# Acoustic emission data based modelling of fracture of glassy polymer

Subrat Senapati<EMAIL_ADDRESS>Department of Applied Mechanics,
Indian Institute of Technology Madras, Chennai-600036, India  Anuradha
Banerjee<EMAIL_ADDRESS>Department of Applied Mechanics, Indian Institute
of Technology Madras, Chennai-600036, India  R.Rajesh<EMAIL_ADDRESS>The
Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai-600113,
India  Homi Bhabha National Institute, Training School Complex, Anushakti
Nagar, Mumbai-400094, India

###### Abstract

Acoustic emission (AE) activity data resulting from the fracture processes of
brittle materials is valuable real time information regarding the evolving
state of damage in the material. Here, through a combined experimental and
computational study we explore the possibility of utilising the statistical
signatures of AE activity data for characterisation of disorder parameter in
simulation of tensile fracture of epoxy based polymer. For simulations we use
a square random spring network model with quasi-brittle spring behaviour and a
normally distributed failure strain threshold. We show that the disorder
characteristics while have marginal effect on the power law exponent of the
avalanche size distribution, are strongly correlated with the waiting time
interval between consecutive record breaking avalanches as well as the total
number of records. This sensitivity to disorder is exploited in estimating the
disorder parameter suitable for the experiments on tensile failure of epoxy
based polymer. The disorder parameter is estimated assuming equivalence
between the amplitude distribution of AE data and avalanche size distribution
of the simulations. The chosen disorder parameter is shown to well reproduce
the failure characteristics in terms of the peak load of the macroscopic
response, the power-law behaviour with avalanche dominated fracture type as
well as realistic fracture paths.

##  I Introduction

Brittle materials, such as concrete, wood, bone, coal etc., have inherent
disorder in the form of micro-cracks, crystal defects, micro-voids,
inclusions, micro-structural variation etc. Under mechanical stress, the
inherent disorder often results in random independent nucleation of damage at
multiple sites that subsequently exhibit correlated intermittent growth
leading to rapid final failure. Acoustic signals, elastic stress waves emitted
by fracture processes, are valuable real-time information about the evolution
of damage within the material from its initial state to its final collapse or
failure [1, 2, 3, 4]. While this information has been qualitatively widely
utilised in identification of type of fracture mechanisms, their onset and
progression, in both brittle as well as ductile materials, [5, 6, 7, 8] the
quantitative analyses of acoustic emission (AE) data have focussed primarily
on establishing the existence of scale free behaviour and the associated
power-law statistics [9, 10, 11, 12]. Predictive indicators from analyses of
experimental AE data are vital in life assessment of existing structures [13,
14, 15], however, utilising AE data for design and development of robust
structural systems requires predictive fracture models that incorporate
disorder by utilising AE data in characterisation of its model parameters.

Stochastic discrete fracture models such as fibre bundle model [16, 17, 18],
random fuse model [19, 20, 21], beam bending model [22, 23, 24, 25], random
spring network model (RSNM) [26, 27, 28] and other discrete element models
[14, 29, 30] have been shown to be effective in reproducing several
statistical signatures observed in AE data of fracture experiments. Disorder
in these models has been typically included by taking threshold for failure to
be statistically distributed [31, 32, 33, 34], wider the distribution, higher
the disorder. The avalanche of damage observed in the models, follows scale
free power law statistics, as observed in experimental AE data [26, 21, 35,
36]. It has also been shown that the disorder plays a significant role in the
type of fracture: nucleation type for low disorder, avalanche type for
moderate disorder and percolation type in the limit of high disorder [33, 37,
38]. Apart from statistical signatures, discrete models have also shown to
reproduce experimental macroscopic response as well as realistic fracture
paths [38, 39, 40], however, the disorder parameter was assumed on the basis
of comparison with features of the macroscopic stress-strain response.

More recently, AE data has also been analysed with the objective of finding
statistical signatures of imminent collapse or final failure. The power-law
exponent characterising the avalanche distribution has been shown to have a
crossover between off-critical and near-critical regime both in experiments
[13, 41] as well as simulations based on fibre bundle model [42], random fuse
model [42], a discrete element model of sedimenting spheres [14], as well as
spring network model [43]. The crossover in exponents has been seen to
correspond to the change in the spatial location of AE events at different
stages of deformation [9, 10, 44, 45]. Initial stages have randomly
distributed independent AE event sources that result in larger exponent in the
size distribution. In the approach to final failure, the location of AE
sources are concentrated in a band or process zone and are spatially
correlated, the associated avalanche distribution exponent is lower in
comparison [41]. Indications of imminent failure are also evident in the
statistics of record breaking avalanches, record becomes those avalanches that
have the highest amplitude compared to all previous avalanches of the time
series. Record breaking avalanches form a subset of events of the complete
time series of the avalanche size. Statistics of record breaking avalanches
reveal that near final failure the record avalanches occur rapidly. The
resulting reduction in waiting time interval between two consecutive record
breaking events is a strong indicator of imminent collapse [13, 41, 14, 46,
47, 48, 43]. While real time AE data is useful in life assessment and
prediction of failure of an existing structure, it has not been utilised as an
input for predictive fracture modelling in the design and development stage of
structures.

In tensile fracture simulations of a representative material, is it possible
to estimate the disorder characteristics by utilising the statistical
signatures of AE data? In the present work, we address this question through a
combined computational and experimental study of the fracture of an epoxy-
based polymer. We perform fracture simulations using a square random spring
network model where the spring behaviour is taken to be quasi-brittle, with
linear and non-linear regimes, and a normally distributed failure strain
threshold. Through a parametric study, we identify the sensitivity of the
power law exponent of the avalanche size distribution, the waiting time
interval between consecutive record avalanches, as well as the total number of
records to the disorder characteristics of the springs. This sensitivity to
disorder is then exploited in estimating the disorder parameter suitable for
reproducing experimental data from the tensile failure of epoxy-based polymer
assuming equivalence between the amplitude distribution of AE data and
avalanche size distribution of the simulations. The effectiveness of the
chosen disorder parameter is demonstrated by showing the simulations reproduce
the experimental failure characteristics in terms of the peak load of the
macroscopic response, the power-law behaviour with avalanche-dominated
fracture type, and realistic fracture paths.

##  II Experiment

###  II.1 Specimen preparation

Figure 1: The dimensions (in mm) of the mould used for preparing the dog-bone
specimen.

(a)

(b)

Figure 2: Experimental setup showing (a) the sensors attached to the gauge
section of the specimen and (b) the AE acquisition apparatus.

Dog-bone specimens are prepared from epoxy based resins using silicon rubber
moulds with dimensions as shown in Fig. 1. Initially, the LY556 (Bisphenol A
Diglycidyl Ether, DGEBA) resin and the HY951 (Triethylene tetraamine, TETA)
curing agent are poured in a beaker at a ratio of 10:1 by weight. The solution
is mixed for eight minutes using hand mixing method followed by a magnetic
stirrer and then allowed to settle for two minutes. The epoxy solution is
slowly poured into the mould, which has an inbuilt riser to compensate for
shrinkage. The filled epoxy is cured in two stages; in the pre-curing stage,
the mould is kept at room temperature for 3 hours, followed by curing in an
oven at $60^{\circ}$C for 8 hours. The cured specimens are polished with 800
and 1000-grit grade sandpapers. The specimens are moulded in three batches
[B1, B3, B4]. Finally, aluminium tabs are pasted on the upper and lower
portions of the specimen to avoid local crushing at the grippet regions.

###  II.2 Mechanical testing

The dog-bone specimens are uniaxially tested by a servo-hydraulic type fatigue
testing machine (DMG group) in displacement control mode at a rate of 0.1
mm/min. The crackling signals emitted by the specimens during the tensile test
are captured by two piezoelectric sensors, as shown in Fig. 2. We apply grease
on the sensor surface to maintain good acoustic coupling between the surfaces
of the sensor and specimen. The sensors (R15$\alpha$), manufactured by
Physical Acoustic Corporation (PAC), USA, operate between 50-400 kHz and at a
resonant frequency of 150 kHz. Two 1283 AE nodes (frequency responses between
1kHz to 1MHz) acquire the AE signals in real-time, and an inbuilt pre-
amplifier amplifies the signals by 40 dB. The amplitude of these signals is
computed in volts using the AEwin software. Before starting the test, the
surface coupling between the sensor and specimen surface is checked and the
attenuation in AE signals is minimised by repeated pencil lead break tests
[49]. The noises from the surroundings and the machine set-up are filtered by
setting a threshold of 31 dB.

(a)

(b)

Figure 3: (a) Response of a typical quasi-brittle material, and (b) stress-
strain response from the experiments.

During the experiment, the stress-strain data and the time-synced AE
parameters are collected from the fatigue machine and AEwin software,
respectively. A typical quasi-brittle stress-strain behaviour of such
materials is shown in Fig. 3(a), and the actual stress-strain responses of the
specimens are in Fig. 3(b). The deviations from linear elastic response of the
specimens are well within the range reported in the literature [50], the
average strength of the specimens is approximately $57$ MPa and the observed
variations in the strengths are expected due to variations in inherent defects
across the specimens.

Figure 4: (a) Schematic representation of a dog-bone specimen, (b) gauge
section of the specimen is discretised with a square lattice spring network,
and (c) the connectivity with neighbours of each lattice point.

##  III Model and details of simulation

In this section, we describe the structure and working principles of the
random spring network model, followed by the characterisation of quasi-brittle
behaviour exhibited by the material. Finally, we will estimate the model
parameters for reproducing the constitutive behaviour and failure pattern of
the epoxy-based glassy polymers.

###  III.1 Model description

In RSNM, we discretise the domain as highlighted in Fig. 4(a) with a square
lattice, as shown in Fig. 4(b). Each lattice point ($p$) in the network is
connected with its nearest neighbours ($q$) and diagonal neighbours ($r$) with
extensional springs. The lattice point also interacts with a set of two
adjacent neighbours by torsional spring (Fig. 4(c)).

The spring constants are determined by equating the strain energy density of
the continuum with the potential energy density of the spring network [51].
The potential energy $\mathrm{\Phi}$ stored in the spring network due to
elastic deformation has contributions from both the extensional
$\mathrm{\Phi_{ext}}$ and torsional $\mathrm{\Phi_{rot}}$ springs:

$\mathrm{\Phi=\Phi_{ext}+\Phi_{rot}}.$ (1)

The net potential energy stored accounting for the eight extensional springs
of every lattice site is

$\mathrm{\Phi_{ext}}=\sum_{\left\langle
ij\right\rangle}\frac{1}{2}k_{ij}\left(|\vec{r}_{i}-\vec{r}_{j}|-a_{ij}\right)^{2},$
(2)

where $\vec{r}_{i}$ and $\vec{r}_{j}$ are the position vectors of points $i$,
and $j$, $\left\langle\right\rangle$ denotes all pairs connected by extension
springs and $a_{ij}$, $k_{ij}$ are the undeformed length and stiffness of the
extensional spring connecting them. The net potential energy stored in the
torsional springs is

$\mathrm{\Phi_{rot}}=\sum_{\left\langle
qpr\right\rangle}\frac{1}{2}c_{qpr}\left(\theta_{qpr}-\frac{\pi}{4}\right)^{2},$
(3)

where $c_{qpr}$ is the stiffness of the torsional spring and for an undeformed
lattice, the angle subtended between $q$,$p$ and $r$,
$\theta_{qpr}=\frac{\pi}{4}$. The summation $\langle qpr\rangle$ is over all
possible combinations of the triads in the lattice. The potential energy
density $\mathrm{\Phi}$ of the lattice is equated with the strain energy
density of the continuum to evaluate the elastic parameters, Young’s modulus
$E$ and Poisson’s ratio $\nu$ in terms of lattice parameters as [51]:

$\displaystyle E=\frac{8k\left(k+\frac{c}{a^{2}}\right)}{3k+\frac{c}{a^{2}}},$
(4)
$\displaystyle\nu=\frac{\left(k-\frac{c}{a^{2}}\right)}{3k+\frac{c}{a^{2}}},$
(5)

where $c=c_{qpr}$, and $k$ is the stiffness of diagonal springs. The resulting
stiffness of both vertical and horizontal springs would be $2k$.

When the network is deformed, the positions of the particles are updated using
Newton’s equation of motion to achieve static equilibrium. The resultant force
on a particle $p$ due to its interactions with its neighbours is given by

$\vec{a}_{p}=-\nabla_{\vec{r}_{p}}\phi,$ (6)

where mass is set to unity. To achive equilibrium as well as to dampen excess
oscillations, an extra dissipative term $-\gamma\vec{v}_{p}$ is also added.
Once $\vec{a}_{p}$ is known, the updated position vector of the lattice point
$p$ at time $t+\delta t$ is computed by the Verlet algorithm, using the last
two position vectors $\vec{r}_{p}(t)$, $\vec{r}_{p}(t-\delta t)$ and
$\vec{a}_{p}$ as:

$\vec{r}_{p}(t+\delta t)=\vec{r}_{p}(t)\left(2-\gamma\delta
t\right)-\vec{r}_{p}(t-\delta t)\left(1-\gamma\delta
t\right)+\vec{a}_{p}(\delta t)^{2},$ (7)

where the $\vec{v}_{p}$ is evaluated using the backward difference formulae
$\vec{v}_{p}(t)=(r(t)-r(t-\delta t))/\delta t+O(\delta t)$. Within a loading
step, Eq.$\left(\ref{eq7}\right)$ is iterated till the net kinetic energy of
the system goes below a predefined threshold. Equilibrium is re-verified by
ensuring that the magnitude of the net force at the top and bottom rows are
equal within numerical error. If any spring breaks at the end of the
equilibrium step, then the system is re-equilibrated. This iterative process
continues within the loading step until no more springs break for the applied
increment in strain.

###  III.2 Modelling the constitutive behaviour

The constitutive behaviour of the epoxy based polymer is quasi-brittle and has
two distinct regimes, as shown earlier in Fig. 3(b). In the model we take the
response to be linear elastic upto the elastic limit, $\sigma_{y}$, and beyond
that the response is taken to be quasi-brittle. The response of the material
is modelled assuming a power-law strain hardening constitutive relation,

$\begin{split}\epsilon&=\frac{\sigma}{E},\sigma\leq\sigma_{y},\\\
&=\frac{\sigma_{y}}{E}\left(\frac{\sigma}{\sigma_{y}}\right)^{\frac{1}{n}},\sigma>\sigma_{y},\end{split}$
(8)

where $E$ is the elastic modulus, $\sigma_{y}$ is the elastic limit and $n$ is
the power-law hardening exponent. For the spring behaviour, in the linear
regime $(\epsilon\leq\epsilon_{y})$, the spring constants are directly
calculated from $E$ and $\nu$, using Eq.$\left(\ref{eq4},\ref{eq5}\right)$.
But when the strain in a spring crosses $\epsilon_{y}$, firstly, the secant
modulus $E_{nl}$ corresponding to the strain $\epsilon+\delta\epsilon$ is
evaluated as [52]:

$E_{nl}\left(\epsilon+\delta\epsilon\right)=E\left(\frac{\epsilon+\delta\epsilon}{\epsilon_{y}}\right)^{n-1}.$
(9)

While unloading, to avoid any changes to the secant modulus, it is updated as:

$E\left(\epsilon+\delta\epsilon\right)=\mathrm{min}[E_{nl}\left(\epsilon+\delta\epsilon\right),E\left(\epsilon\right)].$
(10)

Finally, the spring constants in the nonlinear regime are calculated from the
updated secant modulus $E$ and $\nu$ as per
Eq.$\left(\ref{eq4},\ref{eq5}\right)$ [52].

In the model, a spring fails when it stretches beyond a threshold strain, and
simultaneously, the rotational springs associated with the broken spring are
also taken to be broken. The failure strain of the springs, $\epsilon_{f}$, is
assigned from a normal distribution, and its probability density function
$P\left(\epsilon_{f}\right)$ is defined as

$P\left(\epsilon_{f}\right)=\frac{1}{\sqrt{2\pi(\Delta\overline{\epsilon}_{f})^{2}}}e^{-\frac{\left(\epsilon_{f}-\overline{\epsilon}_{f}\right)^{2}}{2(\Delta\overline{\epsilon}_{f})^{2}}},$
(11)

where $\overline{\epsilon}_{f}$ is the mean and
$\Delta\overline{\epsilon}_{f}$ is the standard deviation of the distribution,
where $\Delta$ is a multiplicative factor, controlling extent of disorder.

###  III.3 The model parameters

The highlighted gauge section of the dog-bone specimen Fig. 4(a) of size
approximately $\mathrm{42\times 10}$ $\mathrm{mm^{2}}$ is discretized with
$NY\times NX$ lattice points having lattice spacing $a$. Hence, for the
lattice spacings of a = 0.1, 0.2 and 0.4 mm, the $NY\times NX$ are $420\times
102$, $210\times 52$ and $105\times 26$, respectively. For simulating uniaxial
tension, quasi-static tensile load is applied at a rate of $0.000436$ mm/step,
as shown in Fig. 4(b). The boundary conditions of the loading configuration
are: vertically upward displacement is applied to the top row, vertical motion
of the bottom row is restricted, and both the horizontal and vertical motion
of the left bottom corner point is restricted. The material parameters, $E$ =
3.12 GPa, $\nu$ = 0.26, $n$ = 0.71, and $\epsilon_{y}$ = 0.0055, used for
simulating the constitutive behaviour are calculated by fitting
Eq.$\left(\ref{eq8}\right)$ to the stress-strain data for the B1S3 specimen.
The damping constant $\gamma$ = 0.5 was found to suitably dampen the excessive
oscillations of the system. In the following parametric study, we will only
vary $a$, $\overline{\epsilon}_{f}$, and $\Delta$, keeping the remaining
parameters and geometry unchanged.

##  IV Results and Discussion

We begin with a computational parametric study on the effect of disorder on
the fracture response, avalanche distribution and record statistics of quasi-
brittle material. The parametric study will assist us in selecting an
optimised degree of disorder suitable for modelling the glassy epoxy based
polymer. Finally, the macroscopic responses, such as the elastic response,
damage pattern, final breakage behaviour, and damage rate, will be compared
with the corresponding experimental results to benchmark the effectiveness of
the methodology.

###  IV.1 Parametric study

(a)

(b)

(c)

(d)

Figure 5: Stress-strain curve of a typical realisation with its corresponding
avalanches and records for (a) $\Delta$ = 0.05, (b) $\Delta$ = 0.15, (c)
$\Delta$ = 0.2, and (d) $\Delta$ = 0.3.

$\Delta$ = 0.05

(a)

$\Delta$ = 0.05

(b)

$\Delta$ = 0.3

(c)

$\Delta$ = 0.3

(d)

$\Delta$ = 0.3

(e)

Figure 6: Contour plots showing the evolution of fracture surfaces between
consecutive records at different stages of loading. The grey, blue and red
area represents the unbroken bonds at the end of the consecutive records,
failed bonds before the consecutive records and the bonds fail between the
consecutive records, respectively. (a) and (b) show the fracture surface
evolved between [$1^{st}$, $2^{nd}$] and [$2^{nd}$, $3^{rd}$] records,
respectively, for $\Delta$ = 0.05. Fig (c), (d) and (e) show the fracture
surface evolved between [$4^{th}$, $5^{th}$], [$8^{th}$, $9^{th}$] and
[$9^{th}$, $10^{th}$]records, respectively, for $\Delta$ = 0.3.

To establish the sensitivity of the statistical signatures of the avalanche
data to the RSNM model parameters, we first perform a parametric study.
Initially, we look at the effect of disorder on the macroscopic response and
the associated avalanche time series. The results for a range of $\Delta$ =
0.05, 0.15, 0.2 and 0.3, are shown in Fig. 5. The avalanche size, represented
by $s$, is the number of springs broken for a given incremental strain. In the
figure, we also highlight the record-breaking avalanches in red circles. A
record-breaking avalanche is an avalanche whose size surpasses those of all
previous avalanches. To study the avalanche and record up to the macroscopic
failure strain of the material, we have chosen $\overline{\epsilon}_{f}$ =
0.03, 0.045, 0.057, and 0.074 for $\Delta$ = 0.05, 0.15, 0.2 and 0.3,
respectively. The $\overline{\epsilon}_{f}$ are chosen such that for each
$\Delta$, the stress-strain response averaged over 100 realisations closely
reproduces the macroscopic failure strain. In spite of the randomness of the
avalanche sizes and their time of occurrence, the macroscopic response is
smooth for the chosen range of disorders. For low disorder, very few
avalanches are observed and they are confined to the deformation regime close
to final failure. With increasing disorder, the overall count of avalanches
increases and avalanches are observed in several stages of deformation,
including in the initial elastic regime for $\Delta$ greater than $0.2$. The
record-breaking events for low disorder appear to occur rapidly till final
failure. However, for the higher disorder, $\Delta$=$0.3$, we see a regime
after the initial few records when there is a long gap before the next record-
breaking event. In this stationary regime, the fracture process is dominated
by the uncorrelated nucleation of micro-cracks, which are a consequence of the
higher disorder. With further deformation as the fracture process approaches
final failure, the subsequent record-breaking events occur more rapidly when
the damage starts to develop longer-ranging correlations.

(a)

(b)

Figure 7: (a) The probability of an avalanche of size $s$ and (b) the average
waiting time of record avalanches for different $\Delta$. In (a), the data
have been shifted vertically such that the data overlap for intermediate $s$.

To bring out the difference between the manner of damage propagation at
different stages of deformation we look in detail at the spatial patterns of
damage. For ease in distinguishing we colour code the springs that break in
the given duration as red while the springs that were already broken before
the duration are coloured blue. The remaining unbroken springs are coloured
grey. At lower disorder ($\Delta$ = 0.05) the total number of records, for the
typical realisation considered, is only $3$. In the duration between 1st and
2nd record very few springs break, appearing as isolated independent red
regions, see Fig. 6(a), while the rapid propagation of damage is seen between
the 2nd and 3rd record, Fig. 6(b). For higher disorder, which has distinct
regimes in the waiting time intervals observed during the failure process, for
the record that has the longest waiting time (between 4th and 5th record) in
the contour plot in Fig. 6(c) the new events (red regions) are largely
independent with very few in the neighbourhood of existing damage sites (blue
regions), implying that in this duration the springs break throughout the
domain like independent nucleation sites largely oblivious to the prior defect
structure. In the waiting time for the last record, however, most of the red
regions are localised as seen in Fig. 6(e) and several are part of a large
cluster having connections with blue regions. This localisation of avalanches
plausibly causes the waiting time between records to decrease. Once many
springs are broken creating damage in different locations, it becomes easier
to break the intermediate springs connecting the damaged areas, which in turn
triggers more breakages, thus making it easier for larger avalanches to occur.

The role of disorder on different statistical signatures, the avalanche size
distribution and the waiting time interval between record breaking events, are
presented in Fig. 7. The probability of an avalanche of size $s$ is P(s) and
its distribution follows a power law of the form: $P(s)\sim s^{-\tau_{n}}$ for
$s>>1$. In Fig. 7(a), to compare the slopes of P(s), the data for different
disorders is shifted along the $y$-axis so that the data for intermediate $s$
overlap. For large avalanche sizes the power law exponent seems to be similar
for all disorders and is approximately $2.3$. This suggests that for the range
of disorder considered here, the effect of disorder on the exponent of
avalanche size distribution is not sensitive enough to characterise the
disorder from the exponent.

To understand the effect of disorder on the rate of occurrence of record
breaking avalanches, we define the waiting time between two consecutive
records $r_{k}$ and $r_{k-1}$ as:

$\tau_{k}=t_{k}-t_{k-1},$ (12)

where $t_{k-1}$ and $t_{k}$ are the timestamp of the ${k-1}^{th}$ and
${k}^{th}$ record events. In Fig. 7(b) the variation of $\tau_{k}$ with $k$,
averaged over 100 realisations, is shown for different disorder $\Delta$. The
maximum of the range of $k$ indicates the highest number of records among all
the realisations, and the red circles show the average of the number of
records. From the record statistics of the simulation data, the waiting time
between record breaking events at lower disorder ($\Delta$=$0.05$) is
initially large and shows a rapid monotonic decrease with increasing k. In
contrast, for higher disorder ($\Delta$=$0.3$) the waiting time is initially
very short, it becomes longer with increasing $k$ upto $k=3$ where it peaks.
With further increase in $k$, the waiting time reduces implying that as the
system approaches final failure, record breaking events occur with shorter and
shorter waiting time. While approaching to final failure acceleration in
occurrence of record breaking events is seen in all cases of disorder, only
high disorder exhibits the initial deceleration resulting in a peak. The peak
shifts to the left as the disorder is reduced. The effect of disorder is also
seen to affect the overall count of record breaking events. For higher
disorder the total number of record breaking events is higher.

(a)

(b)

Figure 8: (a) The probability of an avalanche of size $s$ and (b) the average
waiting time of record avalanches for the $\Delta$ = 0.2 and different lattice
sizes $a$.

In the RSNM simulations, lattice spacing is also an input parameter. To
evaluate the sensitivity of the statistical signatures to the lattice spacing
for a specific degree of disorder ($\Delta$ = 0.2), the lattice spacing of a=
$0.1,0.2$ and a=$0.4$ mm were chosen for the simulation. These lattice
spacings fall within the acceptable range of RVE size suitable for modelling
fracture in epoxy based polymer [53]. The avalanche distributions for all
lattice spacings, as shown in Fig. 8(a), overlap each other implying minimal
effect on the power law exponent of the distribution. The waiting times
however are affected by the lattice parameter $a$. Increasing system size
(same as decreasing lattice spacing) seems to increase the overall number of
records and the behaviour shows qualitative similarity to the effect of
increasing disorder as seen in Fig. 8(b). This implies that the effect of the
system size and the disorder on waiting time interval is coupled and thus, the
estimate of disorder parameter for a specific material system would be
dependent on the choice of the system size. From the parametric study, we
conclude that records are more sensitive to disorder than avalanche
distribution, thus making records more suitable for characterising material
disorder.

###  IV.2 Experimental results

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9: Force-time response with its corresponding avalanches and record
bursts from experiments, for specimens: (a) B1S1, (b) B1S3, (c) B3S1, (d)
B3S3, (e) B4S2 and (f) B4S4. The blue and red circled blue dots correspond to
avalanche and record avalanche, respectively.

(a)

(b)

Figure 10: (a) Avalanche distribution in Amp (volt) for three batches.(b)
Average waiting time of records for three batches. [The x-intercept of the
vertical dashed line indicates the average of the number of records
($k_{avg}$) detected in the specimens B1S1, B1S3, B3S1, B3S3, B4S2, and B4S4.]

For estimation of the disorder parameter in the model we examine the AE
activity data for the equivalent statistical signatures. From the tensile test
data of 6 samples prepared in three different batches (B1, B3, B4), the time-
synced force (kN) and amplitude (volt) data, presented in Fig. 9, shows
characteristics of quasi-brittle fracture, such as multiple events spread
through the deformation to final failure which may be attributed to nucleation
of uncorrelated micro-cracking in the early stages, followed by the growth and
interactions of the micro-cracks represented by the medium amplitude AE
events, and finally, the complete fracture of the specimen corresponding to
the highest amplitude event that coincides with the final catastrophic load
drop in the specimen. The quasi-brittle fracture signatures are also observed
in the rate of record events (highlighted by the red circles). These AE
signatures will be analysed next for characterising the degree of disorder.

The avalanche distributions of the collective amplitudes recorded from the
specimens of each batch are presented in Fig. 10(a). The avalanche
distributions of all the batches are very similar to each other and follow a
power law distribution: $P(A)\sim A^{-\tau_{A}}$ with $\tau_{A}\approx 2$. The
waiting time for the records, obtained from averaging only the non-zero
entries for each record, are presented in Fig. 10(b). All the batches exhibit
a peak (at 3rd or 4th record). The batch B4, however, shows anomalous
behaviour with two additional peaks. The anomaly is due to the specimen B4S2
exhibiting large cluster of AE activity in the initial stages (at time ranging
between 100-200 secs) which is not observed in any other specimen. In further
analysis, the experimental data from B4S2 is ignored.

###  IV.3 Experimental validation

Figure 11: The variation of waiting time averages for the experiment record
bursts are compared with the corresponding records from simulation for
$\Delta$ = $0.275$.

It is understood from the parametric study that the waiting time interval and
the total number of records have significant sensitivity to the disorder
incorporated in the model. From the experimental data of all batches
(excluding B4S2), we find the average waiting time for every record by
considering only the nonzero $\tau_{k}$ (see Fig. 11). A distinct peak is
observed at $k=3$, and the averaged total number of records to be between 5
and 6. Comparison with the simulation results in Fig. 7(b) suggests the
$\Delta$ to fall between $0.2$ to $0.3$. Iteratively the combination of
$\overline{\epsilon}_{f}$ = 0.0715 and $\Delta$ = 0.275 is found to closely
reproduce the features of the experimental data, as seen in Fig. 11. The
lattice spacing is chosen to be $\mathrm{a=0.2mm}$, same as the initial
parametric study.

(a)

(b)

Figure 12: (a) The stress-strain response of the B1S3 specimen from the
experiment is compared with the averaged stress-strain response from the
simulation for $\Delta$ = 0.275, and (b) their corresponding avalanche
distributions. In the avalanche distribution, the unit of $s$, $A$, $P(s)$ and
$P(A)$ is arbitrary and for coinciding the middle portion of the graphs, the
graphs are shifted by maintaining the same slope.

(a)

(b)

Figure 13: (a) The fracture surface in the gauge section of the failed
specimens and (b) the simulated fracture path for different realisations.

(a)

(b)

Figure 14: (a) In the time or displacement vs. the cumulative damage growth
graph, first, both the x and y data are normalised with their respective
maximums, (b) then the normalised avalanches are averaged over the
realisations for the simulation and the normalised amplitudes are averaged
over the samples in the experiment

The effectiveness of the chosen model parameters is further examined by
comparing the simulation data with other features of the experiments. The
macroscopic stress-strain response averaged over 250 realisations, shown in
Fig. 12(a), well reproduces experimental curve in its linear and non-linear
regime. The averaged strength from simulations is within $2\%$ of the average
experimental strength. The avalanche size distribution, P(s), and the AE
amplitude distribution, P(A), both exhibit power law behaviour in the large
avalanche size regime, the exponents being 2 and 2.3.

Further, we examine the ability of the model to simulate realistic fracture
paths. In Fig. 13(a) we note that several types of fracture paths are seen in
experiments, nearly flat (B1S1, B1S3, B3S1), tilted (B3S3), branched (B2S2,
B4S4). Such features are well reproduced in a few selected realisations from
the simulations in Fig. 13(b). The model, thus, for the estimated disorder
parameter is able to simulate close to realistic fracture paths.

In Fig. 14, over the course of deformation to final failure the progressive
growth of damage is evident in the cumulative amplitude and cumulative
avalanche size from experiments and simulations respectively. The cumulative
data is normalised with the final value for comparison of the rates. The
comparison of averaged experimental data and simulations, shown in Fig. 14(b),
further confirms that incorporating a suitable degree of disorder in RSNM can
adequately simulate the damage nucleation and growth activity in glassy epoxy.

##  V Conclusion

In the present work, we show that the statistical signatures of AE activity
during the failure process can be utilised for characterisation of disorder
parameter in simulation of tensile fracture of epoxy based polymer. For
simulations we use a square random spring network model with nearest and next-
nearest interactions. The spring behaviour is taken to account for quasi-
brittle behaviour with linear and non-linear regimes and a normally
distributed failure strain threshold. In the initial parametric study, we show
that the disorder characteristics while have no noticeable effect on the power
law exponent of the avalanche size distribution, are strongly correlated with
the waiting time interval between consecutive record breaking avalanches as
well as the total number of records. This sensitivity to disorder is exploited
in estimating the disorder parameter suitable for experiments on tensile
failure of epoxy based polymer. Assuming equivalence between the amplitude
distribution of AE data and avalanche size distribution of the simulations, we
choose the $\Delta$ to be $0.275$ on the basis that the waiting time interval
characteristics and total number of record breaking events are seen to
correlate well between experiments and simulations for it. The chosen disorder
parameter is next shown to well reproduce the failure characteristics in terms
of the peak load of the macroscopic response, the power-law behaviour with
avalanche dominated fracture type as well as realistic fracture paths.
Utilising AE data for characterising disorder has enormous potential in
fracture modelling of brittle, heterogeneous material systems as it can
overcome one of the major drawbacks of local approach to fracture modelling:
difficulty in estimation of model parameters from direct experimental data.

## References

  * [1] P. Han, Y. Zhao, C. Zhang, X. Wang, W. Wang, Effects of water on mechanical behavior and acoustic emission characteristics of coal in brazilian tests, Theoretical and Applied Fracture Mechanics 122 (2022) 103636.
  * [2] W. Hu, J. Zhang, Effect of growth rings on acoustic emission characteristic signals of southern yellow pine wood cracked in mode I, Construction and Building Materials 329 (2022) 127092.
  * [3] T. Y. Guo, L. N. Y. Wong, Cracking mechanisms of a medium-grained granite under mixed-mode i-ii loading illuminated by acoustic emission, International Journal of Rock Mechanics and Mining Sciences 145 (2021) 104852.
  * [4] C. Van Toen, J. Street, T. R. Oxland, P. A. Cripton, Acoustic emission signals can discriminate between compressive bone fractures and tensile ligament injuries in the spine during dynamic loading, Journal of biomechanics 45 (9) (2012) 1643–1649.
  * [5] Q. Lin, B. Wan, Y. Wang, Y. Lu, J. F. Labuz, Unifying acoustic emission and digital imaging observations of quasi-brittle fracture, Theoretical and Applied Fracture Mechanics 103 (2019) 102301.
  * [6] L. Salminen, A. Tolvanen, M. J. Alava, Acoustic emission from paper fracture, Physical Review Letters 89 (18) (2002) 185503.
  * [7] X. Wang, J. Xu, Q. Yue, X. Liu, Tracing fracture damage evolution and identifying damage patterns in cast steel using advanced acoustic emission analysis, Engineering Fracture Mechanics 293 (2023) 109680.
  * [8] L. Botvina, A. Bolotnikov, I. Sinev, E. Beletsky, Acoustic emission, damage and fracture mechanisms of structural steel under mixed-mode loading, Engineering Fracture Mechanics 292 (2023) 109635.
  * [9] Y. Zhao, H. Liu, K. Xie, E. K. Salje, X. Jiang, Avalanches in compressed sandstone: crackling noise under confinement, Crystals 9 (11) (2019) 582.
  * [10] E. K. Salje, H. Liu, L. Jin, D. Jiang, Y. Xiao, X. Jiang, Intermittent flow under constant forcing: Acoustic emission from creep avalanches, Applied Physics Letters 112 (5) (2018).
  * [11] A. Planes, L. Mañosa, E. Vives, Acoustic emission in martensitic transformations, Journal of Alloys and Compounds 577 (2013) S699–S704.
  * [12] C. Maes, A. Van Moffaert, H. Frederix, H. Strauven, Criticality in creep experiments on cellular glass, Physical review b 57 (9) (1998) 4987.
  * [13] X. Jiang, H. Liu, I. G. Main, E. K. Salje, Predicting mining collapse: Superjerks and the appearance of record-breaking events in coal as collapse precursors, Physical Review E 96 (2) (2017) 023004.
  * [14] G. Pál, F. Raischel, S. Lennartz-Sassinek, F. Kun, I. G. Main, Record-breaking events during the compressive failure of porous materials, Physical Review E 93 (3) (2016) 033006.
  * [15] Z. Danku, F. Kun, Creep rupture as a non-homogeneous poissonian process, Scientific Reports 3 (1) (2013) 2688.
  * [16] H. E. Daniels, The statistical theory of the strength of bundles of threads. i, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 183 (995) (1945) 405–435.
  * [17] A. Hansen, P. C. Hemmer, S. Pradhan, The fiber bundle model: modeling failure in materials, John Wiley & Sons, 2015.
  * [18] S. Pradhan, A. Hansen, B. K. Chakrabarti, Failure processes in elastic fiber bundles, Reviews of modern physics 82 (1) (2010) 499.
  * [19] L. de Arcangelis, S. Redner, H. Herrmann, A random fuse model for breaking processes, Journal de Physique Lettres 46 (13) (1985) 585–590.
  * [20] S. Zapperi, A. Vespignani, H. E. Stanley, Plasticity and avalanche behaviour in microfracturing phenomena, Nature 388 (6643) (1997) 658–660.
  * [21] S. Zapperi, P. Ray, H. E. Stanley, A. Vespignani, Avalanches in breakdown and fracture processes, Physical Review E 59 (5) (1999) 5049.
  * [22] E. Schlangen, E. J. Garboczi, Fracture simulations of concrete using lattice models: computational aspects, Engineering fracture mechanics 57 (2-3) (1997) 319–332.
  * [23] F. Raischel, F. Kun, H. J. Herrmann, Simple beam model for the shear failure of interfaces, Physical Review E 72 (4) (2005) 046126.
  * [24] R. V. Sagar, B. R. Prasad, S. Nazreen, R. Singh, Modeling mode-i fracture process in concrete at meso-scale: Computational aspects of lattice model and a comparison between results of two dimensional lattice simulation and acoustic emission measurements, Engineering Fracture Mechanics 210 (2019) 257–278.
  * [25] J. G. Van Mier, M. R. van Vliet, T. K. Wang, Fracture mechanisms in particle composites: statistical aspects in lattice type analysis, Mechanics of Materials 34 (11) (2002) 705–724.
  * [26] P. Ray, Breakdown of heterogeneous materials, Computational materials science 37 (1-2) (2006) 141–145.
  * [27] S. Kale, M. Ostoja-Starzewski, Elastic-plastic-brittle transitions and avalanches in disordered media, Physical review letters 112 (4) (2014) 045503\.
  * [28] S. Zapperi, P. Ray, H. E. Stanley, A. Vespignani, First-order transition in the breakdown of disordered media, Physical Review Letters 78 (8) (1997) 1408.
  * [29] F. Kun, I. Varga, S. Lennartz-Sassinek, I. G. Main, Approach to failure in porous granular materials under compression, Physical Review E 88 (6) (2013) 062207\.
  * [30] F. Kun, I. Varga, S. Lennartz-Sassinek, I. G. Main, Rupture cascades in a discrete element model of a porous sedimentary rock, Physical review letters 112 (6) (2014) 065501.
  * [31] H. J. Herrmann, A. Hansen, S. Roux, Fracture of disordered, elastic lattices in two dimensions, Physical Review B 39 (1) (1989) 637.
  * [32] M. Sahimi, J. D. Goddard, Elastic percolation models for cohesive mechanical failure in heterogeneous systems, Physical Review B 33 (11) (1986) 7848.
  * [33] S. Senapati, A. Banerjee, R. Rajesh, Role of composition in fracture behavior of two-phase solids, Physical Review E 107 (5) (2023) 055002.
  * [34] W. Curtin, H. Scher, Brittle fracture in disordered materials: A spring network model, Journal of Materials Research 5 (3) (1990) 535–553.
  * [35] P. K. V. Nukala, S. Zapperi, S. Šimunović, Statistical properties of fracture in a random spring model, Physical Review E 71 (6) (2005) 066106.
  * [36] S. Zapperi, P. K. V. Nukala, S. Šimunović, Crack roughness and avalanche precursors in the random fuse model, Physical Review E 71 (2) (2005) 026106.
  * [37] A. Shekhawat, S. Zapperi, J. P. Sethna, From damage percolation to crack nucleation through finite size criticality, Physical review letters 110 (18) (2013) 185505.
  * [38] D. Kumar, A. Banerjee, R. Rajesh, Interplay between disorder and hardening during tensile fracture of a quasi-brittle solid, Proceedings of the Royal Society A 478 (2262) (2022) 20210934.
  * [39] D. Boyina, T. Kirubakaran, A. Banerjee, R. Velmurugan, Mixed-mode translaminar fracture of woven composites using a heterogeneous spring network, Mechanics of Materials 91 (2015) 64–75.
  * [40] A. Mayya, A. Banerjee, R. Rajesh, Role of porosity and matrix behavior on compressive fracture of haversian bone using random spring network model, Journal of the Mechanical Behavior of Biomedical Materials 83 (2018) 108–119.
  * [41] X. Jiang, D. Jiang, J. Chen, E. K. Salje, Collapsing minerals: Crackling noise of sandstone and coal, and the predictability of mining accidents, American Mineralogist 101 (12) (2016) 2751–2758.
  * [42] S. Pradhan, A. Hansen, P. C. Hemmer, Crossover behavior in burst avalanches: Signature of imminent failure, Physical review letters 95 (12) (2005) 125501.
  * [43] S. Senapati, S. Roy, A. Banerjee, R. Rajesh, Record statistics based prediction of fracture in the random spring network model, arXiv preprint arXiv:2312.01158 (2023).
  * [44] E. K. Salje, X. Jiang, Crackling noise and avalanches in minerals, Physics and Chemistry of Minerals 48 (2021) 1–15.
  * [45] I. Iturrioz, G. Lacidogna, A. Carpinteri, Acoustic emission detection in concrete specimens: Experimental analysis and lattice model simulations, International Journal of Damage Mechanics 23 (3) (2014) 327–358.
  * [46] V. Kádár, Z. Danku, G. Pál, F. Kun, Approach to failure through record breaking avalanches in a heterogeneous stress field, Physica A: Statistical Mechanics and its Applications 594 (2022) 127015.
  * [47] Z. Danku, F. Kun, Record breaking bursts in a fiber bundle model of creep rupture, Frontiers in Physics 2 (2014) 8.
  * [48] S. Roy, Record statistics of emitted energies—prediction of an upcoming failure, Physica A: Statistical Mechanics and its Applications 627 (2023) 129129\.
  * [49] A. E976-15, Standard guide for determining the reproducibility of acoustic emission sensor response, Tech. rep., ASTM International, West Conshohocken, USA (2021).
  * [50] J. R. d’Almeida, S. N. Monteiro, The influence of the amount of hardener on the tensile mechanical behavior of an epoxy system, Polymers for Advanced Technologies 9 (3) (1998) 216–221.
  * [51] L. Monette, M. Anderson, Elastic and fracture properties of the two-dimensional triangular and square lattices, Modelling and Simulation in Materials Science and Engineering 2 (1) (1994) 53.
  * [52] D. Kumar, A. Banerjee, R. Rajesh, Crushing of square honeycombs using disordered spring network model, Mechanics of Materials 160 (2021) 103947.
  * [53] J. Lemaitre, Local approach of fracture, Engineering Fracture Mechanics 25 (5-6) (1986) 523–537.