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Experimental data-driven uncertainty quantification for the dynamic fracture
toughness of particulate polymer composites
A. Sharma1, T. Mukhopadhyay2,*, V. Kushvaha3,*
1Department of Civil Engineering, Indian Institute of Technology Jammu, J&K, India
2Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, India
3Department of Civil Engineering, Indian Institute of Technology Jammu, J&K, India
*Corresponding author: tanmoy@iitk.ac.in (TM), vinod.kushvaha@iitjammu.ac.in (VK)
Abstract: This paper presents an experimental investigation supported by data-driven approaches
concerning the influence of critical stochastic effects on the dynamic fracture toughness of glass-filled
epoxy composites using a computationally efficient framework of uncertainty quantification. Three
different shapes of glass particles are considered including rod, spherical and flaky shapes with coupled
stochastic variations in aspect ratio, dynamic elastic modulus and volume fraction. An artificial neural
network based surrogate assisted Monte Carlo simulation is carried out here in conjunction with
advanced experimental techniques like digital image correlation and scanning electron microscopy to
quantify the uncertainty and sensitivity associated with the dynamic fracture toughness of composites
in terms of stress intensity factor under dynamic impact. The study reveals that the pre-crack initiation
time regime shows the most prominent effect of uncertainty. Additionally, rod shape and the aspect
ratio are the most sensitive filler type and input parameter respectively for characterizing dynamic
fracture toughness. Here the quantitative results based on large-scale data-driven approaches
convincingly demonstrate using a computational mapping between the stochastic input and output
parameter spaces that the effect of uncertainty gets pronounced significantly while propagating from
the compound source level to the impact responses. Such outcomes based on experimental data
essentially bring us to the realization that quantification of uncertainty is of utmost importance for
developing a reliable and practically relevant inclusive analysis and design framework for the dynamic
fracture of particulate composites. With limited literature available on the determination of fracture
toughness considering inertial effects, the present work demonstrates a novel and insightful
experimental approach for uncertainty quantification and sensitivity analysis of dynamic fracture
toughness of particulate polymer composites based on surrogate modeling.
Keywords: Stochastic dynamic fracture toughness; Uncertainty quantification in dynamic fracture;
Sensitivity analysis of particulate composites; ANN assisted stochastic experimental characterization
of composites
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1. Introduction
Particulate polymer composites (PPCs) are an emerging class of polymer composites that has
many potential applications in aerospace, automotive, marine and electronic industries due to the
synergistic combination of mechanical characteristics like high strength and lightweight [1–3]. Such
industrial applications demand for prior knowledge of the fracture behaviour due to the inherent brittle
nature of these particulate composites [4]. The kind of reinforcement phase used in these composites
plays a vital role in determining the fracture behaviour of the resulting composite material. Mica, silica,
alumina, carbon nanotubes (CNT) and layered silicates are a few examples of inorganic particulate
fillers which are known to enhance the strength and stiffness properties of polymers [5]. We provide a
brief review of the mechanical properties of such particulate composites in the following paragraphs.
An enhancement was observed in alumina-filled epoxy composites in terms of mode-I stress
intensity factor by Sandeep et al. [6]. Another research group [7] studied the fracture toughness of
CNT reinforced epoxy composites under quasi-static and dynamic loading conditions. Critical stress
intensity factors were found to be improved after performing a non-ionic surface treatment on the CNT
fillers. The nature of the fillers such as the geometric properties (size, and shape), volume fraction,
filler dispersibility and interfacial bonding significantly influence the stress distribution within the
composite [8–10]. Additionally, the crack growth of such composites is loading rate-dependent [11,12]
and in this context, several researchers have studied the aspect of fracture toughness of fiber reinforced
polymer composites under static and dynamic conditions [13–18].
However, very few have attempted to study the fracture toughness of particulate polymer
composites under dynamic loading conditions. Determining the fracture toughness under dynamic
loading conditions is typically achieved by using a split Hopkinson pressure bar or gas gun setup [19–
21]. The onset of crack propagation is captured using a high-speed camera and a technique known as
digital image correlation (DIC) is often used to determine the parameters of dynamic fracture [22].
Owing to the technical complexity of this experimental setup, it is difficult to carry out large-scale
experiments on the account of time and cost. Moreover, in contrast to the quasi-static approach,
experimental methodology for the dynamic conditions wherein the inertial effects are taken into
account, is yet to be standardized [23,24]. In this view, it is imperative to devise some alternative
techniques/models to efficiently predict the dynamic fracture toughness of particulate polymer
composites with limited physical experimentation.
In the recent years, machine learning (ML) has emerged as a promising tool for the predictive
modeling of polymer composites with significant computational efficiency [25–31, 60-64]. Out of
different ML algorithms, Artificial Neural Network (ANN) is a well-known predictive technique used
successfully by various researchers to model the mechanical behaviour of polymer composites [32–
36]. ANN is a universal function approximator that has the capability of handling large covariate
spaces with significant level of accuracy [37]. The efficiency of this technique lies in its ability to solve
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highly complex problems by making use of non-linear mapping functions [38]. Therefore, to reduce
the experimental and computational effort, we aim to exploit artificial neural networks for reliable
predictions of dynamic fracture toughness.
A realistic analysis and design framework for particulate composites should account for the
possible uncertainties due to the inherent inhomogeneity and multiphase nature of the PPCs. Moreover,
such unavoidable variation in process parameters due to the varying physical properties of matrix and
reinforcement, degree of polymerization, environmental conditions and filler dispersion can
significantly affect the ultimate response of PPCs. To ensure the accurate assessment of the ultimate
composite performance and to avoid the deviation from the expected material behaviour, uncertainty
quantification is critically important [39]. The effects of the uncertainties in Young’s modulus, aspect
ratio, volume fraction and radius of the clay platelets was studied, revealing that among all the
considered parameters, variation in Young’s modulus affects fracture toughness the most. Another
research group [40] proposed a stochastic framework based on radial basis functions to quantify the
uncertainty in the micromechanical properties of polymer composites. They also performed a
sensitivity analysis to determine the impact of uncertainty in the input parameters on the natural
frequency of polymer composites. The uncertainties can be categorized as model uncertainties and
parameter uncertainties. The model uncertainties arise from the oversimplification of the physics
involved while the uncertainty in the parameters arises from stochasticity in the inputs [41]. The
uncertainties in the inputs (often correlating directly to the manufacturing uncertainties) have more
influence and their propagation is complicated due to the ineffable relationships between the
parameters.
Expressing complex stochastic input-output relationships requires statistical approaches where
the results can be computed with a variability bound in the inputs to provide the confidence interval of
the potential outputs. Several statistical approaches such as Monte Carlo simulation, perturbation
method, surrogate-based modeling etc. have been explored in this context in numerous engineering
problems [42–47]. One of the prevalent methods is the Monte Carlo (MC) technique for the
quantification and propagation of uncertainties due to its simplicity and high statistical accuracy up to
a large extent of input variability [48–50]. The MC simulation method is a sampling-based approach
that generates thousands of samples corresponding to the random input variables as per their
probability distribution and subsequently, the probabilistic distribution of the output quantity of
interest is characterized. However, the downside of the standard Monte Carlo method is its slow
convergence and a large number of realizations (~104) are required to attain the desired accuracy. To
mitigate the computationally expensive nature of MCS, the possible avenues could be parallelization
of the Monte Carlo simulations [51] or utilization of surrogate modeling approaches [52,53]. Even
though parallelization may be able to reduce the time for Monte Carlo Simulations, it still requires
high computational effort. Note that it does not have any relavance to Monte Carlo simulations using
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experimental data. On the other hand a surrogate model can effectively replace the expensive
simulation models or physical experiments based on a limited optimum sample evaluation. Thus in
case of experimental stochastic characterization, based on a few experimental tests a computational
mapping can be established between the stocahstic input parameters and the output quantity of interest
using surrogate modeling. Subsequently, the surrogate model can be exploited for predicting the output
parameter corresponding to any random combination of the input parameters within the design domain
and the Monte Carlo simulation can be performed efficiently.
In the current paper, an ANN based uncertainty quantification approach would be presented to
quantify the stochastic variability in the dynamic fracture toughness of glass-filled epoxy composites
due to the inevitable random stochasticity in the material and geometrical properties (such as aspect
ratio, dynamic elastic modulus and volume fraction). The gap between the necessity of large-scale data
generation for Monte Carlo simulation and the limitation of carrying out multiple experimentations is
proposed to be addressed by adopting ANN based surrogate modeling approach here. With limited
literature available on the determination of fracture toughness considering inertial effects, the present
work would demonstrate a novel and insightful experimental approach for uncertainty quantification
and sensitivity analysis of dynamic fracture toughness of particulate polymer composites based on
surrogate modeling. Hereafter this paper is arranged as, section 2: Experimental framework; section 3:
Methodology; section 4: Results and discussions; section 5: Conclusions.
2. Experimental framework
2.1 Specimen preparation
The materials used for preparing the glass-filled polymer composite were an epoxy system and
glass particles in three different shapes viz. spherical, flake and rod (refer to Fig. 1). The epoxy system
comprised of a Bisphenol-A resin and an amine-based hardener, purchased from Buehler, U.S.A. The
composite sheets were cast using glass particles in a volume fraction of 0%, 5%, 10% and 15%.
Further, these sheets were machined into rectangular specimens of size 60 mm × 30 mm × 9 mm with
a notch of length 6 mm at the center of each specimen. The physical properties of these composite
specimens were measured using pulse-echo techniques.
2.2 Details of dynamic fracture test
The schematics of the dynamic fracture test are given in Fig. 2. A gas gun setup with high
pressure cylinder was used for launching the projectile. The impact of the striker (diameter, 25.4 mm;
velocity ~16 m/s) onto the specimen generated a compressive stress wave which was propagated
through the specimen. The stress wave was responsible for the in-plane deformation in the specimen,
which was measured using the technique called, digital image correlation. A black and white random
granular pattern was created on the specimen surface and a high-speed digital camera (Cordin 550)
was used to capture the images of the specimen before and after the event of impact. The striker when
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Fig. 1: Schematic diagram of the particulate polymer composite under consideration. Rectangular
specimens of epoxy composite reinforced with glass fillers along with the microscopic images of glass
fillers obtained using a scanning electron microscope (scale bar: 50 m). (A) Spherical shaped glass
fillers (B) Flake shaped glass fillers (C) Rod shaped glass fillers.
came in contact with the long bar, completed the electrical circuit to cause the delay generator to trigger
the high-speed camera. Subsequently, the camera triggered the high energy flash lamps and initiated
the capturing of images after every 3.33 µs (300,000 frames per second). Hence, every image in the
undeformed state had a corresponding one in the set of deformed images. Thus obtained images for
the composite with 10% volume fraction of rod shaped glass particles are shown in Fig. 3.
Further using DIC, these sets of images were correlated, and the in-plane displacement
components namely crack opening and crack sliding were obtained using the following
asymptotic expressions [22]
.
(1)
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(2)
In the above-mentioned equations, () are the crack-tip in polar coordinates, is
for plane
stress where is the shear modulus and is the Poisson’s ratio. The coefficients and of
the leading terms (when ) are the mode-I and mode-II dynamic stress intensity factors,
respectively. To evaluate the displacement components after the crack started propagating, the
following equations were used [54].
.
(3)
.
(4)
Fig. 2: Gas gun setup. Schematic diagram of the experimental setup used for determining the dynamic
fracture toughness.
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Fig. 3: DIC Images. Images for epoxy composites reinforced with 10% Vf of rod shaped glass fillers
with the displacement contours of crack opening (uy) (in the middle) and crack sliding (ux) (in the
extreme right) Displacement contour units are in mm on the x- and y-axis.
where,
(5)
Here () and () are the cartesian and polar coordinates respectively, represents the
instantaneous crack speed, is the longitudinal wave speed and is the shear wave speed for the
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composite material. Using these displacement components, dynamic fracture toughness in terms of
stress intensity factor was computed within a time regime of -30 µs to 30 µs. The negative sign here
does not represent the global time, rather it is a sign convention used by the authors to represent the
time instants before the crack initiates. In this context, nine-time instants are considered before the
initiation of crack (referred to as the pre-crack initiation regime) and nine are considered after the
initiation of crack (referred to as the post-crack initiation regime). When the time instant approaches
to zero, the crack initiates.
3. Methodology
3.1 Surrogate modeling
In the present work, artificial neural network is used as a surrogate model for predicting the
dynamic fracture toughness of glass-filled epoxy composites, wherein a feed-forward multilayer
perceptron (MLP) with a back-propagation learning algorithm is implemented. Back-propagation
involves the fine tuning of network weights based on the error calculated in each iteration and hence
helps in improving the prediction capability of the model. ANN is a non-parametric mathematical
model which is comprised of three main layers (input, hidden and output) and many interconnected
processing units, commonly known as neurons. The neurons of one layer are connected to the neurons
of the next layer and are responsible for summing up the incoming information along with the synaptic
weights. The propagation of information between the neurons of different layers is determined by an
activation function. These functions filter out the information of every neuron based on its relevance
for the model’s prediction and help in normalizing each neuron’s output within a specific range.
In the current study, aspect ratio (AR), dynamic elastic modulus (volume fraction () and
time () are used as the input parameters and the dependent variable i.e., stress intensity factor (SIF) is
the output parameter. To improve the network training, ‘standardized’ technique is used to rescale the
covariate space. The entire available data is partitioned in training and testing datasets following a
partition ratio of 70:30. The used ANN model follows an automatic architecture wherein one hidden
layer with two neurons is used. Hyperbolic tangent and identity are used as the activation functions for
the hidden and the output layer respectively. Considering the computational efficiency, gradient
descent is used as the optimization algorithm [55] with hyperparameters like initial learning rate = 0.4,
momentum = 0.9 and batch as the training type. Learning rate and momentum are two very important
configurations of the ANN model. Learning rate determines the pace and degree to which the model
can be changed in response to the error calculated from the updation of synaptic weights at each time,
while momentum controls the instabilities of the network caused by a very high leaning rate. The data
in the range of aspect ratio 1 (spherical fillers) and 80 (rod shaped fillers) is fed into the above-
mentioned neural network. The performance accuracy is statistically evaluated by calculating mean
absolute percentage error (MAPE) and the coefficient of determination () as:
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(6)
(7)
where is the number of data points, is the true value, is the predicted value and is the mean
of the true values. After obtaining a satisfactory network performance with the selected architecture,
SIF is predicted for the flake shaped glass fillers corresponding to aspect ratio in the range of 6.
3.2 Surrogate based stochastic approach
The stochastic response of dynamic fracture toughness of glass-filled epoxy composites is
investigated under the inherent uncertainty in the input parameters (aspect ratio, dynamic elastic
modulus and volume fraction of fillers). Fig. 1 gives an idea of uncertainty in the aspect ratio of fillers
as the resultant picture shows an inconsistency in the particle shape. Since fracture in this study is a
dynamic event, stochasticity in the input parameters is introduced corresponding to each time instant.
The considered cases of uncertainty (stochastic variation) here are as follows:
i. Uncertainty in aspect ratio only:
ii. Uncertainty in aspect ratio and dynamic elastic modulus:
iii. Uncertainty in aspect ratio and volume fraction:
iv. Combined uncertainty in aspect ratio, dynamic elastic modulus and volume fraction:
,
Here the symbol
indicates the parametric stochasticity, while n represents the number of data
points.
To account for these cases of uncertainty, Monte Carlo simulation (MCS) method is integrated
with the ANN model for generating a large-scale dataset based on limited experimental results.
Although the PPCs are macroscopically isotropic, the inhomogeneities in the composite are
responsible for the complex mechanical behavior governed by coupled parameters. The development
of satisfactory computational models accounting for uncertainties in such coupled parameters using
MCS is very expensive in terms of time and cost. Obtaining solutions using the conventional numerical
approaches requires thousands of simulations. Thus we exploit experimental data that captures the
stochastic variations realistically. A flowchart for the proposed ANN assisted MCS methodology is
given in Fig. 4.
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Fig. 4: Flowchart for ANN assisted MCS methodology. Steps followed for implementing the
proposed framework of uncertainty quantification using experimental data.
A sample space of 10000 × 3 is generated for each time instant using a pseudo-random
distribution within the design space [56]. We have used a large sample size (~104) to make sure the
outcome of MCS is based on converged data. Depending on the range of available data and physical
intuition from experimental observation, a certain stochastic band is selected for the aspect ratio,
volume fraction and dynamic elastic modulus. The degree of stochasticity (DOS) for the considered
band is 30%, 20% and 10% for aspect ratio, volume fraction and dynamic elastic modulus respectively.
DOS in aspect ratio is a direct result of the fact that it is very difficult at the manufacturer’s end to have
a high degree of precision when it comes to the filler size distribution and volume fraction. Also, there
are significant chances of variability while coming down to a specific filler shape. Usually the buyer
gets a technical sheet from the manufacturer, stating the average size of the filler without any tolerance
band [57]. The degree of stochasticity is primarily decided on the basis of available data and practical
experience from experimental observation. Here we have considered a lesser degree of uncertainty in
the dynamic elastic modulus as the degree of control in this parameter to avoid measurement error is
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relatively more compared to the other parameters under consideration. The considered DOS in volume
fraction and elastic modulus comes mainly from the inevitable measurement error and manufacturing
variability. Even though we have considered the above-mentioned DOSs in the present analysis,
following the proposed generic ANN based stochastic methodology other DOSs can also be
considered, if necessary. Using the degree of stochasticity, an input space is created and fed into the
developed ANN model corresponding to which stochastic responses of SIF are obtained. These
stochastic responses resulting from the forward propagation of uncertainties are then statistically
characterized by computing their probability density function (PDF) at each time instant. The
computed probability density function reflects the statistical moments along with the likelihood of
stress intensity factor at a given time instant.
3.3 Sensitivity analysis
To examine the model robustness and evaluate the effect of individual or joint contribution of
the parametric uncertainties on the model response, a global sensitivity analysis is performed further.
Among several available variance-based methods, the variance decomposition-based Sobol sensitivity
analysis is known to quantify the contributions of each input parameter and their interactions to the
overall model output variance accurately. The output variance is decomposed into summands of
variances using the combinations of input parameters in increasing dimensionality. A model of the
form, can be decomposed into terms of increasing dimensionality as follows
[58] :
(8)
where,is model output,
, represents Input parameters, is the number of input
parameters. Similarly, the variance of the output can be decomposed as:
(9)
The decomposed variances can also be represented as:
(10)
Normalizing the equation with unconditional variance of output, we get,
(11)
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This decomposition results in normalized indices more commonly known as Sobol sensitivity indices.
Measure associated with the first term of equation (11) is known as the first order sensitivity index
(and accounts for the effect of on the output of the model. The subsequent terms give a measure
of higher order indices that interpret the parametric interactions and the effect of a single input
parameter along with all its possible interactions, leading to total effect sensitivity index (. The
expressions for these two indices are given as:
(12)
(13)
where is the i-th parameter and denotes the matrix of all factors but . In the present study, the
same has been implemented using a python library [59].
4. Results and discussions
4.1 Results of the surrogate model and the experimental findings
The proposed architecture of ANN model is used to predict the SIF histories corresponding to
different aspect ratios and volume fractions of glass fillers. A study highlighting the optimal balance
of bias and variance corresponding to the different sizes of the training dataset is performed and the
results for the same are shown in Fig. 5 (convergence study). Here the experimental data in the range
of aspect ratio 1 and 80 was used to train the network, and the experimental and predicted SIF values
were compared as shown in Fig. 6. The used ANN model was found to have a good prediction accuracy
as the mean absolute percentage error and the coefficient of determination were found to be 3.8% and
0.99 respectively. Later the SIF histories for aspect ratio 6 were predicted using this trained neural
network and the results for the same are shown in Fig. 7.
The results shown in Fig. 6 and Fig. 7 demonstrate the ability of the proposed ANN model to
efficiently handle the complex relationship of the input parameters and predict the SIF response of the
composite accurately for unforeseen scenarios. Such ANN model can also provide a clear
representation of output space in terms of the interactive plots of different input parameters. Fig. 8 is
one such representation where the distribution of SIF at three different time instants is given over the
entire domain of aspect ratio and volume fraction (note that only based on experimental
characterization it will be practically impossible to investigate such detailed interactive effects due to
the expense and time involved with such endeavor). It is observed that with an increase in the volume
fraction and aspect ratio of glass fillers, the SIF increases with time, which in turn increases the
resistance of the composite material to crack growth. It is evident that irrespective of the aspect ratio,
an increase in the volume fraction of the fillers increases the crack initiation toughness (SIF
corresponding to t = 0). However, the resistance to crack growth is more dominant for rod-shaped
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Fig. 5: Effect of the size of training dataset on prediction capability of ANN model. Percentage
error in the predicted values of the testing sample in terms of MAPE and standard deviation (S.D.)
fillers (AR = 80) with a higher volume fraction. The reason for the rod shaped fillers to have the
maximum fracture toughness is the highest dissipation of energy in this case owing to the underlying
failure mechanisms. Due to the relatively higher aspect ratio, rod shaped fillers can lead to matrix
cracking, filler-matrix interfacial separation, breakage of fillers along with filler pull out. Considering
the ~800 μm length of these fillers, crack bridging takes place and contributes in resisting the further
propagation of cracks. While in the other two filler types, crack bridging and energy dissipation is
found to be comparatively less. Considering the tensile strength of the glass fillers to be ~ 3 GPa, more
failure breakages lead to higher energy dissipation resulting in higher fracture toughness.
4.2 Uncertainty quantification
Having the ANN based predictive framework validated in the preceding subsection, here we
exploit the ANN model further for quantifying the effect of stochasticity in material and geometric
attributes at different time instants. The uncertainty quantification of dynamic fracture toughness of
glass-filled epoxy composites (leading to complete probabilistic characterization) is carried out
considering the stochastic effects in aspect ratio, dynamic elastic modulus and volume fraction. The
main idea of uncertainty analysis is to describe the complete set of possible outcomes corresponding
to the random/uncertain input space with the associated probability distributions. The probability of
having stochastic output values within a certain range is characterized by a probability distribution.
Following this probabilistic approach, here the uncertainty in the response of SIF history is described
in terms of probability density function (PDF) and stochastic bounds. The PDF response of SIF after
introducing the aforementioned degree of stochasticity in the input parameters as per the cases
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Fig. 6: Validation of the prediction capability of ANN model (spherical and rod shaped glass
fillers). Comparison of experimental and predicted values of stress intensity factor (A) for 5% Vf , (B)
for 10% Vf and (C) for 15% Vf when the ANN model is trained with the data corresponding to aspect
ratio 1 and 80.
specified in section 3.2, for rod, flake and spherical shaped fillers is shown in Fig. 9, 10 and 11
respectively. In these figures, probabilistic values of SIF are normalized with respect to the
corresponding deterministic SIF values, as presented in Fig. 6 and 7. These figures are the illustrations
of the probabilistic analysis conducted using the ANN model to predict the SIF history corresponding
to seven time instants. Probabilistic characterization of the stochastic response of stress intensity factor
corresponding to the cases where input stochasticity is considered as the coupled effect of all factors,
is presented for only 10% volume fraction of glass fillers for the sake of brevity.
(A) (B)
(C)
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Fig. 7: Validation of the prediction capability of ANN model (flake shaped glass fillers).
Comparison of experimental and predicted values of stress intensity factor using the dataset
corresponding to aspect ratio 6 for (A) 5% Vf , (B) 10% Vf and (C) 15% Vf .
As a reference to the individual or combined input variation, a solid line is drawn at the mean
level in all the PDF plots indicating the variability in the crack propagation. Skewness in the shape of
PDF plots indicates the increase in the non-linear relationship between the input and output space due
to the introduced uncertainty. The 3D plots of the PDF clearly indicate that as the time progresses, the
effect of introduced stochasticity in the input space has a less pronounced effect on the SIF response.
The input parameters are primarily responsible for the interfacial bond strength that contributes in the
energy dissipation properties of the composite, which in turn helps in resisting the crack initiation.
However, once the SIF reaches its critical value, a crack is initiated and as the crack progresses the
effect of stochasticity in the input space is less influential for the fillers with lower aspect ratio (spheres
and flakes) while sizeable influence can be observed for the rod shaped fillers. To account for the
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Fig. 8: ANN based predictions of SIF in three different time regimes over the complete domain
of aspect ratio and volume fraction. (A) within the pre-crack initiation regime (at t = -19.98 µs), (B)
when the crack initiates (at t = 0 µs) and (C) within the post-crack initiation regime (at t = 19.98 µs).
propagation of introduced stochasticity, uncertainty bounds for different filler shapes corresponding to
different stochastic cases are presented further in Fig. 12 - 14, as discussed in the following paragraphs.
The load transfer from the polymeric matrix to the filler reinforcement is the basic mechanism
for the working of a polymer composite material. This load transfer mechanism is governed by the
interfacial strength of the composite. Filler reinforcements embedded in the polymeric matrix cause
perturbations at the interface between the matrix and the reinforcement. The extent of these
perturbations depends on the geometry and the volume fraction of the fillers [41]. Hence the overall
composite behavior in terms of strength, stiffness and toughness strongly depends on the state of the
polymer-filler interface. The stochastic variability bounds in the composite fracture toughness due to
the effect of considered stochasticity in the input parameters are shown in Fig. 12, 13 and 14.
Additionally, such a framework of uncertainty quantification facilitates a ready assessment of the
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Fig. 9: Probability density function plot for rod shaped glass fillers. PDF plots accounting for the
stochastic response of stress intensity factor for variation in only aspect ratio (A) at 5% Vf,, (B) at 15%
Vf, and PDF responses of normalized SIF history for 10% Vf of rod shaped glass fillers when a
stochastic variation is introduced (C) only in the aspect ratio, (D) simultaneously in aspect ratio and
dynamic elastic modulus, (E) simultaneously in aspect ratio and volume fraction and (F)
simultaneously in aspect ratio, dynamic elastic modulus and volume fraction. Thus we show the results
for the individual stocahsticity in aspect ratio for 5%, 10% and 15% Vf, while different compound
effects of stocasticity are shown considering 10% Vf . The corresponding deterministic results are
shown in figure 5.
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Fig. 10: Probability density function plot for flake shaped glass fillers. PDF plots accounting for
the stochastic response of stress intensity factor for variation in only aspect ratio (A) at 5% Vf,, (B) at
15% Vf, and PDF responses of normalized SIF history for 10% Vf of flake shaped glass fillers when a
stochastic variation is introduced (C) only in the aspect ratio, (D) simultaneously in aspect ratio and
dynamic elastic modulus, (E) simultaneously in aspect ratio and volume fraction and (F)
simultaneously in aspect ratio, dynamic elastic modulus and volume fraction. Thus we show the results
for the individual stocahsticity in aspect ratio for 5%, 10% and 15% Vf,, while different compound
effects of stocasticity are shown considering 10% Vf . The corresponding deterministic results are
shown in figure 6.
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Fig. 11: Probability density function plot for spherical shaped glass fillers. PDF plots accounting
for the stochastic response of stress intensity factor for variation in only aspect ratio (A) at 5% Vf,, (B)
at 15% Vf, and PDF responses of normalized SIF history for 10% Vf of spherical shaped glass fillers
when a stochastic variation is introduced (C) only in the aspect ratio, (D) simultaneously in aspect ratio
and dynamic elastic modulus, (E) simultaneously in aspect ratio and volume fraction and (F)
simultaneously in aspect ratio, dynamic elastic modulus and volume fraction. Thus we show the results
for the individual stocahsticity in aspect ratio for 5%, 10% and 15% Vf,, while different compound
effects of stocasticity are shown considering 10% Vf . The corresponding deterministic results are
shown in figure 5.
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Fig. 12: Uncertainty bound for rod shaped fillers. Response bands for SIF history after introducing
stochasticity (A) only in the aspect ratio (DOS = 0.3), (B) simultaneously in aspect ratio (DOS = 0.3)
and dynamic elastic modulus (DOS = 0.1), (C) simultaneously in aspect ratio (DOS = 0.3) and volume
fraction (DOS = 0.2) and (D) simultaneously in aspect ratio (DOS = 0.3), dynamic elastic modulus
(DOS = 0.1) and volume fraction (DOS = 0.2). The corresponding deterministic results are shown in
the insets.
confidence in the experimental predictions for further industrial adoption. Considering the first
stochastic case (only AR), among the three filler types, rod-shaped glass fillers are found to exhibit
maximum variation in the SIF response. The geometrical aspect of the fillers is directly related to the
interfacial strength and hence has a direct impact on the failure mechanisms. Considering the larger
aspect ratio of rod-shaped fillers, any variability in the size will lead to a significantly varying SIF
response while for the other two filler types, the variability in the SIF response is relatively less owing
to the inherited shape and lower aspect ratio (it is least for spherical-shaped fillers). When stochasticity
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Fig. 13: Uncertainty bound for flake shaped fillers. Response bands for SIF history after introducing
stochasticity (A) only in the aspect ratio (DOS = 0.3), (B) simultaneously in aspect ratio (DOS = 0.3)
and dynamic elastic modulus (DOS = 0.1), (C) simultaneously in aspect ratio (DOS = 0.3) and volume
fraction (DOS = 0.2) and (D) simultaneously in aspect ratio (DOS = 0.3), dynamic elastic modulus
(DOS = 0.1) and volume fraction (DOS = 0.2). The corresponding deterministic results are shown in
the insets.
is introduced simultaneously in two parameters (AR and Ed ; AR and Vf), as shown in Fig. 12 (B), (C)
- 14 (B), (C), an increase in the variability bound is found. Also, it is evident from the above figures
that combined uncertainty in the input parameters (Fig. 12 (D) – 14 (D)) results in higher variability
(wider bound) in the stochastic SIF history compared to the other cases.
The deviation in the stochastic SIF responses from the deterministic SIF values for all the filler
types at three different time instants is shown in Fig. 15. Three time instants are chosen in a way that
one time instant (t = -19.98 µs) represents the time regime before crack initiation, another time instant
(t = 0) indicates the time at which the crack initiates and the third one (t = 19.98 µs) represents the time
regime after the initiation of crack. Considering that the procedure will remain the same, other time
22
Fig. 14: Uncertainty bound for spherical shaped fillers. Response bands for SIF history after
introducing stochasticity (A) only in the aspect ratio (DOS = 0.3), (B) simultaneously in aspect ratio
(DOS = 0.3) and dynamic elastic modulus (DOS in = 0.1), (C) simultaneously in aspect ratio (DOS =
0.3) and volume fraction (DOS in = 0.2) and (D) simultaneously in aspect ratio (DOS = 0.3), dynamic
elastic modulus (DOS = 0.1) and volume fraction (DOS = 0.2). The corresponding deterministic results
are shown in the insets.
instants could also be readily investigated. Note that the considered three time instants, each
representing one time regime, are chosen here to present typical numerical results. The rod shaped
fillers show maximum deviation from the deterministic SIF values in all the three time regimes. In
case of spherical shaped glass fillers, once the crack initiates, the deviation caused by introducing
uncertainty simultaneously in the aspect ratio and dynamic elastic modulus has shown a less prominent
effect on the SIF when compared with the deterministic values. Also, in the pre-crack initiation regime,
in case of spherical fillers, the combined uncertainty in the aspect ratio and volume fraction, results in
a comparatively lesser deviation in SIF while in the post-crack initiation regime, this deviation is in
almost the similar range with flake shaped fillers. Although the dynamic elastic modulus is one of the
key parameters in describing the fracture toughness, not much variation is seen in this factor for the
23
Fig. 15: Stochastic variation of SIF at different times. Deviation of mean stochastic SIF responses
from the deterministic values when there is uncertainty (A) only in the aspect ratio (DOS = 0.3), (B)
simultaneously in aspect ratio (DOS = 0.3) and dynamic elastic modulus (DOS = 0.1), (C)
simultaneously in aspect ratio (DOS = 0.3) and volume fraction (DOS = 0.2) and (D) simultaneously
in aspect ratio (DOS = 0.3), dynamic elastic modulus (DOS = 0.1) and volume fraction (DOS = 0.2).
different shapes of fillers. However, the filler shape is observed to have more control over the kind of
crack interaction that takes place and consequently affects the ultimate fracture toughness of the
composite [11]. The sensitivity of different such critical input parameters is further quantified in the
following subsection.
4.3 Sensitivity analysis
In order to quantitatively characterize the importance of each input parameter along with the
consideration of parametric interactions, a global sensitivity analysis as discussed in section 3, is
performed. This facilitates the identification of the input parameter with the most crucial effect on the
crack initiation toughness. The sensitivity analysis is carried out on the rod shaped fillers here as these
fillers have shown the most pronounced effect of uncertainty. Therefore, to account for the
24
Fig. 16: Global sensitivity analysis for crack initiation toughness. (A) Sobol’s first order sensitivity
index to highlight the individual impact of each parameter on the variance of crack initiation toughness
(B) Sobol’s total effect sensitivity index highlighting the overall impact of a single input parameter
involving all its interactions with other parameters (refer to equation 13). The results are presented for
the instance of crack initiation (t = 0).
probabilistic distribution of the input space and the parametric interactions, Sobol’s first order and total
effect indices are calculated. Parametric interactions are a result of the non-additive effect of nonlinear
components for the prediction of crack initiation toughness. Keeping in view the inherent uncertainties
and the resulting random input fields, Sobol’s sensitivity indices act as performance measures of the
random output variable for achieving adequate control and reliable manufacturing process. Based on
the output variance decomposition, the statistically most important parameter is discovered and the
results are shown in Fig. 16.
Out of the three critical parameters under consideration, aspect ratio is found to have the highest
value of Sobol’s first order sensitivity index followed by dynamic elastic modulus and then the volume
fraction. The reason for this trend could be attributed to the resulting change in internal stresses and
strains within the composite in response to the aspect ratio variations, which in turn affects the overall
toughness of the composite. Additionally, the analysis with the consideration of all the possible
parametric interactions of any one parameter with the remaining two, indicates the same order of
sensitivity indices. In general, the quantitative analyses considering both individual and interaction
effects of sensitivity reveal that the aspect ratio of the glass fillers is the most influential parameter,
leading to the highest variability impact on the crack initiation toughness of particulate polymer
composites.
5. Conclusions
This work creates a computational bridging between limited experimental observations and
large-scale data-driven approaches like Monte Carlo simulation to quantify the effect of inevitable
25
uncertainties in the dynamic fracture toughness of particulate composites. This is achieved by
exploiting artificial neural network as a surrogate of the original physical experiments based on
advanced techniques like digital image correlation and scanning electron microscopy. The source
uncertainty in critical geometric and material parameters like aspect ratio, dynamic elastic modulus
and volume fraction are captured at different time regimes and subsequently the ANN model is coupled
with Monte Carlo simulation for efficient propagation and quantification of uncertainty in dynamic
fracture toughness. The key findings of this experimental data-driven uncertainty quantification for
dynamic fracture toughness of particulate glass-filled epoxy composites are as follows:
i) The effect of uncertainty in the input space has the most pronounced effect on the stress intensity
factor in the pre-crack initiation regime for all the considered stochastic cases.
ii) Stochasticity in aspect ratio in combination with the dynamic elastic modulus affects the crack
initiation toughness more compared to the other possible individual and compound scenarios of
uncertainty.
iii) The rod-shaped glass fillers have shown the most prominent effect of uncertainty on fracture
toughness in all the considered stochastic cases.
iv) Aspect ratio is found to be the most sensitive input parameter in response to the inevitable
stochasticity.
In general, the numerical results convincingly demonstrate using a computational mapping
between the stochastic input and output parameter spaces that the effect of uncertainty gets pronounced
significantly while propagating from the compound source level to the impact responses. The
comprehensive inferences drawn based on experiment-informed large-scale data-driven analyses, as
presented in this study, would allow the researchers and designers to make an informed decision on
the optimum design of particulate composites including the effect of inevitable source-uncertainties
along with a sense of the manufacturing quality control required for different critical input parameters,
leading to an inclusive analysis and design paradigm.
Acknowledgments
AS and VK would like to acknowledge the financial support received from DST‐
SERBSRG/2020/000997. TM would like to acknowledge the support received through the Science
and Engineering Research Board (Grant no. SRG/2020/001398), India.
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