THE FRACTURE-QUASIPARTICLE CONNECTION

The Fracture-Quasiparticle
Connection Looking at
Fracture in a New Light
Dann Passoja
New York, New York

2
[THE FRACTURE-QUASIPARTICLE CONNECTION LOOKING AT
FRACTURE IN A NEW LIGHT]
[Type text] [Type text]2
Table of Contents
Introduction ....................................................................................Error! Bookmark not defined.
Prime Objective of this Work.........................................................Error! Bookmark not defined.
Geometry in Fracture........................................................................................................................5
Fractals .............................................................................................................................................5
Engineering Mechanics: Consideration of Linear Elastic Fracture Mechanics on the Macroscopic
Scale .................................................................................................................................................9
The Atomic Structure and Force................................................................................................13
From the Atomic Coordinates to the Laboratory Coordinates..................................................15
The Plastic Zone.........................................................................................................................22
The Physics of Fracture from the Quantum Scale to the Macro
Scale................................................................................................................. 26
A Historical Note........................................................................................................................27
Insulators and Conductors..................................................................... 27
The Structure of a Generic Crack...................................................................................................27
The Griffiths Equation - the “Gold Standard”................................ 29
The Quantum/ Classical Boundary-Phonons.............................................................................30
Cracks are Fractal .....................................................................................................................31
Griffiths Equation with Phonons and Entropy ...............................................................................34
Quantum Mechanics and the Schroedinger Equation ................ 38
Something Other than G Emerges from this Derivation ................................................................41
Surface Plasmons.......................................................................................................................44
The Crack as a Source of Electrons, Phonons and Surface Charges ........................................45
A Crack- A Source of Phonons and Electrons in Insulators......................................................46
The Microstructure of a Crack .......................................................................................................48
The “Particle in The Tight Crack Tip”......................................................................................50
Statistical Mechanics ................................................................................ 52
The Distribution of Broken Bonds at a Crack Front.................................................................54
Statistical Mechanics .................................................................................................................55
Another Example of Size Distributions ......................................................................................60
Insulators and Polarization .................................................................... 64
The Dipole Moment....................................................................................................................64
Fracture in Insulators......................................................................................................................66
Statistical Physics of Bond Failure in Polar Insulators.............................................................70
Berry’s Phase..................................................................................................................................75
Applied Fields in Simple Geometries.........................................................................................77
Internal Fields.................................................................................................................................79
G,Γ and γ -Theory and Experiment.................................................... 84
Theoretical and Experimental Values of G ....................................................................................85
Alkali Halides and Insulators .........................................................................................................85
On the many meanings of G,Γ and γ that will be encountered in this work ..............................85
The Alkali Halide Data Base..........................................................................................................87
General Organization of the Analytical Work on the Alkali Halides ........................................99

The Fracture-Quasiparticle Connection Looking at Fracture in a New Light
September 2015
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Bulk Modulus vs Polarizability Base-Line Relationship.........................................................101
Bulk Modulus vs
1
ao
3
Base-Line Relationship..........................................................................102
Band Gap Energy vs Ionization Energy Base-Line Relationship ............................................104
Band Gap Energy vs Polarizability Base-Line Relationship ...................................................105
Band Gap Energy vs Electron Affinity Base-Line Relatioship.................................................106
Bulk Modulus ao vs Eionoz Base-Line Relationship ...................................................................109
Γ vs Δ atom radius Theoretical Relationship...........................................................................112
Γ2 vs Eioniz Theoretical Relationship.........................................................................................114
Γ vs [ Ecohesion - (Eanion+Ecation)] Theoretical Relationship ........................................................117
Prime Objective of this Work1
The objectives of this work are to study how the solid state physics of fracture in
insulators an delineate some of the fundamental influences that solid state
phenomena have on a material’s fracture behavior. The fundamental concept
that I hope to elucidate is to establish relationships existing between the various
structural scales of matter and fracture
Fracture is complicated, wide ranging, technically challenging and
ubiquitous. It is an extremely complicated physical process that takes place far
from equilibrium. The physics is complicated because a crack concentrates
force at quite small distances and couples to the atomic structure. Once the
crack goes critical it grows quickly, while still being coupled to the atomic
structure. The crack has surfaces that increase in time. In insulators the surfaces
are active externally to the environment and internally to the band structure and
to the solid’s polarization.
The microstructure and its relationship to quantum behavior has been difficult
to rationalize for various reasons. One in particular is that fractures are fractal
and this, therefore, presents a substantial problem in dealing with the
relationship between classical and quantum behavior Where does classical
behavior end and quantum behavior begin?. Indeed a crack’s a macroscopic
structure, an observable on the macro scale, is best described statistically by
expected values, for example, <x> and similarly, expected values are at the
heart of quantum mechanics <x>.
Another schism that invades this space is one of dealing with irreversible
physical processes that are finite in contrast to ones that are defined on a
continuum. Limits to integrals are not well defined, and when taking them, they
1

4
can become discontinuous at certain scales requiring that a new set of
boundary conditions be considered.
Some Preliminary Comments
After preparing this work and then reading it several times I realized that
I had presented two different physical viewpoints: classical and quantum. I had
done this purposely , not to confuse the reader but to be honest and to include
viewpoints of both the scientists and engineers. The viewpoints seemed to fall
into- engineer macroscopic-classical, scientist- atomic quantum. I shouldn’t
dwell too much on these classifications because they aren’t rigid. They do,
however, reveal something that’s important about fracture: scaling phenomena
play an important role in it. No longer will it be possible speak of the macro
scale as though it is totally disconnected from the atomic scale.
Fracture’s Role in Our Lives
Someone might think that something like fracture is quite far removed from
their lives. Nevertheless, perhaps they’ve seen signs of them in various places
mostly when things get old and start falling apart. Maybe a sign of neglect and
old age that we’d all like to ignore. I’d have to agree with them to a point, for
the most part fractures aren’t too important in people’s lives, and they’re
passive . However, sometimes they’re far from passive, and then they do play a
role in some people’s lives, threaten them, in fact, they start earth quakes and
Tsunamis. People notice them when they become threatened by them for
example: when an aircraft crashes for mysterious reasons simply because a
crack opened up in the fuselage, or when a ship had to be abandoned because
a crack opened up in one of its bulkheads, or when one the cracked continental
plates began sliding and earth quakes began to occur.
On larger scales, the earth’s surface and the seafloor remain extensively
fractured but most of them are geologically quiet (the cracks are often many
miles long) and planets’ surfaces and their moons are also extensively cracked
showing signs of their history.
So, yes, we probably do notice fractures indirectly when we are threatened
by their causative influences on our lives. By themselves, fractures usually aren’t
a problem they just act as middlemen.
If someone takes the time to look carefully they’ll find evidence of fractures
all around them in: street surfaces, sidewalks, buildings, dishware, windows
there are many of them and most of them are associated with the degradation

September 2015
of our environment. Cracking seems to be a sign of one of Nature’s aging
mechanisms.
Geometry in Fracture
Fractals
Fractures of all types have been found to be fractal i.e. they can be
described by fractal geometry. Fractals can be measured or created in various
ways. A deterministic fractal is shown below:

6
A famous deterministic fractal known as the Koch “Snow Flake” is shown
above. It is constructed from triangles that have a certain size ratio relative to
each other. By placing them at prescribed positions at each step of the
construction a well define fractal construction can be constructed.. Scaling the
sizes of the triangles is determined be the “self similarity ratio”.
A more loosely defined representation of a fractal starts by considering a
space that collapses onto itself. It’s possible to consider it in this manner or that
it’s an operation that an observer with a microscope makes as he/she increases
the magnification as he/she makes his/her observations. In order for these
observations to verify the existence of a fractal they must have the following
mathematical properties:
where n= the number of objects observed
at each magnification
m= the magnification
Df= the fractal dimension that is related
to the self similarity ratio
Additionally, the objects must be self similar (or self affine) in a manner that
can be directly observed or determined to be so mathematically. Statistical
fractals are common in nature but it’s not possible to observe their dimensions
directly. Such statistical fractals are frequently associated with fractures that
show a wide range of scaling behavior. Fractures are simultaneously connected
to: the atomic scale, the nano scale, the micro scale and the macro scale. It is for
this reason that they obey fractal geometry.
nm
−Df
= 1

September 2015
A Preliminary Peek at Fractals, Fourier Space and
Wavenumber Space
Fourier space and the Fourier transform form the position<-> momentum
gateway in quantum mechanics. It is what’s behind the scenes, a supporter of
almost everything that quantum mechanics expresses. It’s possible (and helpful)
to visualize how a fractal, (here the Serpinski carpet) might appear in
wavenumber ( reciprocal, or “k” space)
Constructions such as those based on inverse length or “k” space are
representations of physical operators, namely momentum, that are conjugate to
length. Quite often fractal constructions seem to have a similarity to them.
Shown above is a fractal known as the “Serpinski Carpet” that has eightfold
symmetry that is downsized and repeated to infinity (that is, in physical terms
“dilatational invariance”). The other construction is an adaption of the carpet- a
Seripinski Carpet in k space. The construction is based on sixfold symmetry in
which every point has a reduced copy of double diffraction ( commonly

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observed in electron microscopy, where using the (000) beam for observation a
a diffracted beam (say a (111 ) )is excited becoming a secondary source of
diffraction spots having sixfold symmetry. To be a fractal, however, the
secondary diffraction spacing mast be larger than the primary spacing.
Fractures too have been analyzed and described in k space. I’ve found
that both metals and ceramics have scaling laws that form the relationship
between the crack height amplitudes and wave vector, k that are all of the form:
Y
Yo
⎛
⎝⎜
⎞
⎠⎟
2
=
ko
k
⎛
⎝⎜
⎞
⎠⎟
6−2Df
The fractal dimension determines the observed scaling behavior.
Having the fracture information in the form of a spectrum has been
indispensible when it is used in terms of physics. Later in this paper it will be
used to calculate the fracture entropy of a solid. Using spectra for describing
macro, nano-scale and atomic phenomena facilitates extremely useful insights
to be had when it comes to fracture because there’s a continuity of language.

September 2015
Engineering Mechanics: Consideration of Linear
Elastic Fracture Mechanics on the Macroscopic Scale

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To support the table above in order to maintain clarity:
σ =
KIC
c
φ
where
σ = the yield stress
KIc= the fracture toughness
c= the crack length

September 2015
where G= the work of fracture
ε= the strain
E= the elastic modulus
Materials exhibit various stress-strain behavior upon loading, this is under
the purview of many disciplines and has been for years. Mechanical Engineers,
Materials Scientists, and others as well, study this (and have studied) material
behavior because of its importance to us all.
When a solid is loaded by applying a force the bonds are stretched and a
strain results. There’s reversible elastic deformation and permanent irreversible
plastic deformation. It can be shown that the solid’s atomic bonds are
responsible for what is observed. The energy that’s responsible for what’s
observed is the strain energy density:
It is these measures of the mechanical quantities that can be related to
fracture by
G=
K2
E
1−ν( )
K = σ c
G
c
= Eε2
The determination of these mechanical quantities takes place with equipment
that makes measurements on an extremely large scale relative to the atomic
scale nevertheless, the experimental outcomes are influenced by what
transpires on the atomic scale.
G is also a materials constant for it is a measure of toughness. Toughness is
determined by how much energy is absorbed by a material if it should foster a
crack that’s under load. A material having a high toughness and a high yield
strength is highly desirable. Tough materials like metals typically have a
G~1000-2000Joules/meter2
with yield strengths ~ 300 GPa; G for insulators
(“tough ones”) are a maximum of 50 Joules/meter^2. Metals plastically deform,
stretch and absorb energy, but insulators don’t.
Loading and fracture takes place under non-equilibrium conditions non
adibatically meaning that fracture takes place exchanging heat irreversibly with
ΕSED = Eε2
=
σ 2
2E
=
G
c
=
energy
volume

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2
its surroundings. Determining all of the energy lost during fracture can be a
complicated affair. The input is the strain energy density introduced by an
applied external load to a notched sample of material. The outputs are heat,
sound, light, vibrations, fracture surfaces, disruption of the atomic and meta-
atomic microstructures. Energy is lost in the outputs.
Everything begins with an applied load, a material, support and a notch
either prepared or not. An example of a typical crack (shown below) is
embedded in the microstructure and is acted upon by the applied load.
This was an example of a brittle inclusion particles, MnSiO4 , in a welded
ductile steel matrix. This is an excellent example of ductile/brittle behavior that
is easy to remember.
The crack initiated and failed below the yield stress of the matrix. The matrix
around it held it in place until it too failed at a higher stress starting with the
particle as a nucleus. The Griffith equation (derived later) can be used to obtain
the yield stress. The stress on the particle at fracture was:
σ *
inclusion =
Eγ
πc
γ = G =
K2
E
1−ν( )!
105
( )
2
0.7
2x105
< c >! 0.5 µm ! 1.27x10−4
inches
σ *
= 9,400 psi = 135.8 GPa

September 2015
This was the stress in the ductile matrix that was related to the brittle fracture
of the particle. The strain in the matrix exceeded the fracture strain in the brittle
inclusion.
The Atomic Structure and Force
Since “atomism” is behind all natural phenomena it seems pertinent to
begin by considering a force acting on the atomic scale. Energy vs distance
relationships can be used to determine the cohesive energy of a great many of
atomic structures such as ionic solids i.e. alkali halides . This equation is
valuable since the force displacement relationship for alkali halides can be
determined by differentiating the energy curve.
F(r) =
∂U
∂r
= N
2α
r
−
Ze
−
r
ρ
ρ
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
Shown below is a theoretical force distance curve computed for NaCl an alkali
halide. This was a simple well known structure that will be a member of the set
of materials that will be used in this study.
Figure 1
Identity Spacing/Size Angstroms Comments
Bohr radius 0.529177 A fundamental cut-off
spacing
U r( )= N zλe
−
r
ρ
−
αe2
r
⎛
⎝
⎜
⎞
⎠
⎟

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Max in Force Curve 0.4987 The spacing at the force
maximum
Atom Diameter 0.795 The atom diameter taken
from ad Na /2+ Cl/2
Nearest Neighbor
Spacing
2.820 The NN distance in the
NaCl lattice
Lattice Spacing 5.426 The spacing of the NaCl
lattice
Atom Diameter Na 1.9 The (uncoordinated)
diameter of the Na atom
Atom Diameter Cl 0.79 The (uncoordinated)
diameter of the Cl atom
Evidently the force, as calculated, acts on NaCl but it appears that there
needs to be a statement regarding how it does so because it’s not possible to
identify a unique loading point with any confidence.
There are also some other considerations that need to be taken into account
if a more realistic picture of load transfer between the macro and atomic scale
is to be made.
The arguments, as presented, are appealing and, for the most part, are correct
but they too are incomplete. Since:
1. The presentation is in concepts and not in terms of
materials-for example: if that is a force distance curve
actually applies to atoms how did they prepare and hold
the specimen?
2. Is the model used for E(x) the only one possible? Why use
one over the other. I chose the one that I did because I
intend to study the alkali halides in this work and I thought
that it was correct and relevant.
However, there are others
• The Morse Potential
• The 6 12 Potential
• The Rittner Potential
• The V-S-4 Potential
• The Modified Rittner Potential
• The Gohel-Trivedi Potential
• The V-S-3 Potential

September 2015
And they all have their advantages/disadvantages. Some stem from educated
guesses followed by empirical successes with scientific justifications for their
success. Some are just accurate, tested, have been in use for many years and
are easy to use. However these potential energy curves can be useful and, even
though it might be difficult to get a reasonable answer without a struggle, there
should always be a struggle when it comes to doing good science.
From the Atomic Coordinates to the Laboratory
Coordinates

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This is a load displacement curve (in blue) that is observed
Q The atomic force-distance relationship doesn’t look anything like the load
displacement curve measured in the laboratory as shown above - how are they
related to one another?
Part of the problem is associated with the fact that the force distance is for a
single atom whereas in the laboratory the load is applied over the entire cross
section. You can make an estimate how many atoms this is ~ for a 1 cm2
. Each
atom has it’s own mechanics to follow. The atoms get “out of step “ (lose phase
coherence) and so the lab test measures something of an ensemble average of
things.
Rationalization of Classical Coordinates
In attempt to make things more understandable and somewhat relevant I’ll
consider the atomic forces acting along a line in the plane of the test. I consider
the coordinates along the line as being fixed and move the force curve along
the line and in so doing form a convolution integral i.e.
F = kx = mω2
x
F x − xo( )Δ x( )
xo=0
xm
∫ dx = mω2
x2
= E

September 2015
Setting things up this way makes the right hand side of the equation equal to
the energy (a constant) which makes everything tidy since energy is path
independent. I’ d like to know what Δ is all about because that is what scales us
from the atomic to the laboratory coordinates. By making the integration over
large enough distances I’m doing just that.
I’d like everything to be neat and tidy but it’s not. I wouldn’t have expected it
to be simple, nevertheless, although rudimentary, the mathematical approach is
limited and probably is too much of an oversimplification. Shown below is the
function Δ for ω and x.. It appears to
be a spectrum (of sorts) along the ω =0 axis. As yet it is only an illustration of an
interesting idea. The idea being that the force- distance curve is an incomplete
description when working in terms of the laboratory coordinates.
It appears that the application of the load isn’t very well defined and it needs
further definition. However, it’s not possible to specify the application of a load
at a point on the atomic scale. Not being able to determine exactly where the
load is ap

1
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plied is typical of the shortcomings of classical mechanics. Enter quantum
mechanics that says ...
• “it’s just not possible to determine the load point on the atomic
scale. Even if you reduce the scale somewhat you can only obtain
statistical measures such as the expected value of force <F> and
its standard deviation <σforce>.
• Where you intend to apply the load will depend on the
Uncertainty Principle. Which says, in so many words, no one can
be certain of where the load point is
So the force-distance curve gave us a feeling of confidence and a sense of
reality but it lacks credibility, it cannot be verified because it’s only a theoretical
construct.
Mechanical and Electromagnetic Forces
The force- distance curve doesn’t provide a number of things which need to
be seriously considered if we are to speak in macroscopic terms . I’m sure that
as we work to encompass an even wider range of phenomena in mechanics it
still won’t be enough, because it never is.
Here are some more realizations of reality:
1. If a load is applied to a material (ductile) sooner or later it
will either bend permanently or break. Strain is evident in
the form of plastic deformation and accompanies fracture
2. The force-distance curve shown for NaCl in Figure 1 doesn’t
portray a realistic load displacement curve because the
curve usually does some irreversible and unpredictable
things that are non-linear.
3. Permanent strain and plastic deformation occurs beyond
the elastic limit
4. An electron diffraction pattern of a metal under load (see #3
below ) shows that the crystal structure is intact with some
additional streaking coming from another phase that’s
forming and deformation that’s occurring along preferential
planes.
5. In order to handle a number of small strains the approach is
to use terms that are linear in strain and adopt the edict

September 2015
“add small strains” εi
i
N
∑ but this can only be applied to
“small strains”.
6. Something that is important to remember: strain
determines what is happening to the microstructural
coordinates of a material.
7. The load displacement behavior is not reversible meaning
that if you take a specimen and load it to particular point of
load and strain and stop, record everything, then apply a
new load moving to a new strain point, and record
everything, then, it’s not possible to return to where you
started. A path in load- strain space isn’t reversible.
Basically, you did some work and couldn’t recover it.

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September 2015
This is an interpretation of the above electron diffraction pattern.

2
2
The Plastic Zone

September 2015
Visualization of the macro constituents acting during the formation of a crack.
This should put into perspective in relation to the ductile behavior of metals and
to the brittle behavior of ceramics.
Two equations have been found, one applies to steels that exhibit ductile
fracture by means of inclusions that are ~0,1 µ in diameter and the other
applies to brittle fracture of insulators (ceramics).
Some of the relevant experimental work that has been done to date is
summarized in the following table. Both equations are similar and show that the
work of fracture G is related to some distance that dictates the fracture behavior
in steels or ceramics. Ceramics G values are ~ 1000-5000 ergs/cm^2 and steels
are about 20 times higher than that. The “characteristic length” in ceramics is in
the 5-50 A range and the inclusion spacing on the fracture surface of steels
is~1-10µ . So it would appear that by means of empirical observations that a
(generic, non-descript) spacing of some kind is responsible for the observed G.
These distances emerge due to statistical sampling at the crack tip and
therefore they are best determined to be for steels- an average value:
x = Ns x2
e−Nsx2
∫ dx
and their statistical properties can be measured on a fracture surface.
Nevertheless, these experimental observations do indeed work in certain
applications but they are quite a way from being incorporated into or
establishing some new physical laws because they have not shown to be
associated with the canonical variables of physics. In that respect the canonical
coordinates are momentum and position.

2
4
Material
Conductor/Non-
Conductor
Fracture Equation Structural
Terms
Energy Terms
Steels
Ductile Fracture
(Conductor)
Passoja-Hill
EDuctile = kεvλ
εV ! 5⋅1013 ergs
cm3
The energy term
εV could be
determined from
the fracture surface
measurements
λ =directly
measured
mean
inclusion
spacing on a
fracture
surface
Mechanical
measurements-
The area under the
stress strain curve up to
fracture was measured
and it compared
favorably with the
values as determined
from the fracture
measurements
εV ! 1− 5⋅1013 ergs
cm3
Ceramics
Brittle Fracture
(non-
Conductor)
Mackin-
Mecholsky-
Passoja
G =
1
2
ED*
ao
D*
= the fractal
dimension
increment
E= the elastic
modulus
a is a “characteristic
length”
5A ≤ ao ≤ 50A
ao =not
directly
measured
but inferred
from
experimental
data
Equivalent energy
density would be
related to the strain
energy density at
fracture but it wasn’t
measured

September 2015
Specimen Dynamics of Fracture in a Nanoshell
1. The impulse response of the specimen-detector
assembly before fracture
2. Representation of the collapse of a transverse wave
due to fracture
3. Emission of longitudinal compression waves due to
fracture
4. Reflection of emitted waves off the ends of the
specimen and their interference with other reflected
waves.

2
6
The Physics of Fracture from the Quantum
Scale to the Macro Scale
There’s an enormous amount of published information regarding the
fracture of solids. Fracture phenomena displays wonderfully complicated
behavior and to engage an analysis that tries to cover so many things is
probably not a good idea. A limited view might help this problem but probably
the best approach is a flexible one- one that uses what knowledge is available
and realizes this as a limitation.
I think that it’s safe to say that even today, after all the work that’s been done
on fracture the elusive relationship between a solid’s structure and the fracture
behavior is only partly known. For example, at present, starting from first
principles, it’s not possible to construct the fracture surface of 6061T6
aluminum. This is a practical problem of interest because this alloy is used in
military aircraft. Given that task it would also be useful to predict the fracture
toughness, and the behavior of this alloy in a fatigue test. To put it another way:
There’s an intelligent (hidden) beneficent being in all of us who might say:
• If you plan to fly in that aircraft will the wings fail when
you’re aloft? I mean, honestly, the very wings on the aircraft
that you’re planning to fly in. I’m not interested in any
statistical data (quite obviously) for an explanation because
you should know what’s going to happen for sure.
• Find someone could monitor the most important things, the
ones that would most likely cause the plane to crash when
you’re up!
Even if the wings were taken off the plane and scrutinized it would not be
possible to determine what would happen ...and if we were to use a microscope
that allowed us to see things at the atomic level we’d know even less about
everything (Heisenberg’s Uncertainty Principle) and would have to spend more
and more time searching for some meaningful observations which may or may
not be meaningful.
So the answers to problems such as the “Fracture Problem” certainly can’t
be determined by a limited number of observations made at high

September 2015
magnifications. Furthermore, observational “noise” enters analyses of all
physical problems at many different scales. The noise usually is intrinsic and
can’t be removed in a simple manner.
At present, these things can’t be done, but scientists and engineers are
trying to find these things out because they are important problems.
A Historical Note
Was improved toughness ever an important technological development in
history? Yes, Damascus steel swords come to mind. The metallurgy started in
India and migrated to the Middle East where the fabrication of swords having
superior hardness and toughness were produced. They had a decisive effect in
many battles with the European conquerors of the Crusades.
Insulators and Conductors
This work will be concentrated on insulators and alkali halides (which also
are insulators but have some wonderful and interesting properties in their own
right). I chose to study insulators because of their extremely low electrical
conductivity. Metals fail and absorb a great deal of energy and ehhibit ductile
fracture. Any bond failures in metals are healed by the free electrons. In
comparison a metal’s strain to failure makes insulators look anemic in
comparison. The other property that interested me was that insulators are brittle
and their work of fracture are notoriously low.
As someone with a background in solid state physics it seemed to me that
the differences in the solid state properties could help to explain the fracture
properties of insulators.
The Structure of a Generic Crack
A crack can be described as geometric object that is a surface that forms in a
physical continuum and reacts with the continuum in a number of ways. The
crack owes its existence to the mechanical forces that created it. However, with
a first glimpse a crack isn’t really isn’t a “thing”, it’s not a physical object because
it has no mass.
The mechanics that created the crack operates on scales that can start with
the atomic and can range upward to kilometers.

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That’s really not much of a surprise once you stop and think how fracture
happens. The entire process can be pictured like this: on loading the
mechanical state of an object (yet to obtain a propagating crack) is changed so
that the stored elastic strain energy (from bending, shock loading etc.) reaches
a critical value at some crack somewhere. This begins a cascade of events first
damaging things on a local atomic scale and, as the crack grows, it soon
envelops larger and larger scales. If the crack size is sub critical it may advance
but not freely propagate, but should it ultimately reach criticality, failure and the
formation of the fracture surface will occur. The fractal relationship is present
since the crack is connected on the micro and the atomic scale by means of the
crack front.
The surfaces that form must ultimately separate on the atomic scale after
deformation with distributed bond breaking in the vicinity if the crack tip
damage zone. Indeed the nano scale ↔ atomic scale interaction is an essential
part of the fracture process.

September 2015
An evaluation of the energies, momenta and displacements is in order but
with a realistic set of eigenfunctions including eigenstates that includes
everything involved with a fracture is daunting . One of the major problems that
occur at the onset of fracture is that only a small number of atoms (compared to
the entire volume containing the crack) are responsible for the crack’s behavior,
in other words, a single bond, or a representative of a single bond, does not
explain things but neither does a fairly large number of them. Mathematically
there’s a problem of convergence. The analytical side of things is more like
working with a DFT rather than a Fourier series.
The crack samples ~ 1016
atoms at the crack tip for every atomic extension it
makes. Nevertheless, with some critical thought and by approaching fracture as
being composed of a statistical ensemble of atoms followed by quantum
mechanics and statistical mechanics it’s possible to make some realistic
theoretical statements. Nevertheless a crack is a statistical representation of a
geometric object embedded in a solid. It’s a finite thing that lends itself to
counting what’s there in order to gain its secrets.
In developing theories of fracture that encompass a broad range of
materials and phenomena fracto-emission, light emission, acoustic emission etc
by making the direct observation of the fracture(ed) object and recording its
mechanical properties often ends up short when it is attempted to include all
the contributions of energy. That’s because the list of energy contributions is
usually incomplete. Furthermore, evaluations are usually done after the fact by
studying the fracture surfaces, the “burned out skeletons” of fracture. For
example, in insulators a crack and its surfaces generate fields and dipoles that
perturb the solid in a fundamental way. The fields disappear once the fracture is
completed. Conductors and insulators fracture quite differently and they must
be handled in a different manner. This type of behavior in either case isn’t
included in the classic Griffiths Equation. But the Griffiths Equation is the “Gold
Standard” and it’s the best place to start.
The Griffiths Equation - the “Gold Standard”
Historically, the Griffiths equation forms the “Gold Standard” in fracture
mechanics It has withstood the test of time and even with the fruits coming from
LEFM and the measurements of the toughness, KIc it still creates a valid scientific
framework for understanding fracture.
Historically, fracture has relied on the Griffiths equation for its description. It
has done well for the fracture of homogeneous materials. Fracture toughness
and linear elastic fracture mechanics has superseded and improved on Griffiths

3
0
considerably but it’s still possible to show that the fundamentals of the Griffiths
are still part of LEFM.
I’m sure that someone has incorporated entropy into fracture equations
before but the theory doesn’t seem to be taught this way. So, by using the
Gibbs free energy in the standard manner, stating that there’s a balance
between the surface created (a positive term) and the stored strain energy
density lost (a negative term) that is associated with fracture it’s possible to write
(1
Under the assumption (here) that temperature is constant-(which it’s not!). The
entropy is included, otherwise fracture is presented as a reversible process
which isn’t realistic. The structural terms are included
The Quantum/ Classical Boundary-Phonons
A crack is initiated, it propagates and then it stops usually when the part that
contains it comes apart. Consider an interval when the crack is moving at
steady state velocity (some fraction of the speed of sound). In our case It’s an
insulator so most of it’s behavior is elastic with a logic of the type “it’s either
fractured or it’s not”, The incremental steps leading to fracture are in the
Angstrom range so there’s nothing like plastic flow that would change the
fracture’s logic. To be sure, quantized elastic waves of the lattice, phonons, are
part of this process. As the crack generates them it does so incoherently but it
can also do things coherently if the parasitic resonances of the fixture and other
outside noise sources are suppressed. The crack acts as a phonon source that
isn’t always keeping things in phase, instead the phonons wave shapes are
shifted in time and interfere destructively with each other. Sometimes, that is.
When there’s coherency at the source, there’s a potential for the existence of a
phonon having a quantum number n. That’s related to the energy of the system:
Where υ is ≅ the Debye frequency ≅ 1013
Hz.
Like any other wave a phonon has a frequency and amplitude but it’s just
quantized in terms of n. The running crack emits elastic waves of the form
and it displaces a volume element a distance u of the material. There is ½
kinetic energy and ½ potential energy that is in the volume element. The kinetic
energy density is
ΔEf = −ΔEv + ΔEa − TΔS( )
E = hω n +
1
2
⎛
⎝⎜
⎞
⎠⎟
u = uo cos Kx( )cos ωt( )

September 2015
when it’s time averaged this becomes:
Which somewhat justifies the fact that a classical elastic waves are related to
phonons that are quantized.
where the x terms are the amplitudes squared. This represents a boundary
between classical and quantum mechanical behavior.
Cracks are Fractal
The evidence has been accumulating cracks can be described by fractal
geometry. Whether the size of mountains and coastlines or as small as what
cannot be seen by the unaided eye, “cracks are fractals” as the scientists would
say.
There are many ways to describe the geometry of cracks and it certainly isn’t
simple, so to have arrived at a time when we can have a precise and useful
description of a crack’s geometry is certainly an enormous improvement of what
we had before .
A crack spectrum on the macroscopic scale has a characteristic shape- it has
a high amplitude low frequency with a high frequency low amplitude shape
meaning that it resembles an echo, but one that, as time goes on, you would
hear the whispers of something like Alvin the chipmunk.
KEρ =
1
2
ρ
∂u
∂t
⎛
⎝⎜
⎞
⎠⎟
2
KEρ = !∫
1
2
ρ
∂u
∂t
⎛
⎝⎜
⎞
⎠⎟
2
dV =
1
4
ρVω2
uo
2
sin2
ωt( )
1
8
ρVω2
uo
2
=
1
2
n +
1
2
⎛
⎝⎜
⎞
⎠⎟ !ω
x2
=
h
mω
n +
1
2
⎛
⎝⎜
⎞
⎠⎟ = xq
2
n +
1
2
⎛
⎝⎜
⎞
⎠⎟
x2
x
2 = n +
1
2
⎛
⎝⎜
⎞
⎠⎟

3
2
This figure has some 1dimensional traces of potential crack paths.
The next figure reveals a slightly more sophisticated representation of “crack
path amplitudes”
The lower figure shows a crack and some of the potential paths it could have
taken instead of the one it took. There’s a distribution function to the right of the
crack showing a continuous distribution of crack paths that the crack could have
taken.
This is a familiar problem in statistics and in Statistical Mechanics . The crack
has the potential of making choices of where it ends up. There are N virtual
cracks and k places to put them. The probability distribution is related to the
entropy the probability distribution is shown above.

September 2015
There are N virtual cracks
that fill a proportion of boxes in phase space. The trajectories of the boxes in
phase space reveal the dynamics of the crack’s evolution (and virtual cracks)
mapped on a set of canonical coordinates x and p. The virtual crack paths were
drawn as random sets of straight lines. By inspection it’s possible to see the
relationship between the crack’s tortuosity and entropy.
It is entropy that will allow us to scale between the atomic and the
macroscopic scales. The number of paths that someone might take when
traveling between points A and B is huge if someone cared to scrutinize every
possibility by using an ultra high resolution microscope. At lower magnification
some of the paths disappear (not really!) so choices become more limited,
however, “if your stride widens” so that you can miss the gaps ( or use a ruler
without a fine spacing on it so that you can shorten the path) . There’s entropy
that goes along with this, so our walking along the paths needs to be accounted
for in a different manner than one that’s dependent upon magnification.
There are a few fundamentally important ideas that emerge from arguments
such as these:
1. Entropy is of fundamental importance in understanding
physical behavior in general. Bridging the gap between the
atomic and the macro scales would require that special
statistical considerations be used to determine which
features are the important ones in the set of the countably
infinite lineaments that exist on the atomic scale.
2. Quantum mechanics doesn’t require us to see things like
electrons in order to know that they are everywhere. In fact
it negates the idea of calling a particle an “it”.
3. It is of fundamental importance that magnification be taken
from it’s place of disregard and be established as an
essential idea, hiding, unrecognized, behind all of our
concepts of large and small. It is magnification that puts us
in the driver’s seat as observers. I think that it’s about time
S = pi ln pi
i=1
k
∑
WN p1, p2...pk( )=
N!
p1N( )! p2N( )!... pkN( )!
pk

3
4
that we realize that magnification might be as important as
entropy is.
Griffiths Equation with Phonons and Entropy
By using some of the ideas above it’s possible to make (superficial perhaps,
but it provides a valuable learning experience). Everything’s the same as it has
been with the exception of entropy.
ΔG = −
4π
3
r3 σ 2
2E
⎛
⎝⎜
⎞
⎠⎟ + 4πr2
γ −TΔS
This leads to the standard equation for the critical crack size.
The out-of plane amplitudes that a crack has-i.e. its “roughness” is related to
work of fracture, G for a host of materials-including ceramics, steels aluminum
etc. This “roughness” can be determined analytically and is known as the fractal
dimension.
The spectral method of measurement is able to characterize the out of plane
fluctuations of the cracks. The fluctuations are related to the entropy since a
crack having high toughness would have a high entropy and a crack that was
flat would have low toughness and would have low entropy. These attributes of
the fracture geometry can be used to evaluate the entropy in the following
manner: as derived previously, in the quantum realm, the relationship between
the y amplitudes and the quanta, n is
yo
y
⎛
⎝⎜
⎞
⎠⎟
2
= n +
1
2
⎛
⎝⎜
⎞
⎠⎟
K, the wavevector is related to 1/x an inverse distance- it is an analytical
measurement and not related to the momentum so that
kox
kx
⎛
⎝⎜
⎞
⎠⎟
6−2Df
=
1
xo
1
x
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
6−2Df
=
x
xo
⎛
⎝⎜
⎞
⎠⎟
6−2Df

September 2015
The entropy is
S = P[n]log[P[n]]
n
∑
and using the above equations it is
S = n +
1
2
⎛
⎝⎜
⎞
⎠⎟
x
xo
⎛
⎝⎜
⎞
⎠⎟
6−2Df
⎛
⎝
⎜
⎞
⎠
⎟ log n +
1
2
⎛
⎝⎜
⎞
⎠⎟ +
x
xo
⎛
⎝⎜
⎞
⎠⎟
6−2Df
⎛
⎝
⎜
⎞
⎠
⎟
n
∑
Since this is an equation that is based on a quantized harmonic oscillator, the
terms are summed over n instead of integrated.
Shown below is a 3D plot of the fractal Dimension, the number of quanta
and the energy (in ergs on the y avis)

3
6
A High Value of the Work of Fracture is Dependent on the
Presence of High Frequency Terms in the Fracture’s
Geometry
Electrical Engineers know that there can be a huge amount of power in a
seemingly insignificant electrical signal- the power is in much of low amplitude
high frequency components. So too it might be that high toughness resides in
the low amplitude high frequency spectral components.
A Surface and its Conjugate
A glimpse of a fracture might give the impression that the very things that
created it were in a state of chaos: brittle fracture doesn’t look chaotic to the
eye but it does at high magnification. That a fracture in a “crystalline” material
shouldn’t look chaotic is just a matter of observation. It’s come to our
knowledge that the very things that make fracture appear to be chaotic to the
eye are also operable on the grand scale like the cracks that form and look
chaotic in a glacier.
Unifying the observations such as those above is what fractographers do.
They work with both quantitative and qualitative information hoping to find
some commonality among their observations. As with any other science
classification of structural details is where everyone begins.
Here are a few items involving fracture that are concerned with the nature of
the structure of fracture surfaces.
Q: How well do the matching surfaces fit together after they’ve been
fractured?
A: It’s been found that they don’t. It’s not a good match at all
Q: How does this happen?
A: Fracture takes place on different scales. But structural details might
not match on a large scale just by adding up many small things. Matching
might occur coincidentally on a large scale; with many infinitesimal
displacements coming together and making up the difference in a large
scale displacement. The problem with such an assessment is associated with
phase matching. There doesn’t appear to be coherent reinforcement of the
small scale deformation and displacement on the atomic scale that can be
readily associated with the macroscopic scale.
Fractals has presented the first evidence of scaling of fracture that has the
possibility of bringing the different scales together. The autocorrelation

September 2015
function of the profile shows nothing of merit. The cross spectrum and the cross
correlation function show spectral lines and some periodic structure. This would
indicate that both fractures contain information of interest. Once they’re
separated it’s not possible to find the right conditions (including he right energy
too) to put them back together with perfect matching.
The figures below show some aspects of this problem in an extremely
simple way:

3
8
This is where we all begin; the fracture occurs and two pieces emerge
(normally) and additional surfaces- the fracture surfaces. The newly fractured
surfaces are somehow related to each other. Attempting to fit them back
together isn’t possible. It’s possible to add energy and rejoin them-weld them-
but that would not satisfy our conditions.
There are several ways through the confusion but most of them don’t
include fractography as their basis. This is not because fractography isn’t useful
it’s only part of the answer. The frontier lies at the atomic level.
Quantum Mechanics and the Schroedinger
Equation
Rather than present a lengthy exposition on quantum mechanics I’ll begin
with an abbreviated one that I believe is relevant to fracture. I discovered this
(path) relationship after many tries, some successful, some not. I have the wave
function etc and can compute various quantities etc but I’d rather present
something that’s useful and relevant to this work on fracture. Incidentally, some
of this might appear to be obvious but I didn’t find it to be so. For example, why
not use an altered version of the harmonic oscillator? I tried many of those.

September 2015
Some worked, some didn’t but some of the successful ones presented
unrealistic results.
Some of my ruminations...
How about another totally different type of potential energy?
Which one applies to fracture?
Fracture’s not a harmonic oscillator having “constants of the motion” so
maybe it’s not the SWE at all.
Fracture has a finite lifetime and is irreversible because you can’t fit the
pieces back together after it’s happened. It’s an irreversible work
phenomena.
I finally found something, made a choice, and it seemed to be helpful.
I’m not suggesting that it’s the only choice nor am I suggesting that this is the
only correct choice because fracture’s too complex and others might work just
as well.2
Is there any Connection to be Found Between Quantum Mechanics and the
Griffiths Equation?
Yes, but more work is needed. Here is what I’ve found so far.
The Shroedinger Equation
The Harmonic Oscillator has to be reconsidered in terms of G
The term for the compliance in the harmonic oscillator should be changed in
the following way:
The usual familiar terms involving k, the compliance are:
F = kx
ω =
k
m
k = mω2
(11
2
Γ is a provisional term for G that will be used throughout this work. The work
will attempt to establish a scientific basis for Γ and, at some juncture prove that
it is equal to G.

4
0
dividing the compliance k the numerator by area and multiplying by length
squared leaves everything where it stands except
mω2 α
x2
mω2 α
x2
x2
= Γx2
yields the term, Γ which can be used in the following manner.
Constructing a Hamiltonian that consists of a Coulomb term and a harmonic
oscillator term establishes a useful basis and allows Γ to be found.1
h2
2m
∂2
ψ
∂x2
−
e2
x
ψ + Γx2
ψ + mω2
x2
= Eψ
Γ =
e2
4x3
+ mω2
Γ*
=
27e2
32aBohr
3
+ mω2
and this results in an equation for Γ .
I’ll be using the length form of these equations in this work ie.
Γ x( )=
e2
Ω
+ mω2

September 2015
The Energy
The energy curve has a shallow minimum (for arbitrary constants) that appears
Figure 3 The plot of energy vs distance for the Hamiltonian that was used.
An Equation for G Emerges from this Derivation but
there’s Something Else, Something Unexpected
I have found that the equation
mω2
=
e2
r 3
is associated with quasiparticles. Everything lines up, the units, the order of
magnitudes but there’s nothing that is associated with surfaces when it comes
to quasiparticles and vice versa. Without too much concern I’ll have to
hypothesize that quasiparticles are indeed part of fracture but they just don’t
stay around very long, they escape leaving the fracture behind. The work of
fracture what this term is all about. Not being able to observe the fundamental
particles of Nature in a scattering experiment is common in Physics. The best
example of this is the electron; after all of these years we still can’t observe an
electron.
To put everything simply: the quasiparticles (are at least part) of the
fracture process but they have a short lifetime and they scatter off of various

4
2
things in their environment phonons etc. They lose coherence with their source
quickly and just cease to exist.
After going over some of the numbers in the equations above I’d have to
say that they’re what the fracture story is all about. But rather than have you take
this on faith (which I never do when it comes to Science). I’ll continue on in this
work and reveal some supporting evidence.
iQuasiparticles
Plasmons-are based on the collective oscillations of electrons
Plasmons are oscillations of the free electron gas in conductors but are also
present in semi conductors and insulators. The ion cores do not participate in
this process due to the fact that they are more massive having more inertia and
therefore they aren’t able to follow the electrons’ movements. The lattice ions
are screened by the electrons to maintain stability but the ion masses are quite
large compared to the electrons. They are free to move about but over very
small distances. This is a classical calculation about plasmons using the ion
cores for the masses.

September 2015
This is based on the oscillations of the ion cores and it’s too low.
To use this for plasmons and electrons in an alkali halide NaCl:
o set the terms equal to each other
o use the electron mass
o calculate volume/electron in a unit cell
o Find the frequency of an equivalent plasmon
o Divide by the volume of the unit cell
F = ma
m
d2
x
dt2
= −eEe−iωt
mawω2
= np/ve2
ω =
np/ve2
maw
Using NaCl as an example
maw = 4.683⋅10−23
np/v =
4
5.62⋅10−8
( )
3 = 2.53⋅1022
ω =
2.53⋅1022
2.307⋅10−19
4.683⋅10−23
ω = 1.054 ⋅1013
Γ =
e2
ao
3
+ mωD
2
vs Difference in first Ionization Energy
E = hω = 0.044eV
meω2
=
e2
Ω
ω =
e2
nv
me
= 1.182⋅1015
Hz
E = hω = 4.888eV
ΓNaCl = 2444
ergs
cm2

4
4
There’s an additional, extremely interesting aspect to these expressions.
These expressions are identical to the ones used to describe quasiparticles. The
equation for :
(18
which describes the classical behavior of quasi particles eg. plasmons. The
following figure expresses some general aspects of fracture and the possibility
of its particle like behavior.
Surface Plasmons
During fracturing, the fracture surface atoms lose their restoring Coulomb
field out of the plane. They are driven to oscillate by the fracture’s elastic
waves, its phonons. The surface atoms no longer have a 3 dimensional
support around them and, lacking that, their dipole fields become expanded
out of the fracture plane. In this manner,the dipole ocsillate and generate a
surface plasmon.
Γ
Γ = meω2
+ e2
nv

September 2015
Fracture, Quasi Particle Behavior or both?
Bringing quasiparticles into fracture, needs justification and some
experimental proof. I maintain that our familiarity with fracture is only a
temporary and partial representation of what fracture is all about. A crack’s
behavior in real time created the fracture surface but it’s probably only part of
the story.
Quantum particles and their properties is the purview of quantum
mechanics. Certainly, quantum mechanics has been very successful in a great
number of scientific endeavors. As I suggested at the beginning of this work
it’s sometimes not very easy to transcribe classical mechanics and into the
“Quantum Language” . In particular, phenomena of higher dimensions are
difficult to handle by means of the Shroedinger Equation. We appreciate that
the fracture puzzle has several pieces that must be assembled with care starting
with the most important ones, energy and geometry.
The Crack as a Source of Electrons, Phonons and Surface
Charges
A propagating crack in insulators generates acoustic waves, stress waves,
photons, fractured atomic bonds which cause electrons to enter the conduction
band and holes to appear in the valence band and, perhaps, quasiparticles.
There will be electrons having undefined states ( formed from fractured states)

4
6
within the band itself. The states in the band pose a problem because their
numbers and type are not well defined.
As it propagates, the crack develops an effective mass. This is easy to
understand in the case of insulators. Being unstable, the crack breaks bonds.
Since this is an insulator any disrupted charges will show up on the crack’s
surfaces . These charges interact with other lattice charges by means of a
Coulomb interaction. Elastic loading displaces the lattice atoms from their
equilibrium sites thus generating internal fields. This will cause an internal
charge flow (if possible) so as to eliminate the field. The crack will now interact
with its environment and become more massive.
This is a phonon that has just been initiated by the crack it is portrayed in a
different color in order to distinguish it as being different. It’s easy to see that it
has a wavelike character due to its amplitude but it also has a different solid
state structure too.
A Crack- A Source of Phonons and Electrons in Insulators
Crack Direction
Phonon Direction

September 2015
The involvement of a crack occurs by interactions of electrons with the
uncompensated broken bonds on the crack’s surfaces. Then there are
interactions with the polarization induced charged field from the
uncompensated internal fields of the host material. But all of this stems from the
E r( )= EC + EPE + EPC + EP
c
+ EC−a*
c
+ EC−c*
c

4
8
fracture’s influence on everything and it’s not just the particles alone that
behave in this manner-it’s the particles and, addition, the crack that’s
responsible for this behavior. The crack is therefore a participant. It is an
intermediary for several energy transfers.
The Microstructure of a Crack
Crack’s aren’t “things” but they do develop an effective mass.This is
something that they acquire when they begin to propagate and become
influenced by the complex microstructure. They have trouble growing easily
and have a tendency to oscillate and to leave rough surfaces behind them.
There are scales where a crack’s onset of roughness falls into patterns. Such
patterns can depend on a material, the sample geometry and loading
geometry. These are regions where a crack becomes unstable relative to a
plane. The ratio of the plastic zone size relative to the microstructural unit size
can be related to these transitions.
Another destabilizing influence on the crack is related to the fact that the
fracture ends up with two surfaces-(generally) at every point of its propagation
that is, the crack leaves two surfaces behind (it’s actually just one surface until
separation occurs) that influence its movements.
This is best described by the “Mexican Sombrero” example shown below.
Consider an atom vibrating as a harmonic oscillator: sooner or later it has to
be in one or the other potential wells on either side of the fracture path when
the two surfaces separate. It’s really not just one atom but all the atoms will
undergo similar movements under the influence of the crack’s strain energy
density field. There soon will be two different fields acting separately. In this
case there would be two harmonic oscillator potentials that could be identified
with the crack but joined together as the crack is propagating.

September 2015
These potential energy curves are to be thought of as looking directly at the
crack tip. The dancing ball is to be thought of as an atom that has to make a
decision about which one of the harmonic oscillator PE curves it will be in once
the crack has passed. This is part of a calculation known as the “anharmonic
oscillator”.

5
0
The “Particle in The Tight Crack Tip”
An extremely narrow space, such as the one that exists at the tip of a crack at
the onset of propagation, should give rise to quantum resonances. It is known
that confinement of energy on the atomic scale gives rise to such resonances.
The simplest one to envision is the particle in a box. Shown below is a depiction
of this idea with the energy levels shifted in order to account for the crack’s
being wedge shaped.
The resonance is a source of an electric field. Since this is an insulator, the
crack sets up a field that ranges outward into the solid and polarizes the atoms
throughout it. The solid attempts to eliminate the field ...I’ll leave this where it is
because there’s more on this later.

September 2015
Phonon Scattering Off of the Crack Tip
The crack generates phonons as it moves and interacts with its environment.
The emitted phonons can react both globally and locally in the following way
1. Because the object that contains them is finite and their
lifetime is finite they make a limited number of traversals
within the specimen before they are dissipated
2. During their lifetime they interact with electrons and holes
that have resulted from fracture and reside, somewhat
immobile, on the surfaces of the crack. Any field that the
fracture creates in the volume of the solid emerges on the
solid’s surfaces.
3. Complex electron->electric field->phonon->strain occur
during fracture causing the crack tip to undergo some
significant changes. Charge oscillations and strain
oscillations similar to Friedel oscillations form and alter the
fundamental structure of the crack tip.

5
2
Statistical Mechanics
The use of statistical mechanics to determine the fate of bonds in the vicinity
of the crack tip is based on three assumptions:
• that the number of bonds that fail at any one instant are
small,
• that their lifetime out of equilibrium is short
• that they lose their identity in the ensemble after they’ve
failed.
A crack’s influence takes place a small distance ahead of it. There bonds are
broken or, alternatively, the material is overloaded below the applied stress
due to the crack tip singularity. A unit fracture process that is large relative to
the atomic spacing would be dependent on the averaging of statistical
fluctuations on the atomic scale.
There must be a transition region where some of the bonds have started to
fail but have not done so as yet. In this region it makes no sense to speak of a
single bond that is responsible for the fracture “event” any more than it should
be called an “event” because these things are under the purview of statistical
mechanics and quantum mechanics where such phenomena are statistical in
nature.

September 2015
Analyzing a crack is usually done with a certain scale in mind (as shown
above). Should it be deemed necessary the volume of analysis can be adjusted.
It’s very useful to establish the number of atoms that a straight crack touches
in a bar of silicon that’s 10 centimeters wide with (as above) an analytical area
50 ao silicon:
Item Number of atoms
Plane of silicon 6.78x1017
Crack tip 7.13x1015
The mean atomic spacing at the crack front is ~ 1.184 A. This is presented as a
reference.
Essential Aspects of Bond Failures
Lattice Construction
Atoms , at first are isolated but experience interactions which change the
symmetries and energies of their wave functions. As more atoms become
involved broadening and overlap of the wave functions occurs. Due to filling of
the states the interatomic distances decrease. At a certain point it is
energetically favorable to have the entire process emerge as a condensate- a
periodic soild. This indicates a well defined lattice spacing and a build-up of the
crystalline structure. With a further decrease in this distance, energy bands
form. The presence of periodicity gives rise to energy gaps which interfere with
electron propagation
Fracture a Deconstruction
Deconstruction of a crystal is a far more complicated process because there
are far more choices that can be made by an atom in order to make the
deconstruction- in other words the entropy is quite high in this process. The
figure below summarizes the

5
4
The Distribution of Broken Bonds at a Crack Front
The energy of bond failure will be discussed later in this work. Until then
and particularly in this section, failed bonds are simply considered to be
no bonds or bonds. A distribution (sequence) along a line taken perpendicular
to the fracture plane and perpendicular to the crack front might appear as
Broken Bonds............ Broken/Unbroken Bonds..........Unbroken Bonds
00000000000000101011011110111101110101011101111111111
In the following figure I’ve included a distribution of stretched bonds in
order to appreciate the complexity of the statistics at the crack front.

September 2015
The width of the transition region will be considered to be a result of a
balance between crack speed and the bond failure rate. This has never been
determined so the width that I used was based on an assumption.
The bluish part of the figure below is where everything begins (no bonds
broken) and the reddish area is where all the bonds are broken. For this reason
it’s the cumulative probability distribution that describes the distribution of
bond breaking.
There are two different ways to approach this problem. One is in the purview
of statistical mechanics that takes a broad approach and scrutinizes everything
as being part of a thermodynamic system. The other is just a statistical counting
method that could be plugged into the statistical mechanics construct if was
desired. I wanted to keep these two methods separate so that the statistics
could be clearly understood.
Statistical Mechanics
Following the figure below assume that there are two bond energies
broken, Ub and unbroken Uu. Let the energy of the system be U then U/ε atoms

5
6
are unbroken and (N-U/ε) are broken.The number of ways of choosing U/ε
atoms from the total number N is
Ω =
N!
U
ε
⎛
⎝⎜
⎞
⎠⎟! N −
U
ε
⎛
⎝⎜
⎞
⎠⎟!
S = kB lnΩ = kB ln N!( )− ln
U
ε
⎛
⎝⎜
⎞
⎠⎟!− ln N −
U
ε
⎛
⎝⎜
⎞
⎠⎟!
⎡
⎣
⎢
⎤
⎦
⎥
⎛
⎝⎜
⎞
⎠⎟
S =
U
ε
− N
⎛
⎝⎜
⎞
⎠⎟ kB ln 1−
U
Nε
⎛
⎝⎜
⎞
⎠⎟ −
U
ε
kB ln
U
Nε
⎛
⎝⎜
⎞
⎠⎟
F = U −TS
F = U −T [ kB
U
ε
− N
⎛
⎝⎜
⎞
⎠⎟ ln 1−
U
Nε
⎛
⎝⎜
⎞
⎠⎟ − kB
U
ε
ln
U
Nε
⎛
⎝⎜
⎞
⎠⎟ ]
The energy of the system is U =
Nε
1+ e
ε
kBT
This probability is the same as the Fermi Dirac probability distribution.
The Binomial Distribution
The distribution function for the figure below is:
The coefficients are of the Binomial distribution function.
The statistics that I’ve introduced is that of the Binomial distribution. Its
applicable to bond breaking in the zone around the crack front ~ 50 ao (ao is
the lattice parameter) . It should apply to brittle ceramics.
In order to see what the binomial distribution offers this type of problem, an
expansion to the third power of ↓ ↑ 0 in other words the downward arrow
represents a failed bond, the upward arrow is one that has also failed and the
zero is one that has survived. The following expression is an excellent way to
visualize everything ( with z=0)
z+ ↑ + ↓( )
3
=↑↑↑ +3↑↑ z + 3↓↑↑ +3↑ z2
+ 6 ↑↓ z + 3↓↓↑ +z3
+ 3↓ z2
+ 3↓↓ z+ ↓↓↓
p=probability of success
q=probability of failure
n!
k! n − k( )!
pk
qn−k

September 2015
Where z=0 and the terms that survive are:
0+ ↑ + ↓( )
3
=↑↑↑ +3↓↑↑ +3↓↓↑ + ↓↓↓
It’s easy to show that the coefficients are the binomial coefficients and can be
found in the third row of Pascal’s triangle. It’s quite interesting that the arrows
replace the term’s powers.
Pascal’s Triangle
There are several advantages of using Pascal’s triangle along with the
Binomial distribution.
1. Entropy and the binomial coefficients can all be found on Pascal’s
triangle since
row
column
⎛
⎝⎜
⎞
⎠⎟ =
r!
c! r − c( )!
. This is the way almost every
entropy problem is developed.
2. The value of the row sums on the triangle is 2n
= 2 2m
−1( )
n=1
m
∑ for
any m, so this is a cumulative term. Choosing an m is like
integrating to the limit, m.

5
8
1. The maximum value of the even numbered rows on the triangle is
2n!( )2
n!
this is a most probable value. Usually in statistical
mechanics problems it’s not possible to include all the states in an
ensemble so what’s used is the most probable value of a
distribution instead. With this maximum term on the Triangle the
most probable value is ready made for you.
2. Another interesting about the triangle is that it is robust. It is
capable of handling some amazing numbers. The rows of the
triangle are all 2Row
and if you’ve ever worked with the Triangle
you seldom have cause to go much beyond 28
or so, simply for
practical reasons. A power of 2 looks innocent nothing suprising
220
, 240
well now you’re getting into computer managed territory.
How about 279
? how big can that be? Looks innocent... It’s so close
to Avagadro’s number that it’s just unbelievable 1% or so I think.
279
just doesn’t look that large!
This is a Binomial distribution for a sample size of 25 and for a lattice spacing of
one Angstrom.

September 2015
This is the cumulative distribution for the above probability distribution. It is a
map of the broken bonds from the region in front of the crack tip. This is only a
model calculation made to see how things work.
The mean value of a binomial distribution is <x>=np where n = the sample
size and p = the probability of a success on a draw.
In brittle materials it has been found that the following equation applies:
γ =
1
2
ED*
ao
where:
γ= the work of fracture
E= the elastic modulus
For a binomial distribution the average value of x is
<x>=np where n is the sample size and p is the probability of success.
In terms of this model,
ao= the characteristic distance
ao=n alattice
n= the sample size
If the distribution is binomial then
<a>=nalattice ( n=probability of a broken bond)
So it is n that is related to the observed value of <a> on a statistical basis.

6
0
Another Example of Size Distributions
The above figure shows there can be three different microstructural fracture
unit distributions. In this case fracture occurs in the vicinity around the crack tip.
I’ve included three different probability curves just to make the problem
interesting.
There’s also another assumption that brings out some nice aspects of the
problem: the cracks that compose the crack “atmosphere” depend upon the
strain energy density. That sets a position on the probability curves which have
to be integrated from d*
upward in order to get the number of fracture units
that will collaborate in the “main event”
Other Forms of Fracture
Fracture is a very complex process and it takes on many forms. Indeed
figures below represent uncommon and difficult fractures that are commonly
seen in ceramics and high strength metal alloys. They can also be seen in
fractures that take place in foams or porous metals or molecular sieves. All the
failures have something in common -they failed by fracturing- but in the
following example the fracture wa

September 2015
s wonderfully complex:

6
2
The ancillary figures show how the crack profile was deconstructed in order
perform an analysis of strain vs position. This is an unwrapping technique had
to be used in order to flatten the edges. The next figure shows more
unwrapping with the detachment of the microstructural units. There were still
some atoms out of place ( in red) that had to be dealt with. Our goal is shown in
the last figure. Once the structure is flattened and unwrapped it is embedded
with circles which are the basis set. The difference between the circle and the
microstructure is then digitized entered into a computer and with an FFT a
spectrum is found. These techniques are often difficult and challenging
because unlike crystallography, many of the analyses aren’t ordered.

September 2015

6
4
Insulators and Polarization
Fracture takes place in a lattice of atoms surrounded by
electrons some are free, others are not
Increasing the strain energy density applied to an object will cause a crack
within it to approach criticality. The influential (positive) energy inputs to the
crack comes in several forms: increased atomic vibrational frequencies and
amplitudes, scattering of Fermi electrons off the periodic lattice, changes in the
ground state energy due to elastic strain, scattering of phonons and electrons
off of the crack tip ...
Decreasing this heightened strain energy state comes from crack advance:
bond breaking and the formation of new surface (the change in the system’s
energy from the surface is positive but it’s released by having the surface grow)
, new atomic states on the surfaces, changes in the vibrational frequencies,
scattering of the Fermi electrons off of the new surfaces, phonon emission and
phonon-electron interaction (polarons), field disruption and dipole interactions,
surface disruption from field changes on the crack’s surfaces. In light of the
previous paragraph I’d say “you gain some energy and you lose some energy”
there are a variety of different paths to be taken and some are more probable
than others.
The Dipole Moment
When an insulator, a polar material, is subject to an applied field changes
occur in its charge distribution. The modern theory of polarization considers
such changes to occur from the adiabatic current flow coming from
deformation or some other modification of the crystal. Fracture most likely can
be part of this. Borrowing on some essentials of the “modern theory”, where
possible, I’ll try to establish how fracture might fit into this picture.
I’ll state two of the most important problems for trying to fit the fracture of
insulators into the picture
1. The modern theory requires that there be an
adiabatic path in order to evaluate its most
important term the Berry Phase, in other words,
everything happens slowly
2. Everything happens under null field conditions

September 2015
Even with these limitations the modern theory offers an excellent chance to
fit fracture into the picture because it clarifies some of the physics involved.
Fracture separates the bulk of a crystal and introduces new surfaces. The
polarization that ensues would be expected to be:
Pf =
1
Vf
1
dr rρ1
f −1
∫ r( )+
1
Vf
2
dr rρ2 r( )
f −2
∫
This is just a formal statement that the fracture, f , separates the crystal into two
parts. Continuity conditions.
The fracture introduces two new surfaces with surface charge densities:
−Δσ on the left and Δσ on the right. There’s a charge build-up on the surfaces
and that can be ascribed to the amplitudes of the wavefunctions with the
current being ascribed to the phase of the wave functions This changes the
polarization but it isn’t a change identified by Pf.. It can, in all probability, be
identified with fracture.
In order to establish the role of the fracture surface let there be an integral
over a volume element η on the surface
Pη =
1
Vη
dr
η
!∫ rρ r( )
This term is far from exact but is, most likely,
Pη
These issues indicate that they fall outside the definition of a sample’s bulk
polarization. The way this problem was solved for a crack free sample was to
consider the change in ΔPSample that occurs due to an outside influence. Apply a
field and measure the difference in polarization after the field is switched off. It
is the difference in the ground state wave functions that is behind this behavior.
Evidently, fracture would cause changes ΔP1 and ΔP2 . The charge buildup
that occurs on the surfaces due to fracture must now be considered as part of
the change in polarization of the bulk electronic structure
It is the charge flow during fracture that is important and that is
ΔPf = dt
1
Vη
∫ dr j r,t( )
η
∫

6
6
The instantaneous change in polarization is
dP t( )
dt
= j t( )
ΔP = P t( )− P 0( )= dt j t( )
0
Δt
∫
At this point the new theory adopts an adiabatic limit letting j go to zero and
Δt → ∞ . Certainly the influence of fracture could indeed approach these limits.
After fracturing, the pieces would be protected and allowed to remain at rest for
a given period of time.
In the theory a dimensionless time is introduced for convience:
ΔP = dλ
dP
dλ0
1
∫
In order to maintain
ΔP = P Δt( )− P 0( )
which, basically establishes the definition of polarization. It is time dependent
as is fracture. In order to maintain stability there must be a flow of current
through the sample and it is for this reason that polarization is considered to be
a bulk phenomena.
Fracture in Insulators
The following comments on polarization and how it relates to fracture will
help to follow the next few sections of this paper:
1. Charge screening of fracture at initiation. during propagation and
in its final state are important in insulators.
2. The charge build up on the fracture surfaces of insulators makes
them environmentally sensitive
3. The high band gaps found in insulators makes intrinsic electron
compensation limited.
4. The surface and bulk charges are intertwined by time dependence
- the surface state being influenced by charge i.e.the amplitude of
the electron’s wave functions and the bulk being influenced by
electron current, the phase of the electron’s wavefunctions.

September 2015
5. Berry’s phase plays an important role in the fundamental
understanding of polarization of insulators, however, it’s not clear
that it will be helpful in fracture
6. The changes in the charge distribution brought about by fracture
undergoes changes followed by relaxation into a static
polarization configuration owing to the reorganization of the
ground state wavefunctions. This is a fundamental process that
applies to both insulators and conductors. The polarization
difference is related to this.
Dipole Moments and Strain
A polarizable insulator will remain in an equilibrium state until it is exposed
to an electrical field or to a mechanical load. Being polarizable means that a
solid will respond to an electrical field by having its core charges shifted. The
protons shift a little and the electrons shift more giving rise to a charge dipole
that aligns itself with the electrical field. The fundamental Laws of
Electromagnetism are in play here since any internal fields introduced by the
electrical field will be eliminated by carrier redistribution i.e. electrons will act to
remove any internal fields. Now polarizable insulators can have band gaps ~10-
15 eV meaning that they can’t have very much internal carrier flow at room
temperature. But researchers inject carriers in order to get the properties that
they want.
This discussion will be concerned with intrinsic carrier compensation at
room temperature, mechanical loading and fracture.
The reason that there has to be elimination of any internal fields is that there
can’t be any internal unsupported carrier current, there will be current flowing
to eliminate the fields.
An external applied field aligns the dipoles. With its removal the structure
relaxes back into an equilibrium state. The expected values of the dipoles’
properties reach zero. They can be changed again but there’s no way to
determine a starting point only an end point.
to a strain the spacing between its atoms change. Everything appears as elastic
strain until fracture occurs. When the spacing between the bonds change the
equilibrium dipole distribution acts in response to this changing environment .
They will do so as long as there is current flow within the structure, otherwise,
their response is limited and they will act to oppose the applied stress. The <E>
of an insulator in equilibrium is zero otherwise there would be current flow.

6
8
Removing the material from this capacitor will leave charged surfaces. This is
the nature of a crack in insulators.
The crack introduces internal fields that need to be eliminated on grounds
that are based on the Laws of Electromagnetism. The internal field can be
eliminated by internal current flow however, an excess of charge carriers are not
to be expected in an insulator that might have ~ 10 eV band gap or so.. The
presence of internal fields will cause a small internal current to flow but
probably not enough to eliminate the fields. The uncompensated fields will
respond to the applied load as reactive forces acting against it.

September 2015
Should the developing reactive forces increase to extremely high levels and
the internal fields remain uncompensated they can be eliminated by
propagating a crack . This would allow the dipole distribution to return to
equilibrium and to eliminate the internal fields.
Should the developing reactive forces increase to extremely high levels and
the internal fields remain uncompensated they can be eliminated by
propagating a crack . This would allow the dipole distribution to return to
equilibrium and to eliminate the internal fields.
Before the load was applied the nuclei were relaxed throughout the entire
solid, relaxed in some collective equilibrium state and the internal field was
equal to 0. Upon loading the stress altered the atomic structure which caused
internal changes in the field. Some charge migration occurred but due to the
limited carrier concentration in insulators only so much elimination of the field
could occur. Once the internal current flow stops but the applied load
continues to rise, the internal field will become reactive and resist the applied
load.

7
0
Statistical Physics of Bond Failure in Polar Insulators
In polar insulators the dipole distribution without any outside extrinsic
influence is in some equilibrium state. Upon loading, the physical structure of a
material is strained and the material undergoes changes that disrupt the
internal field. Fundamentally, this will soon be mitigated by internal current flow
since internal fields can’t be maintained in a polar material.
As was mentioned before, the addition of an external field will cause a
change in the atomic charge distribution, basically, small changes in the atoms’
electron and nuclear charge distributions will occur thus creating a dipole. The
other way to create dipoles is to distort and to break bonds as what happens at
a fracture surface. The figure below is an example of what is to be expected in
the vicinity of a crack face. The electric fields are shown in red lines. It’s possible
to find the average spacing for the array of atoms. However, wherever there’s a
spacing that’s deviated from the average such as those shown in yellow, a
dipole has developed, in fact whenever an atomic displacement occurs like
those in the vicinity of a crack an extraordinary dipole is formed . This doesn’t
require piezoelectric properties to be present for their existence because the
bonds that have been affected have been destabilized and are far from
equilibrium. In other words, the bonds that are strained an extraordinary
amount have become dipoles and will interact with their environment
differently than they did before. The yellow “atoms” in the figure are to be
associated with the anions and the white “atoms” are to be associated with the
cations. It’s not expected for them to interact in the same way due to size
differences and electronegativity differences .

September 2015
That bonds become stretched and broken in the vicinity of a crack is not
surprising. Nevertheless it’s useful to explore the occurrence of bond breaking
and field alterations in order to understand some of the the crack’s properties.
It’s important to realize that the fracture itself becomes an active partner
influencing the fundamental electronic structure of the material . Both of these
depictions show different things.

7
2
1. The first one is an idealialized crack tip for which strains can
be calculated and ( most likely a periodic Bloch
wavefunction could be merged with the crack tip singularity
in k space.
2. The second one is a distorted structure, derived from,
perhaps, 6 well -defined units which don’t have any
repetitious behavior i.e.brittle fracture of insulators exhibit
very few high frequency structures having long range
correlations.
3. It’s possible to first understand things on a more
fundamental scale, that is, quantum mechanically but
chemically too. What I mean is that ( which is not to say that
chemists don’t use quantum mechanics!!) it would be nice
to have some practical rules that just made sense when it
came to fracture. Indeed we have some but I’d like to have
more! Tune in for the next section of this paper!!
A more de-magnified image of the crack, idealized of course, showing field
lines (in red) where bonds have broken and dipoles have formed irreversibly
and arrows in green showing atomic shifts, dipoles once again, but with a
different origin than the ones that originated on the broken bonds.

September 2015
Mathematics Concerning the Crack
The equations that describe the stress field around the crack have a
singularity at a crack length =0. Essentially the equations of interest are:
G
c
= Eε2
This equation is in terms of
G
c
= Eε2 Energy
Volume
This singularity can be evaluated in the Complex − Discrete plane in the
following manner:
1. First evaluate the region around the crack so that the crack is isolated this
is important only insofar as it isolates the crack.
2. Next the Bloch periodic function u(x) describing the lattice potential must
be considered. It is present as an ancillary function in the complex plane.
3. Integrating around the crack in the complex plane yields:
f z( )dz!∫ = 2πi0 = 0
4. Integrating within the crack so as to include the singularity yields =G2πi
5. The atoms within the crack have been removed so they can’t influence
their neighbors, however, their neighbors will have an influence on them.
The Bloch function must be changed to reflect this. Therefore there are
virtual atoms within the crack.
6. Integrating within in crack using virtual atomic fields yields ( there are n
atoms in the crack (in the figure n=6 for example)
7. Since zz = x
using real coordinates within the crack and letting the virtual atoms’
potentials be replaced by delta functions.
Let g z( )= δ(x)
Then gn z( )
n
∑!∫ dz = δ x( )
n
∑
δn x( )
n
∑ =
n
∑ e−ikx
0
∞
∫
U k( )=
n
∑ ∫ un x( )e−ikx
dx
The crack's opening is Δτ

7
4
making the exponential factor = Δτkx
u x( )∫ e−iΔτkx
dx = U k( )
This is a Fourier transform of u(x). Which is a one dimensional k space
representation of the “Bloch function”. Its properties include broadening in the
x direction with decreasing crack length. The
G
c
function maps as a flat straight
line in k space . It shouldn’t be a problem to bring Bloch’s function into the
analysis in order to find out what influence lattice periodicity has on the stored
energy per unit volume.
The upper figure is a representation of a periodic structure and the bottom
figure is a representation of a cracked plate in the complex plane.
This development was needed because the analysis of Berry’s phase takes
place in k space and also due to the fact that polarization is quantized.

September 2015
This is a K space representation of all the relevant physical dimensions of the
fracture problem. The coordinates are scaled on a log basis so that all of the
dimensions can be represented in a compact form. The circular figure on the
right is an expanded view of things with 1/(crack size) and 1/(specimen size)
shown in the middle. The ordered purple circles represent the 1/(lattice
constant) of the solid being fractured.
Berry’s Phase
Berry’s phase is a quantum mechanical analysis that takes place in k space
for determining the geometric influence on wavefunctions. The concept has
awakened an enormous amount of interest and has found Michael Berry with a
Nobel Prize
There was a great deal of confusion about the polarization of insulating
solids. In the last 10 years things have changed for the better. The Claussius-
Mossotti equation, while still being used, has taken a back seat to solid state
physics and the definition of a quantum of polarization.
Starting with the Bloch form of eigenfunctions:
ψ nk = eikr
unk r( )
Hk unk = Enk ψ nk
getting down to the Shroedinger equation.
λ is a convenient parameter that takes the place of time. After much
development the equation that emerges that is related to the polarization is:

7
6
dP
dλ
=
ie
2π( )3 dk ∇kunk ∂λ unk∫ + C
Peff = ΔPion +[Pel (1)− Pel (0)]
This shows that the polarization has electron and ion core contributions.
Pel λ( )=
e
2π( )3 Im ∫ dk unk ∇k unk
n
∑
this equation is at the heart of Berry’s phase . It is the central result of the
modern theory of polarization. The integral takes place over a closed manifold-
the Brillouin zone, in k space. The sum is done on a discrete mesh of k points
that span the Brillouin zone. The operator ∇k is a derivative taken in k space.
Pn =
1
2π
e
Ω
φn, j Rj
j
∑
Where Rj and Gj are the real space translation vector that corresponds to the
reciprocal lattice vector G.
φn = Imln un,kj
un,kj+1
j=0
M −1
∏
Since this is an imaginary term it refers to time also. The subscripts indicate
that it’s discrete time and it’s only well defined as
MOD 2π
which leads to the conclusion that
Pn
! = Pn +
eR
Ω
since R is a lattice constant. The polarization as defined results in the definition
of a polarization quantum. The quantization comes about due to the MOD term
as it depends on the lattice constant R.
ΔP ! Pλ=1 − Pλ=0( ) MOD
eR
Ω

THE FRACTURE-QUASIPARTICLE CONNECTION

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