm in a war so
this inspire us to think about a how
this type of structure fracture
is the damage also going to be only
localized on the domain war and
does this still hold a deep into the not
only near you as to city region and what
if we have cracks in the structure that
competed with the state of self stress so
to understand that we are getting
used to our simulation protocol too
to try to fracture lattices So
we're tested to two types of mags
the latest is that deform the square
lighters and the deform the lattice and
the in both of these structures we have a
state of self stress domain walk here and
zero all mode domain walk here and
the what we show here is that under
external load all of the stress in
the linear elasticity region goes to the
state of self stressed only awards and and
thus in the samples on the right it's
probably harder to see if we have some
precut cracks in the network and the
interesting lay the cracks are protected
the stress does not go through
the crack tip but instead it goes to
the state of self stress domain was
is attracted by the Domino was and
then let me play a movie of the whole
non-linear process of the fracturing So
here the broken bones are marketed
by a yellow what you see is that
the network was started to
break up of the domain wall and
that the breaking process is very gradual
if you look at a stress strain relation
when you play the movie
again it's a little fast so
you see there are stress on the domain
war under the fracturing start here so
it's a very gradual very controlled
the fracturing process and
even well where the evil while
deep into the last is it original
the top logical protection
is still there and
it's a sense of distress to the domain
war even in the nanny original and
there is no catastrophic failure and
the cracks are protected through
the cracks does not have propagated OK so
now if you want to make actual design
of metamaterials where the cracks
inside of the material can be protected
and you can make of the fracturing
process to be well controlled or to
only happen I did the domain was you can
put multiple domain words in your
structure so let me play this movie.
So here I have sixteen there was and
as you can see that had the gradual
fracturing process mostly our current
of the domain wall until very late into
the process the whole thing breaks into
two pieces so you have a very gradual.
Stress strain relation and
another good thing to have multiple
domain was is that now you can
control the elastic Model S.
because that's a good model US is
a proportional to the number of states of
self stress that can sustain the external
load so it's going to be proportional to
the number of domain was you put in so
this gave you a lot of freedom to design
your material to design the linear
elasticity under the fracturing process.
OK So let me summarize we're talking
about are two different types of models
where they violated the Gryphus zero four
stress concentration I cracked hopes
that these include if I were at work still
as a statistic where you always have
a diversion to learn scale for
the damage events to happen and
also including the top logical Max or
likenesses where the stress is top logical
or protected under the domain was and
it to protect the interior of the material
OK So thank you for your attention.