# Critical crack-length during fracture

Viswakannan R. K<EMAIL_ADDRESS>Subhadeep Roy
<EMAIL_ADDRESS>Department of Physics, Birla Institute
of Technology and Science Pilani, Hyderabad Campus, Secunderabad 500078,
Telangana, India.

###### Abstract

Through controlled numerical simulations in a one dimensional fiber bundle
model with local stress concentration, we established an inverse correlation
between the strength of the material and the cracks which grow inside it -
both the maximum crack and the one that set in instability within the system,
defined to be the critical crack. Through Pearson correlation function as well
as probabilistic study of individual configurations, we found that the maximum
and the critical crack often differ from each other unless the disorder
strength is extremely low. A phase diagram on the plane of disorder vs system
size demarcates between the regions where the largest crack is the most
vulnerable one and where they differ from each other but still shows moderate
correlation.

Disorder plays a crucial role during the failure of heterogeneous media, a
topic that has been studied extensively in last few decades arcangelis78 ;
duxbury86 ; herrmann90 ; pradhan10 . The existence of defects like micro-
cracks and the interactions among them makes the process of crack propagation
more complicated than Griffith’s theory, which was suggested by A. A. Griffith
in 1921 and reported the critical stress of a homogeneous media with a pre-
existing crack of length $l$ to vary as $1/\sqrt{l}$. For heterogeneous media,
on the other hand, Griffith’s criterion produces significant error in
determining the critical stress or surface energy due to the resistance in the
form of an energy barrier that ultimately arrests a propagating crack, widely
known as the lattice trapping or intrinsic crack resistance bernstein03 ;
mattoni05 ; perez00 . Two length scales are observed to emerge as a result of
such lattice trapping: a small length scale related to the dissipation of
energy near the crack tip and the a large length scale associated with the
elastic deformation around the tip bernstein03 ; mattoni05 ; curtin00 ; long21
. Even for perfectly brittle materials, one needs to take into account the
discrete atomistic nature of the interactions and make modifications in
Griffith’s theory broberg99 . A correlation between crack length and nominal
stress is very important for damage control as it can provide necessary
information regarding an upcoming catastrophic failure. Application of such
prediction ranges from laboratory experiments to large scale building blocks
and even geological scales like seismic events wu18 . The micro and meso-scale
heterogeneity not only affects the nominal stress but also the course of
failure by introducing local breaking events, known as avalanches, producing
crackling noises which can be captured in an acoustic emission (AE) experiment
baro13 . The AE process includes the translation of the crackling noises
(during a crack propagation) into bursts and subsequently emitted energies,
known as the acoustic signals or precursors. Such precursors become more
populated in an accelerating manner roy23  as one approaches the global
failure indicated by the increasing rate of deformation petri94  through
gradual accumulation of damages ramos13 .

A classic work discussing the propagation of smaller crack vs larger crack has
been published by Paris in 1963 where the cracks bigger than a critical length
were observed to follow the Paris-law and not the others paris63 . This is
followed by works of Kitagawa and Takahashi kitagawa76  where a scale-free
decay of critical stress for larger crack lengths, similar to Griffith’s law,
was observed with a crack-length independent critical stress for smaller
cracks haddad80 . Later, Taylor taylor81  showed the existence of two length
scales in the work by Kitagawa and Takahashi. A similar existence of length
scales are observed roy22  numerically as well by one of the author of the
present paper in a statistical disorder system, the fiber bundle model
book_fbm , acted by a tensile force in presence of local stress concentration.

Figure 1: (a) The status of a fiber, broken or intact, is represented by white
and black color respectively against the fiber index. (b) The heat map for the
local stress profile - blue color stands for lower local stress while the
yellow for relatively higher stress. (c)-(d) Local stress and threshold values
as a function of fiber indices. The vertical line corresponds to the fiber
that breaks and set instability within the system. We kept $\delta=0.4$ and
$L=10^{2}$.

This paper focuses on the numerical study of a disordered system, the fiber
bundle model book_fbm , as a prototype for failure dynamics in heterogeneous
media. The model consists of $L$ vertical fibers attached between two
horizontal soft clamps pulled apart by a stress $F$, exerting a stress
$\sigma=F/L$ on each fiber. Each fiber has an unique threshold stress chosen
from a uniform distribution spanning from $0.5-\delta$ to $0.5+\delta$,
$\delta$ is the strength of disorder and $0.5$ is the mean of the
distribution. Being stretched beyond this threshold, a fiber breaks
irreversibly and the stress carried by the broken fiber is redistributed among
its neighbouring fibers as per the following rule:

$\displaystyle\sigma_{r}\rightarrow\sigma_{r}+\displaystyle\frac{\sigma_{b}d_{l}}{d_{r}+d_{l}}$
$\displaystyle\sigma_{l}\rightarrow\sigma_{l}+\displaystyle\frac{\sigma_{b}d_{r}}{d_{r}+d_{l}}$
(1)

Here, $\sigma_{b}$ is the stress of the broken fiber, $\sigma_{r}$ and
$\sigma_{l}$ are respectively the local stress of the right and left nearest
neighbor. The stress distribution is made distance dependent to eliminate any
memory/history dependence in the process. We have used the minimum image
convention to include the boundary effect due to the periodic boundary
condition and calculate the actual distances $d_{l}$ and $d_{r}$. Such a local
stress redistribution is a result of the soft membrane which is supporting the
fibers and mimics the nature of stress localization in an elastic media hui13
. The redistribution can induce further rupture events, starting an avalanche,
due to the local stress enhancement until the next threshold is beyond the
redistributed stress. The external force, at this moment, is increased in a
quasi-static manner to break the next weakest fiber and the model evolves
through a number of stress increment and avalanches until all fibers break
suggesting the global failure. The final value of externally applied stress
just before global failure is the critical stress or strength of the bundle.
The critical crack-length is the size of the crack in the last stable
configuration (just before global failure) which propagates and set in
instability starting the final avalanche and hence breaking the rest of the
bundle. The critical crack is one among a number of micro-cracks and
statistically the most vulnerable one. This does not guarantee that this
critical crack will be maximum in length. We will discuss this in details
next.

Figure 2: Comparison between local stress and threshold values at the notches
for the (a) maximum and (b) critical crack. The figures show 100 different
configurations for $L=10^{3}$ and $\delta=0.4$. The ellipse shows a particular
configuration for which the critical and maximum cracks are the same one.

Here we will discuss whether there exists a relationship between the nominal
stress, the stress at which the heterogeneous system breaks, and the critical
crack that initiates the final avalanche. Let’s call the former $\sigma_{c}$
and latter $l_{c}$. The maximum crack length, on the other hand is denoted by
$l_{m}$ and in principle it can be different from $l_{c}$. Through detailed
numerical simulation in a 1d fiber bundle model of size $L$ and disorder
strength $\delta$, we have explored the nature of critical crack length
$l_{c}$ and maximum crack length $l_{m}$ and how they respond as we change
either of the above two parameters, $L$ or $\delta$. To be specific, whether
there is a scope of future failure prediction associated with it. In gist, we
pose a two-fold question here:

  * •

Does the length of the most vulnerable crack contain any information regarding
the nominal stress and vice versa?

  * •

Is the most vulnerable crack always the largest one or does it depends on the
material properties?

A boundary on the $L-\delta$ plane shows the region where the maximum and the
vulnerable cracks are interlinked and where they are mutually exclusive.

Figure 1 shows the micro-cracks which are developed within the 1d chain just
before the global failure. We chose $\delta=0.4$ and $L=10^{2}$. The smaller
system size is adopted to make the micro-cracks more visible. Later we have
used higher system sizes for the numerical simulation. This diagram will be
used to establish the fact that due to local stress concentration, the local
stress to threshold difference at the notches of the micro-cracks is mutually
exclusive from the size of the cracks itself. Figure 1(a) shows the broken and
the intact fibers by white and black color respectively. The biggest white
patch stands for the maximum crack $l_{m}$, which is of length 4 (fiber
indices 74 to 77) for the present configuration. The critical crack, $l_{c}$,
for the same configuration is of length 2 (fiber indices 9 & 10) on the other
hand. The local stress profile for the same is shown in figure 1(b) - lower to
higher local stress as we go from blue to yellow colors. The red vertical line
represents the fiber (fiber index 11) that breaks and set in instability. The
fact that, in-spite of having higher notch stresses ($=0.401639$) around the
maximum crack, it does not propagate, is really counter intuitive and
monitoring the maximum crack can be highly misleading for failure prediction
and damage control. The critical crack propagates in-spite of having a lower
notch stresses ($=0.295405$).

Figure 3: (a) Decreasing trend of $\langle l_{c}\rangle$ and $\langle
l_{m}\rangle$ with critical stress $\langle\sigma_{c}\rangle$ for $L=10^{4}$
and varying disorder. While $\langle l_{m}\rangle$ does not show a particular
trend, $\langle l_{c}\rangle$ shows the following behavior: $\langle
l_{c}\rangle\sim\langle\sigma_{c}\rangle^{-\eta}$, where $\eta\approx 10$. (b)
For $\delta=0.4$ and varying system sizes, $\langle l_{c}\rangle$ remains
constant w.r.t $\langle\sigma_{c}\rangle$ while $\langle
l_{m}\rangle\sim\langle\sigma_{c}\rangle^{-1}$.

This is due to the interplay between the local stress and threshold values of
individual fibers. Figure 1(d) shows that the threshold values at the notches
of the maximum and critical crack are $0.632846$ and $0.298248$ respectively.
Due to this, the local stress to threshold difference around the maximum crack
($0.231207$) is much higher than that of the critical crack ($0.002843$),
making the critical crack much more prone to propagation. Such a stress to
threshold comparison is more evident from figure 2 where the notch stresses
$\sigma_{n}$ and threshold values $t_{n}$ are plotted against $l_{m}$ (see
figure 2a) and $l_{c}$ (see figure 2b) for 100 different configurations. It is
clear that for most of the configurations, $t_{n}>\sigma_{n}$ for $l_{m}$
ranging between 0 to 12 with higher values more populated. On the other hand,
we observe an overlap between $t_{n}$ and $\sigma_{n}$ independent of the
values of $l_{c}$, which spans from 0 to 10 and populated towards the lower
values. We have circled a particular configuration where the critical and
maximum crack is the same one, though such configurations are very rare to
find at a moderate disorder ($\delta=0.4$).

Figure 4: Correlation between average critical crack $\langle l_{c}\rangle$
and average maximum crack $\langle l_{m}\rangle$ for different system sizes
and disorder strength. We observe different behaviors at small and large
length scales. (a) For small scale: $\langle l_{m}\rangle\sim\log\langle
l_{c}\rangle$. (b) For large scale: $\langle l_{m}\rangle$ grows linearly with
$\langle l_{c}\rangle$. The color gradient is on the disorder strength while
point size for the size of the system which varies from $10^{3}$ (smallest
circles) to $5\times 10^{4}$ (largest circles). Figure 5: (a) Probability
$P^{\ast}$ that $l_{c}=l_{m}$ on the $\delta-L$ plane. (b)-(c) The I-II and
II-III phase boundaries are zoomed in for better visibility (d) $P^{\ast}$ vs
$\delta$ for system sizes ranging between $10^{3}$ and $4\times 10^{4}$.
(a)-(d) shows 3 distinct regions: (I) $P^{\ast}=1$ independent of both
$\delta$ and $L$; (II) $P^{\ast}$ is a decreasing function of $\delta$ without
a L dependence; (III) $P^{\ast}$ is a constant independent of $\delta$ with a
slight $L$-dependence. (e) System size effect of $P_{s}$, the saturation of
$P^{\ast}$ at high disorder: $P_{s}=\displaystyle\frac{1.8}{\log L}-0.07$.

Figure 3 shows the correlation between the average critical stress
$\langle\sigma_{c}\rangle$ and cracks, both critical and maximum, developed
within the 1d chain. We keep the system size constant at $L=10^{4}$ and vary
the disorder strength $\delta$ from 0 to 0.5. The critical stress is observed
to decrease with increasing length of critical crack or the maximum crack.
Such reduction of critical stress is observed earlier in real systems like
mild steels under s periodic load taylor84 . In the fiber bundle model, the
critical stress was observed to fall in a same scale-free manner with the
length of the pre-existing crack roy22 . The present scenario is much
different as we are not starting with a pre-existing crack hence the process
includes both crack initiation and propagation through the dynamics of the
model only.

Figure 3(a) registered the critical crack $\langle l_{c}\rangle$ and the
maximum crack $\langle l_{m}\rangle$ as a function of
$\langle\sigma_{c}\rangle$ when we tune the disorder strength $\delta$ keeping
the size of the bundle fixed at $L=10^{4}$. While both $\langle l_{m}\rangle$
and $\langle l_{c}\rangle$ decreases with $\langle\sigma_{c}\rangle$ as
$\delta$ decreases, $\langle l_{c}\rangle$ only follows the trend below as
move towards lower disorder strength:

$\displaystyle\langle l_{c}\rangle\sim\langle\sigma_{c}\rangle^{-\eta}$ (2)

with $\eta\approx 10$. Both $\langle l_{c}\rangle$, and $\langle l_{m}\rangle$
decreases with decreasing $\delta$ as the failure process becomes more and
more brittle-like where the bundle breaks abruptly at a relatively higher
stress. On the other hand, if we keep the disorder strength constant ($=0.5$)
and vary the system size between $10^{3}$ and $5\times 10^{4}$, $\langle
l_{c}\rangle$ remains constant independent of the critical stress. At the same
time, $\langle l_{m}\rangle$ changes as follows:

$\displaystyle\langle l_{m}\rangle\sim\langle\sigma_{c}\rangle^{-1}$ (3)

The maximum crack grows with the system size as a higher system size will
allow the thresholds come closer to each other as per the weakest link of
chain theory ray85  and at the same time increases the density of the strong
as well as weak fibers. The weakest link of chain will allow a crack to
propagate and the higher density of stronger fibers increases the chance of a
large crack getting arrested as a result of higher threshold value relative to
notch stress. The critical crack-length remains same independent of the
critical stress as the critical crack becomes unstable since it did not
encountered a high enough threshold but still experiences the weakest-link-of-
chain event. As system size increases the critical stress decreases but a
crack-length of almost a similar size is proven to be enough to create
instability at the notches.

Next we turn to understand how the two crack lengths, critical and maximum,
are correlated to each other. For this, we have adopted three different ways:
(i) comparing the average values (over $10^{4}$ realizations) of $l_{m}$ and
$l_{c}$ denoted by $\langle l_{m}\rangle$ and $\langle l_{c}\rangle$
respectively, (ii) probabilistic approach for individual realizations and the
$l_{m}$ and $l_{c}$ we find in each realization, and (iii) through a Pearson
correlation function. We will discuss these three approaches one by one next.

Figure 4(a) and (b) shows the correlation between $\langle l_{c}\rangle$ and
$\langle l_{m}\rangle$ both at a smaller as well as at a larger scale. The
color gradient represents the disorder strength and the size of the points
stand for the system size - larger the points higher the size of the bundle.
We have used system sizes ranging in between $10^{3}$ (smallest circles in
figure 4) and $5\times 10^{4}$ (largest circles in figure 4). For small
lengths of both critical and maximum cracks, we observe the following
logarithmic dependence at a sufficiently larger system size

$\displaystyle\langle l_{m}\rangle=a\log\langle l_{c}\rangle+b$ (4)

where $a=0.7$ and $b=6.9$. At a larger scale, on the other hand, such
logarithmic dependence vanishes and instead we observe a linear behavior
between $\langle l_{m}\rangle$ and $\langle l_{c}\rangle$

$\displaystyle\langle l_{m}\rangle=c\langle l_{c}\rangle+d$ (5)

where $c$ increases from 1.9 to 2.3 as system size is increased from $10^{3}$
to $4\times 10^{4}$. For the same increment in $L$, $d$ increases from 1.9 to
5.5.

Next, we study the correlation between $l_{c}$ and $l_{m}$ from a
probabilistic approach in order to draw a phase diagram on the $L-\delta$
plane. This is represented in figure 5. We define $P^{\ast}$ as the
probability that the critical crack and the maximum crack are the same. In
other word this is the probability that the instability within the system is
created by the maximum crack. Fig 5(a) shows the heat map of $P^{\ast}$ on the
$L-\delta$ plane where 5(d) explicitly shows how $P^{\ast}$ varies with
$\delta$ for system sizes ranging between $10^{3}$ and $4\times 10^{4}$.
Specifically, we observe the following three distinct regions.

  * •

Region I: In this region, $P^{\ast}=1$ making $l_{m}$ equals to $l_{c}$ for
each and every configuration. This is an extremely brittle region where the
first fiber initiate global failure each and every time. This boundary between
I and II remains constant at 0.1 independent of the size $L$ of the bundle
(see figure 5b).

  * •

Region II: The probability of $l_{m}$ being equal to $l_{c}$ is less than 1
here and a decreasing function of disorder strength, but do not respond to the
change in system sizes. The boundary, unlike the boundary between I and II, is
dependent on system size - for higher $L$, we have to go to a higher disorder
strength to enter region III. This makes sense since increasing $L$ makes the
failure process more abrupt and the disorder strength has to be increases to
compensate for that.

  * •

Region III: In this region, $P^{\ast}$ reaches its minimum and saturates
afterwards at a constant value $P_{s}$ independent of the disorder strength.
Contrary to region II, here $P^{\ast}$ is a decreasing function of system
size. Figure 5(e) shows the following scaling as we approach the thermodynamic
limit:

$\displaystyle P_{s}=\displaystyle\frac{1.8}{\log L}-0.07$ (6)

The constant value being closer to zero suggest that $P_{s}$ becomes zero as
we approach $L\rightarrow\infty$. This means in the thermodynamic limit, if we
are at region III, we will not be able to find any configuration where the
instability is set on by the maximum crack length.

The final thrust to the correlation study will be calculating the Pearson
correlation function directly as we tune both $\delta$ and $L$ over $10^{3}$
realizations for each set of ($\delta$, $L$). The correlation function defined
below gives us an even better idea of how strong or weak the correlation is in
between critical and maximum crack for individual realizations and not after
taking the average. This way it can be more relatable to the experiments since
a single experiment can be considered as a single realization in our
simulation. The correlation function has the following form:

$\displaystyle
c_{p}=\displaystyle\frac{\displaystyle\sum(x_{i}-\bar{x})(y_{i}-\bar{y})}{\left[\displaystyle\sum(x_{i}-\bar{x})^{2}\displaystyle\sum(y_{i}-\bar{y})^{2}\right]^{1/2}}$
(7)

where $x_{i}$ and $y_{i}$ values represent $l_{c}$ and $l_{m}$ for individual
configurations while $\bar{x}$ and $\bar{y}$ are their average values $\langle
l_{c}\rangle$ and $\langle l_{m}\rangle$.

Figure 6: Pearson correlation coefficient $c_{p}$ as a function of disorder
strength $\delta$ for system sizes ranging from $10^{3}$ to $4\times 10^{4}$.
The inset shows the result for $L=5\times 10^{3}$ with three distinct regions:
(A) constant low correlation, (B) correlation increasing with disorder, and
(C) constant high correlation.

The results are shown in figure 6. The Pearson coefficient $c_{p}$ is an
increasing function of $\delta$. This makes sense since for high $\delta$, the
failure process happens through a number of avalanches giving us enough
information to correlation between $l_{c}$ and $l_{m}$. At a low $\delta$, on
the other hand, the failure process is much more abrupt (like brittle
material) and there are less chances for such prediction. Because of the same
reason, with increasing system size, as the failure process becomes more
abrupt, $c_{p}$ decreases. In the inset of the same figure, we divided the
whole region for $\delta$ into the following three parts:

  * •

Region $A$: $c_{p}$ is almost constant at a low value independent of $\delta$.
$l_{c}$ and $l_{m}$ are loosely correlated in this region.

  * •

Region $B$: $c_{p}$ increases with $\delta$.

  * •

Region $C$: $c_{p}$ remains constant but at a higher value closer to 0.4. The
correlation between $l_{c}$ and $l_{m}$ is decent here.

In figure 7, we have extensively discussed how the region $C$ is equivalent to
the region III in figure 5. Also, the region II in figure 5 can be divided
between $A$ and $B$ if we consider both the nature of probability $P^{\ast}$
and correlation function $c_{p}$.

Figure 7: $L-\delta$ plane showing different regions depending of the
interplay between $l_{c}$ and $l_{m}$.
Region I: $l_{c}=l_{m}$ with a probability 1. $c_{p}$ is undefined here as
there is no fluctuation in the realizations.
Region IIA: combination of region II and region A, making $P^{\ast}$ a
decreasing function of $\delta$ but $c_{p}$ constant at a low value.
Region IIB: combination of region II and region B, making $P^{\ast}$ a
decreasing function but $c_{p}$ an increasing function of $\delta$.
Region III or C: III and C are equivalent giving $P^{\ast}$ constant at a low
value or $c_{p}$ constant at a high value. The boundary of IIB and III (or C)
is drawn from both the study of $P^{\ast}$ (red solid circles) as well as
$c_{p}$ (black hollow triangle).

Figure 7 shows a detailed phase diagram on the disorder vs system size plane
by considering the contributions of both the probability $P^{\ast}$ as well as
the correlation function. This splits the whole $L-\delta$ plane in four
regions: I, IIA, IIB and III (or C). We have discussed all four regions below
with extreme detail. We have also provided the description in a tabular form
below.

Region I | Region IIA | Region IIB | Region III/C
---|---|---|---
$P^{\ast}$ constant at 1 $c_{p}$ can not be defined |  $P^{\ast}$ decreases with $\delta$ $c_{p}$ remains at a constant low value |  $P^{\ast}$ decreases with $\delta$ $c_{p}$ increases with $\delta$ |  $P^{\ast}$ remains at a constant low value $c_{p}$ remains at a constant high value
Table 1: Table showing charectaristic behaviour of the regions: I, IIA, IIB
and III (or C).

We observe region I when the disorder strength is extremely low. In this
region the maximum stress always set in the instability. Not only that
$l_{m}=0$ here, which means the instability sets in from any random point in
the bundle and not from a crack tip. This is due to the low disorder which
makes the failure process extremely fast and the model reaches the global
failure even before the local stress concentration acts in. At a moderate
disorder we find region II which is again divided in two parts IIA and IIB. In
IIA, with increasing $\delta$, $P^{\ast}$ decreases making $l_{m}$ and $l_{c}$
more and more mutually exclusive. At the same time, $l_{m}$ and $l_{c}$ are
loosely correlated here giving a low value of $c_{p}$ independent of $\delta$.
In IIB, $P^{\ast}$ shows the same behavior as IIA but $c_{p}$ gradually
increases here with $\delta$. Finally, in III, $P^{\ast}$ saturates at a low
value independent of $\delta$. Here almost for no realization the maximum
crack is responsible for the instability and $l_{c}$ is most of the time
different than $l_{m}$. At the same time, $c_{p}\approx 0.4$ making $l_{c}$
and $l_{m}$ more than moderately correlated in this region. We want to stress
the fact here that ${\bf III}$ and $C$ are same region on $L-\delta$ plane.
This is evident from the boundary drawn between IIB and III (or C) from the
study of $P^{\ast}$ (red solid circles) as well as $c_{p}$ (black hollow
triangle) and they almost fall on each other.

The present study deals with the fact that during the failure process of a
disordered system, it is not sufficient to monitor the largest crack in order
to predict the instability in the system as often it might not be the
vulnerable one and the instability in the system can be initiated from some
other part, making failure prediction and damage control much more tricky than
it already is. For proper failure prediction, both the knowledge of maximum
and critical crack (the most vulnerable one) will be required simultaneously.
In the present paper, we have numerically studied a fiber bundle model in one
dimension with a varying disorder strength and system size. An inverse
correlation is observed between the strength of the disordered media and two
crack lengths which are prominent through during the failure process -
critical crack length $l_{c}$ and the maximum crack length $l_{m}$. At the
same time, the average $\langle l_{m}\rangle$ maximum crack and $\langle
l_{c}\rangle$ of critical crack are correlated with each other linearly at a
larger length scale and in a logarithmic way for a shorter length scale. Such
a correlation between $\langle l_{c}\rangle$ and $\langle l_{m}\rangle$ on the
$L-\delta$ plane, shows three distinct region. For low disorder, where the
failure process is extremely abrupt, we get $l_{c}=l_{m}$ with unit
probability and with an undefined correlation. With increasing disorder
strength, $P^{\ast}$ decreases and $c_{p}$, the correlation between critical
and maximum crack, increases. In this limit, the maximum and the critical
crack becomes mutually exclusive and the chance that the final trigger comes
from the maximum crack decreases. At the same time, $l_{m}$ and $l_{c}$
becomes moderately correlated ($c_{p}\approx 0.4$ at higher disorder strength)
and we can extract the information about one crack length from the other one
with higher accuracy. The future direction to this work can be a controlled
laboratory experiment monitoring both critical and maximum crack and
correlation between them and use them for real life failure prediction and
damage control.

Viswakannan R. K. thanks Birla Institute of Technology and Science for the
support provided during the work.

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