Prediction of Tensile Strength of Polymer Carbon Nanotube Composites Using Practical Machine Learning Method
Prediction of Tensile Strength of Polymer Carbon Nanotube Composites Using Practical Machine Learning Method
Prediction of Tensile Strength of Polymer Carbon Nanotube Composites Using Practical Machine Learning Method
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Tien-Thinh Le 1,*
1
Université Paris-Est, Laboratoire Modélisation et Simulation Multi Echelle, MSME UMR 8208
Abstract: This paper is devoted to the development and construction of a practical Machine Learning
(ML)-based model for the prediction of tensile strength of polymer carbon nanotube (CNTs)
composites. To this end, a database was compiled from the available literature, composed of 11 input
variables. The input variables for predicting tensile strength of nanocomposites were selected for the
following main reasons: (i) type of polymer matrix, (ii) mechanical properties of polymer matrix, (iii)
physical characteristics of CNTs, (iv) mechanical properties of CNTs and (v) incorporation
parameters such as CNT weight fraction, CNT surface modification method and processing method.
As the problem of prediction is highly dimensional (with 11 dimensions), the Gaussian Process
Regression (GPR) model was selected and optimized by means of a parametric study. The correlation
coefficient (R), Willmott’s index of agreement (IA), slope of regression, Mean Absolute Percentage
Error (MAPE), Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE) were employed
as error measurement criteria when training the GPR model. The GPR model exhibited good
performance for both training and testing parts (RMSE = 5.982 and 5.327 MPa, MAE = 3.447 and
3.539 MPa, respectively). In addition, uncertainty analysis was also applied to estimate the prediction
confidence intervals. Finally, the prediction capability of the GPR model with different ranges of
values of input variables was investigated and discussed. For practical application, a Graphical User
Interface (GUI) was developed in Matlab for predicting the tensile strength of nanocomposites.
1
Keywords: Carbon nanotubes; Nanocomposites; Machine learning; Tensile strength; Gaussian
Process Regression.
1. Introduction
In the face of major challenges in technological innovation as well as from economic and
conventionally, novel materials are done discovered and characterized by experimentation (Pablo et
al., 2019). In general, laboratory experimentation takes a long time (10-20 years), for the development
and testing of materials with appropriate properties (de Pablo et al., 2014). It should also be noted
that these laboratory experiments are generally complicated, and costly both in terms of resources
and equipment. Despite all efforts, it is only possible to test a limited number of ingredients and/or
In order to develop new materials more quickly, several crucial projects have been announced
and received significant attention from researchers and engineers all around the world. In the USA,
in 2011, President Obama introduced the Materials Genome Initiative (MGI), which supports US
research institutions in the discovery, design, manufacturing and deployment of advanced materials
at twice the speed and for a fraction of the cost (i.e. from about 10-20 years to about 5-10 years) (de
Pablo et al., 2014; Green et al., 2017; Schmidt et al., 2019). MGI is frequently a candidate in bridging
the gap between materials experiment and theory. Such a project could also promote hugely data-
intensive and systematic research approaches. Three main contributions are needed to serve MGI’s
purpose: computational tools, experimental resources and digital data (Pablo et al., 2019). Indeed,
such a combination is crucial in accelerating the development of new materials. Computational efforts
could allow researchers to efficiently explore the composition and properties of materials, whereas
2
experimental data are essential for validating numerical models. In fact, huge savings are made in
terms of time and cost of material design by combining information from simulations and appropriate
In terms of polymeric nanocomposites, recently, research and development on this aspect have
become a very important subject in materials science – particularly, polymers reinforced with carbon
nanotubes (CNTs). Typical features of CNTs are included: a significant aspect ratio up to several
thousands, a specific surface area up to 1300 m2/g, a low density, as well as exceptional stiffness
(Young’s modulus up to 1,000 GPa) (Gojny et al., 2005; Harutyunyan et al., 2008; Peigney et al.,
2001; Sinnott et al., 1998; Stan et al., 1999). As demonstrated by various investigations in the
literature, CNTs exhibit crucial capability in improving various properties of polymeric matrices –
for instance, the optical (Singh et al., 2008; Tang and Xu, 1999), electrical (Choi et al., 2003;
McCullen et al., 2007), chemical (Garg and Sinnott, 1998; Liu and Kumar, 2014), electromagnetic
(Nanni and Valentini, 2011; Thomassin et al., 2013), thermal (Kuan et al., 2005; Yuen, Ma, Chiang,
et al., 2007), fire-retardant (Dewaghe et al., 2011; Kausar et al., 2017) and tribological (Song et al.,
2019) properties, especially in terms of mechanical responses (Arash et al., 2014; Breton et al., 2004;
Li et al., 2019; Montazeri et al., 2010; Shirkavand Hadavand et al., 2013; Spitalsky et al., 2010). It
should be pointed out that such improvement in CNT nanocomposites is the global result of an
efficient load transfer from the matrix to the CNTs (Gojny et al., 2005; Schadler et al., 1998). Indeed,
boundary due to (i) strong chemical interactions and (ii) geometric interactions (i.e. high specific
surface area) between the particle surface and the polymer segments near CNTs (Coleman et al.,
3
2006; Spitalsky et al., 2010). Such an interphase region, also exhibiting strong mechanical properties
(Han et al., 2014), successfully transfers the applied load from the matrix to the CNTs, enhancing the
several critical issues need to be addressed in the area of manufacture. Due to the Van der Waals
forces and Coulomb attractions, CNTs attempt to create interwoven networks and to clump together
to form “clusters” (Han et al., 2014; Rao et al., 2013). Consequently, major heterogeneities may result
from in the nanocomposite microstructure. Various techniques have been introduced in the literature
to investigate the distribution of CNTs in the polymer matrix. They mainly include solution mixing,
melt blending and in situ polymerization (Spitalsky et al., 2010). Zou et al. (Zou et al., 2004) have
increased the screw speed when incorporating CNTs into polyethylene in order to obtain a uniform
dispersion. Liu et al. (Liu and Choi, 2012) have compared different organic solutions to improve the
(Dondero and Gorga, 2006) employed the melt fiber spinning processing method to investigate
processing method has been used by McCullen et al. (McCullen et al., 2007) in fabricating
nanocomposites largely depend on processing methods (Coleman et al., 2006; Moniruzzaman and
Winey, 2006).
Another important issue that could considerably affect the effective properties of
nanocomposites is chemical modification at the surface of CNTs. Such modification may be classified
4
into two main categories: non-covalent and covalent bonding between the CNT surface and polymer
(Spitalsky et al., 2010). The first category indicates that polymer chains are simply wrapped around
the surface of the CNTs, whereas in the second category, polymer chains are grafted to the CNTs by
strong chemical bonds. Jin et al. (Jin et al., 2007) have modified the surface of CNTs by using acid
and diamine groups in order to improve the mechanical properties of CNTs in poly(ethylene
terephthalate). They found that both acid and diamine modifications were crucial in increasing the
strength of the nanocomposite compared to the use of only pristine CNTs. In another study, Kwon et
al. (Kwon and Kim, 2005) experimentally demonstrated that acid‐treated CNTs exhibit higher
mechanical performance than pristine CNTs when incorporated into polyurethane. Clearly, the
stronger the chemical bonds between CNTs and polymer that are established, the better load transfer
is obtained.
strategies have been proposed involving both analytical calculation and numerical simulations such
as continuum mechanics (Thostenson and Chou, 2002; Gupta and Harsha, 2014; Wang et al., 2018;
Chen et al., 2011; Hassanzadeh-Aghdam and Jamali, 2019), molecular dynamics (Ansari et al., 2018;
Frankland et al., 2003; Griebel and Hamaekers, 2004; Sharma et al., 2015), and multiscale approaches
(Alian et al., 2015; Banerjee et al., 2016; Choi et al., 2016; Han et al., 2014; Radue and Odegard,
2018; Tran et al., 2019). The Halpin-Tsai equation has also been employed by Kanagaraj et al.
2019) have derived a new form of the Halpin-Tsai formula taking account of random dispersion, non-
5
straight shape and the agglomerated state of the CNTs in the model. In terms of numerical simulations,
the finite element method has been largely employed to calculate the effective properties of CNT
nanocomposites. In this strategy, the interphase can be treated entirely as a load transfer region
between the CNTs and the polymer matrix (Banerjee et al., 2016). However, various assumptions
have been made in considering these approaches. For instance, Golestanian et al. (Golestanian and
Shojaie, 2010) have investigated the mechanical behavior of nanocomposites, assuming (i) perfect
bonding between CNTs/matrix and (ii) elastic properties in the interphase region. In another study,
Han et al. (Han et al., 2014) have modeled the overall mechanical properties of nanocomposites,
neglecting the influence of CNT diameter. Again, Wernik et al. (Wernik and Meguid, 2014) made an
assumption of a uniform distribution of polymer chains around the embedded CNTs. Moreover, such
As mentioned above, the literature review reveals that the overall mechanical properties of
geometry of CNTs, processing method, chemical modification, etc. Hitherto, both analytical and
numerical simulations have not always had the required ability to investigate the relationship between
such input information and the target mechanical properties of nanocomposites. In addition,
numerous limitations may occur when manufacturing real CNT nanocomposites, including
over/under dispersion of CNTs, agglomeration and aggregation resulting in poor behavior, poor
mechanical properties in the interphase region leading to low efficiency in load transfer, etc.
Therefore, it is clear that a more robust method is required in order to better understand and predict
6
the effective properties of such nanomaterials (Kopal et al., 2018; Molina et al., 2019; Yousef et al.,
2011).
The main objective of this study is to develop a quick and robust computational tool based on
machine learning (ML) Gaussian Process Regression (GPR) model to predict the tensile strength of
CNT/polymer nanocomposites. The GPR model was trained and validated against experimental data.
To this end, a dataset concerning 198 configurations was compiled from the available literature. The
input variables of the dataset included mechanical properties of separated phase, density of polymer
matrix, processing method, geometry of CNTs (i.e. average length and diameter), modification
method at the CNT surface, etc. On the other hand, tensile strength of nanocomposite is the output of
the prediction problem. Finally, results of prediction are presented and discussed.
2. Research significance
In this study, for the first time, a ML-based GPR numerical tool was developed and optimized
to estimate the tensile strength of CNT polymer nanocomposites. The model was trained and validated
against experimental data, which were established based on relevant information: (i) type of polymer
matrix, (ii) mechanical properties of polymer matrix, (iii) physical characteristics of CNTs, (iv)
mechanical properties of CNTs and (v) incorporation parameters such as weight fraction of the CNTs.
Moreover, the CNT surface modification method and processing method were also taken into
account, so as to explore the influence of these factors on the nanomaterials’ macroscopic behavior.
For practical reasons, a Graphical User Interface (GUI) was developed in Matlab for predicting the
tensile strength of nanocomposites. Without solving complex mechanical equations, the proposed
GPR model was able to efficiently predict and analyze the macroscopic behavior of the nanomaterials.
The trained model is able to assist the initial phase of nanocomposite investigation and design before
7
3.1. Database
The data used in this work have been collected from the available literature. Details of the
database are presented in Appendix A (see Tables A1, A2 and A3 for summary information). In this
database, 23 different polymers have been used, combined with 22 incorporating methods and 20
CNT surface modifications. These parameters were coded as indicated in Table A2. Table A3
presents the initial statistical analysis of each variable in the database, involving its min, average,
max, standard deviation and coefficient of variation (CV, in %). In addition, Table A4 indicates the
linear statistical correlation coefficient between each pair of variables in the database.
The variables in the database have been selected and categorized based on the following
criteria:
modification method.
instance, Kernel Ridge regression (Mannodi-Kanakkithodi et al., 2016), Recursive Neural Networks
(Duce et al., 2006), Artificial Neural Network (Molina et al., 2019), Radial Basis Function Neural
Network (Kopal et al., 2019) and Graph Convolutional Neural Networks (Zeng et al., 2018). In the
present study, Gaussian Process Regression (GPR) was proposed to predict the tensile strength of
8
polymer nanocomposites. A Gaussian process (GP) is collection of random variables, any finite
number of which have (consistent) joint Gaussian distributions (Rasmussen, 2003). It can be used for
solving non-linear regression (Rasmussen, 1997; Williams and Rasmussen, 1996) and classification
(Nickisch and Rasmussen, 2008; Williams and Barber, 1998) problems. One of the features of GPR
is that it directly defines a prior probability over a latent function. GPR is fully expressed as a
Gaussian process of its mean function m( x) and covariance (kernel) function k ( x, x ') :
The mean vector represents the central tendency of the function f and normally it is assumed
to be zero (Rasmussen, 2003). The covariance matrix describes the structure and shape of the
function. The relation between the input and output variables is defined as:
y = f ( x) + (2)
where is called the independent noise, which is covered by a distribution of a zero mean
= Ã(0, n2 ) (3)
where
y = [ y1 , y2 ,..., yn ]T (5)
9
and I is a M M matrix. From the definition of the Gaussian process introduced in (MacKay,
1998), the marginal distribution L(f ) is defined by a Gaussian with a zero mean and a covariance
Here, we use the term “marginal” to indicate that we are dealing with a non-parametric model.
Observe that Equations (4) and (5) follow the Gaussian distribution; the marginal distribution of y is
to the input variables x* and * as the corresponding noise. The joint Gaussian distribution is then
defined as:
éy ù æéf ù é ùö æ éK y K* ùö
ê ú= ççê ú+ ê ú÷ ÷» Ã
çç ê ú÷
÷
÷
êy* ú ççèêf ú ÷
ê * ú÷ çç0, ê T 2 ú÷
(9)
ë û ë*û ë ûø è êëK * K ** + n ú
û÷
ø
where K * k (x* , x1 ), ... , k (x* , x M ) and K** k (x* , x* ) . Based on the rules for
T
m( x* ) = K *T K -y 1y (10)
2 (x* ) = K ** - K *T K -y 1K * + n2 (11)
In order to determine the inverse of the covariance matrix K y , the Cholesky decomposition
(Higham, 1990) can be applied. The covariance (kernel) function is a very important factor in GPR
10
as it defines the similarity of the data, which has a major impact on the prediction results (Rasmussen,
2003). In this study, the rational quadratic kernel function was employed to predict the tensile strength
-
æ r2 ö
÷
k (xi , x j ) = ççç1 +
2
÷
2÷
(12)
è 2 l ÷
f
ø
where r is the Euclidean distance between two variables xi and x j defined as:
r ( xi x j )T ( xi x j ) (13)
l and f are the characteristic length scale and the signal standard deviation, respectively.
The hyperparameter of the covariance function can be calculated by several methods (Murray and
Adams, 2010).
In this work, the GPR model was trained using the Mean Squared Error cost function, while
the cross-validation was chosen as 5 (Dao, Adeli, et al., 2020; The MathWorks, 2018). It is worth
noting that MSE is commonly used as a cost function when training machine-learning models
The Monte Carlo technique has been widely used in order to take account of randomness of
the input space (Guilleminot et al., 2013; Pham, Nguyen, et al., 2019; Le et al., 2016; Pham et al.,
2020), especially in mechanics of materials. Hun et al. (Hun et al., 2019) studied crack propagation
in heterogeneous media within a probabilistic context of Monte Carlo simulations. In another work,
Capillon et al. (Capillon et al., 2016) studied the influence of random uncertainty in structural
dynamics for composite structures using the Monte Carlo method. Several works have successfully
applied the Monte Carlo method to take account of randomness in mechanics (Le et al., 2015, 2020;
11
Soize et al., 2015; Staber et al., 2019; Tran et al., 2016, 2018). The main idea of the Monte Carlo
method is to reproduce the output responses a certain number of times by randomly choosing values
of the input variables in the input space (Dao, Ly, et al., 2020; Le, 2020). That way, fluctuations in
each sub-space in the input space can be entirely propagated to the output response through the model
mapping (QH Nguyen et al., 2020). In this study, a numerical parallelization scheme was developed
to execute the randomness propagation process. The statistical convergence of the Monte Carlo
100 m
mS
f conv = Sj, (14)
j=1
where m is the number of Monte Carlo iterations, S is the random variable considered and S is the
average value of S.
In the present work, three quality assessment criteria – Coefficient of Determination (R2),
Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE) – have been used in order to
validate and test the developed AI models. R2 allows us to identify the statistical relationship between
two data points and can be calculated using the following equation (Pham, Jaafari, et al., 2019; Ly,
y0, j - y y p, j - y
N
j=1
R= 2 N 2
, (15)
y0, j - y y p, j - y
N
j=1 j=1
12
where N is the number of observations, y p and y are the predicted and mean predicted
values while y0 and y are the measured and mean measured values of Young’s modulus of the
nanocomposite, respectively ( j 1: N ). In the case of RMSE and MAE, which have the same units
as the value being estimated, a low value of RMSE and MAE basically indicates good accuracy of
the models’ prediction output. In an ideal prediction, RMSE and MAE should be zero. RMSE and
MAE are given by the following formulae (Ly, Le, et al., 2019; Ly, Monteiro, et al., 2019; Ly, Pham,
N
RMSE ( y
i 1
0 y p )2 / N (16)
N
1
MAE
N
y
i 1
0 yp (17)
In addition, Willmott’s index of agreement (IA), Mean Absolute Percentage Error (MAPE)
and slope of regression plots have been employed in this study. IA and MAPE are given by the
following equation (Kim and Kim, 2016; TC Nguyen et al., 2020; Qi, Tang, et al., 2019):
y yp
N
2
0
IA 1 i 1
(18)
y
N 2
0 y yp y
i 1
1 N y0 y p
MAPE y
N i 1
(19)
0
13
Figure 1 presents a block diagram representation of the current approach including five main
steps: (1) preparation of data, (2) construction of model, (3) comparison of performance, (4)
Step 1, Preparation of data: In this step, the data were collected from the available literature
Step 2, Construction of model: In this step, the GPR model was trained by using MSE cost
function, Monte Carlo technique was employed for propagation of variability of inputs, and
optimal training rate was also identified (Sections 4.1 and 4.2).
Step 3, Comparison of performance between the GPR model and other methods for validating
Step 5, In this step, post-analysis was performed including regression, uncertainty analysis,
and exploration of current performance and limitations. Finally, a Graphical User Interface
14
Figure 1. Flowchart of the methodology of this research.
15
4. Results and discussion
It was highlighted in previous ML studies that the capability of ML models strongly depended
on the training set size (Qi, Chen, et al., 2019; Qi et al., 2020; Qi, Tang, et al., 2019). Therefore, a
parametric study was first conducted to explore the influence of training set size on the final
performance of the GPR model. Nine training set sizes, denoted by Nrate, were chosen, varying from
10 to 90% in intervals of 10%. For each case of Nrate, 2000 random sampling processes were applied
to take account of the variability of the input space. The convergence of the random sampling
processes is presented in Section 4.1, while the influence of Nrate is shown in Section 4.2.
The statistical convergence of the GPR model in terms of R, RMSE and MAE is analyzed in
this section (also see Equation (14). Figures 2a and 2b show the convergence of R over 2000 Monte
Carlo simulations, for training and testing data, respectively. Figures 2c and 2d show the convergence
of RMSE over 2000 Monte Carlo simulations, for training and testing data, respectively. Figures 2e
and 2f show the convergence of MAE over 2000 Monte Carlo simulations, for training and testing
data, respectively. A high fluctuation was observed when the number of Monte Carlo simulations is
relatively small (less than 400), especially for RMSE and MAE using training data. However, a small
fluctuation was obtained in the case of R. Consequently, random sampling was necessary in order to
estimate the statistical fluctuation with respect to all error criteria. It can be concluded that 2000
Monte Carlo runs is sufficient in order to obtain representative statistical results for all Nrate
configurations, as seen in Figure 2. In addition, the same was observed for IA, MAPE and Slope (but
not shown).
16
17
Figure 2. Convergence of random sampling runs using the training data for (a) R, (c) RMSE,
(e) MAE; using the testing data for (b) R, (d) RMSE, (f) MAE.
18
As reliable results were obtained as shown in Figure 2, this section presents the influence of
Nrate on the performance of the GPR model. Figures 3a and 3b present the evaluation of R as a function
of Nrate, for training and testing data, respectively. Figures 3c and 3d present the evaluation of IA as
a function of Nrate, for training and testing data, respectively. Figures 3e and 3f present the evaluation
of Slope as a function of Nrate, for training and testing data, respectively. The same presentation was
organized in Figure 4, for MAPE, RMSE and MAE, respectively. It should be noted in Figures 3 and
4 that the box plot was adopted to represent the probability distribution of error criteria over 2000
random sampling runs, including median, mean, 25-75% and 9-91% percentiles.
Figures 3 and 4 show significant fluctuation in the case of Nrate = 10, 20, 30, 40, 50 and 60%,
using the training data. Moreover, using these configurations of Nrate, the performance of GPR using
the testing data is low, with respect to R, IA, Slope, MAPE, RMSE and MAE, respectively. When
Nrate is greater than 70%, it is seen that the fluctuation of error criteria becomes smaller using training
data, together with an improved performance of GPR when applied to the testing data. In addition, a
nonlinear (logarithmic) relationship between error criteria and Nrate can be established for the testing
data. This finding was in accordance with the literature (Qi, Chen, et al., 2019; Qi, Tang, et al., 2019).
The optimal Nrate was found to be 80%, exhibiting 158 samples in the training data and 40 samples
in the testing dataset. Finally, Figure 5 shows the 2D histogram between R and other error criteria
19
20
Figure 3. Influence of Nrate using the training data for (a) R, (c) IA, (e) Slope; using the testing data
21
22
Figure 4. Influence of Nrate using the training data for (a) MAPE, (c) RMSE, (e) MAE; using the
23
Figure 5. 2D histogram distributions between R and (a) RMSE, (b) MAE, (c) MAPE, (d) IA and
(e) Slope.
1
In this section, the prediction performance and robustness in the presence of variability in the
input space, of the GPR model, are compared to other methods: Linear Regression (LN), Regression
Tree (RT), Support Vector Machine (SVM), Ensemble Boosted Tree (EBT), Fuzzy Logic (FL), and
Artificial Neural Network (ANN) (Matloff, 2017; Sen and Srivastava, 1997; Shanmuganathan and
Samarasinghe, 2016; Witten et al., 2016). The optimal value of Nrate=80% was selected, as deduced
in the previous section. 2000 random sampling processes were performed at Nrate=80% for LN, RT,
SVM, EBT, FL and ANN. In the case of RT, the minimum number of leaf node observations was 4.
In the case of SVM, a third-order polynomial was used, the values of box constraint and half the
width of the epsilon-insensitive band were 0.7413, and 0.0741, respectively, with automatic selection
of the Kernel scale factor. In the case of EBT, least-squares boosting was used, the number of
ensemble learning cycles and learning rate for shrinkage were 30 and 0.1, respectively. In the case of
FL, the model was generated using fuzzy C-means (FCM) clustering; the number of clusters was 10,
with a Gaussian membership function. In the case of ANN, one hidden layer architecture containing
20 neurons was selected, the model was trained using the Levenberg-Marquardt backpropagation
algorithm, and the activation function was hyperbolic tangent sigmoid. Like the GPR method, LN,
RT, SVM, EBT, FL and ABB were trained using the Mean Squared Error cost function, while the
Figures 6a–6f present the comparison in box plot mode for error measurement with respect to
R, IA, Slope, RMSE, MAE and MAPE, using the testing data. It is worth noting that the same box
plot presentation was also used above to represent the probability distribution of error criteria over
2000 random sampling runs. In Figure 6, the performance of the LN model is poor – beyond the
2
presentation range. Figures 6a–6e demonstrate that the GPR method outperforms other models, with
regard to the mean, median and standard deviation (i.e. highest mean and median values in the cases
of R, IA, lowest mean and median values in the cases of RMSE and MAE, and smallest 25%–75%
quantile range in all cases). On the other hand, the best performing models are ANN and RT with
regard to the Slope and MAPE criteria, respectively. Nonetheless, the performance of the GPR model
is not far short, especially in the case of Slope, where the GPR model produces the smallest 25%–
75% quantile range. It can be stated that the GPR model is the most efficient model with respect to
mean, median and as standard deviation (25%–75% quantile range). It should be noted that the
observation in the 25%–75% quantile range confirms that the GPR model is the most robust model,
as it produces the smallest variation when taking account of variability in the input space through
3
4
Figure 6. Distribution of different error measurements using the testing data: (a) R, (b) IA, (c)
Slope, (d) RMSE, (e) MAE and (f) MAPE. The performance in the case of Linear Regression (LN)
In order to quantitatively appreciate the difference between the GPR method and others, Table
1 compares the mean values of the distributions presented in Figure 6 for R, IA, Slope, RMSE, MAE
and MAPE. The gain values between the results obtained by the GPR model and others are also
calculated using Equation (20). Table 1 shows that the LN method performs poorly. On the other
hand, the GPR model offers the best performance, especially with respect to R, IA, RMSE, and MAE.
%Gain (20)
( ) / 100 in case of: RMSE, MAE and MAPE
others GPR others
Table 1. Detailed comparison between GPR and other methods in terms of mean value over
5
GPR 0.96 0.91 0.98 12.14 7.56 31.73
This section presents the identification of weights of input parameters for predicting the tensile
input in question before training and testing. For instance, the weight of the second input parameter,
the density of matrix, was characterized by training and testing the GPR model with a database where
the second column was set to zero (a constant) in all positions. This procedure for identifying weights
enables us to efficiently find the influence of each input parameter on prediction performance,
especially from a statistical point of view. To this end, 2000 Monte Carlo simulations were conducted
for each case of weight identification (11 input parameters in total). The prediction performance of
these 11 weight identifications in terms of error criteria was then compared to the reference case –
i.e. where full statistical information is available for all inputs. It is worth noting that the greater the
deviation from the reference case, the stronger the influence of the input parameter.
Figure 7 presents the difference in average value of R, IA, Slope, RMSE and MAE, compared
between the weight identification procedures and the reference case, for 11 sorted input parameters.
6
It should be noted that for convenience of presentation, all values were normalized to the maximum
in Figure 7. Original values (i.e. before normalization) are given in Table 2. It can be seen from Figure
7 that CNT surface modification method has the highest weight, as confirmed by R, IA, Slope, RMSE
and MAE. As indicated in Table 2, without statistical information from the CNT surface modification
method, the prediction performance is reduced by 3.37% in the case of R, 1.89% in the case of IA,
5.96% in the case of Slope, 32.49% in the case of RMSE and 28.55% in the case of MAE,
respectively. CNT weight fraction is the second highest weighted input parameter. As indicated in
Table 2, without statistical information from CNT weight fraction, the prediction performance is
reduced by 1.47% in the case of R, 0.79% in the case of IA, 3.14% in the case of Slope, 16.45% in
the case of RMSE and 16.25% in the case of MAE, respectively. Tensile strength of matrix is the
third highest weighted input parameter. As indicated in Table 2, without statistical information from
tensile strength of matrix, the prediction performance is reduced 0.89% in the case of R, 0.56% in the
case of IA, 1.28% in the case of Slope, 10.1% in the case of RMSE and 10.43% in the case of MAE,
respectively. The weights of other input parameters are shown in Figure 7 and indicated in Table 2.
7
Figure 7. Weights of input parameters with respect to different error measurement criteria.
As the output parameter of the prediction function is the tensile strength of the
nanocomposites, there is a direct relationship between such output and the tensile strength of the
polymer matrix used. It is worth noting that the tensile strength of the polymer matrix ranges from
0.49 MPa to 132 MPa, with a mean value of 42.10 MPa and a standard deviation of 38.10 MPa, as
indicated in Table A3. Moreover, 23 polymer matrices were used to create the database in this study.
CNT weight fraction is also one of the highest weighted input parameters. As a reinforcement phase
in the polymer matrix, the role of the weight fraction of CNTs has been demonstrated in various
investigations in the literature, especially in regard to mechanical responses (Arash et al., 2014;
Breton et al., 2004; Li et al., 2019; Montazeri et al., 2010; Safadi et al., 2002; Shirkavand Hadavand
et al., 2013; Spitalsky et al., 2010). For instance, Safadi et al. (Safadi et al., 2002) found an
enhancement in tensile strength for a polystyrene matrix of up to 24.5, 25.7, and 30.6 MPa when
8
incorporating a weight fraction of 1, 2, and 5% CNT, respectively; the tensile strength of the pure
polymer was 19.5 MPa. It is worth noting that the higher the volume of reinforcement in the matrix
phase, the better the load transfer within the system, leading to an overall improvement in the
mechanical properties. The important role of CNT surface modification has also been reported in the
literature. It is noteworthy that as CNTs have a large specific surface area, the stronger the chemical
interactions between the polymer matrix and the fillers, the better the effective properties (Cadek et
al., 2004). Indeed, research in this area could be considered intense, with various significant studies
with a view to improving: (i) the dispersion of the CNTs in the polymer matrix, (ii) the properties of
the CNTs, and (iii) the affinity between the CNTs and polymer matrix (Spitalsky et al., 2010). For
instance, Jin et al. (Jin et al., 2007) proposed to use diamine groups to modify the surface of the CNTs
when being incorporated into a poly(ethylene terephthalate) matrix. Their results showed an
improvement in the interaction between the polymer matrix and diamine-CNTs, leading to a major
increase in the tensile strength of the nanocomposites. In another study, Kwon et al. (Kwon and Kim,
2005) experimentally demonstrated that acid‐treated CNTs exhibit better mechanical performance
than pristine CNTs when incorporated into polyurethane. When the surface of CNTs is modified for
better affinity with the matrix, the properties of the interphase region (i.e. the disturbed area of the
polymer matrix) are also enhanced (Coleman et al., 2006; Spitalsky et al., 2010). The interphase
region successfully transfers the applied load from the matrix to the CNTs, enhancing the overall
mechanical performance of the nanocomposites (Han et al., 2014; Marcadon et al., 2013).
Nonetheless, more data should be gathered in further studies in order to construct a more
9
Table 2. Details of weighting: difference in average value (%).
Input parameter R IA Slope RMSE MAE
method
In this section, the performance of the developed GPR is investigated. For this purpose, error
estimation criteria R, Slope, IA, MAPE, MAE and RMSE are indicated in Table 3 for training, testing
and all data, respectively. The output results using the training dataset as a function of the
corresponding training target are shown in Figure 8a, whereas the predicted results using the testing
dataset versus the corresponding testing target are shown Figure 8b, respectively. Figure 8c shows
the regression graph of all data. In these figures, the linear fit was also highlighted, corresponding to
a slope indicated in Table 3. A slope of 0.960, 0.954 and 0.959 for the training, testing and all data,
10
respectively, was obtained, corresponding to an angle between the linear fit line and the horizontal
line of 43.83°, 43.65° and 43.80°, respectively. It is shown that for three data points, the linear fit is
very close to the diagonal line (i.e. 45°), which confirmed that the correlation coefficient R is very
strong (i.e. R = 0.991, 0.993, 0.991, for training, testing and all data, respectively). The same remark
can be made in respect of IA, which is 0.995, 0.996 and 0.995 using training, testing and all data,
respectively. In terms of RMSE, MAE and MAPE, the GPR model exhibits a high prediction
performance. As indicated in Table 6, RMSE = 5.982 MPa, 5.327 MPa and 5.856 MPa; MAE = 3.447
MPa, 3.539 MPa and 3.466 MPa; MAPE = 10.589, 33.394 and 15.196, using training, testing and all
data, respectively. Close agreement between the predicted and the actual values of tensile strength of
11
Figure 8. Regression graphs between actual and predicted tensile strength: (a) training data,
In addition, uncertainty analysis was also performed to quantify the uncertainty of the GPR
model during prediction. Nine quantile levels of the target tensile strength were introduced, ranging
from Q10 to Q90 with a resolution of 10%. The corresponding data in each level were deduced, and
then used to compute the standard deviation. Figure 9 presents the 68, 95 and 99% confidence
intervals, together with the average curve, for the prediction of tensile strength using GPR,
respectively. In this figure, the number of data in each quantile level are also presented. It is seen that
the higher the value (Q90) of tensile strength, the larger the confidence interval.
12
Figure 9. Uncertainty analysis of GPR model exhibiting 68, 95 and 99% confidence intervals,
respectively.
4.6. Discussion
In this section, the prediction capability of the GPR model is explored at different ranges of
values of input variables. Figures 10a, 10b and 10c show the bar graphs of error as a function of CNT
weight fraction, CNT average diameter and CNT average length, respectively. This error was defined
as error = (predicted-target)/target (in %). In the same context, Figure 11a, 11b and 11c present the
bar graphs of error (in a ranking mode) as a function of the polymer matrix, CNT modification method
and processing method, respectively. The bars were also classified from narrowest to widest values
of average error, including the cases of under (the predicted value is smaller than the target) and over
(the predicted value is bigger than the target) estimations, respectively. Standard deviation was also
highlighted, together with the number of data, in these figures for each case.
Figure 10a shows that for all CNT weight fractions, average error was generally smaller than
10%, except a few cases where wt = 0.075, 0.11, 0.15 and 15%. Regarding the error as a function of
13
the average CNT diameter, average error was smaller than 5% except the case of 5.5 nm diameter.
However, in the case of 75 nm diameter, a significant standard deviation was observed (it should be
noted that this case exhibits only two configurations). In the case of CNT average length, all average
errors were less than 5%, which highlights the strong performance of the GPR model.
14
Figure 10. Error analysis with respect to (a) CNT weight fraction, (b) CNT average diameter and
15
In terms of the polymer matrix used (see Figure 11a), for 18/23 (78%) polymer matrices
investigated in this study, the average error was smaller than 10%. It should be noted that PVA
exhibits a very strong standard deviation, because of its highest deviation, as observed in Figure 8a.
For five polymer matrices – PC, PU, PET, PEO and SBSS – the average error varied from 10 to 20%,
respectively. In terms of the CNT modification method (see Figure 11b), for 17/21 (81%) of
modification methods investigated in this study, the average error was smaller than 10%. For other
cases, the average error varied from 10 to 25%. In terms of processing method (see Figure 11c), for
17/22 (77%) processing methods investigated in this study, the average error was smaller than 10%.
For other cases, the average error varied from 10 to 20%, except the case of melt extrusion (32%, 1
data point). In general, the GPR model exhibits strong prediction capability. In order to improve the
model, more data should be investigated in further research, especially for cases where the error is
high.
16
17
Figure 11. Error analysis with respect to (a) polymer matrix, (b) CNT modification method and
Overall, without solving complex mechanical equations, a GPR model could be trained to
predict the macroscopic behavior of nanocomposites based on existing experimental data. Therefore,
the ML-based technique can assist initial estimating when studying CNT polymer nanocomposites
For further application, a Graphical User Interface (GUI) was implemented in Matlab 2018a
(The MathWorks, 2018). Figure 12 shows the main GUI, which is simple and easy to use. User can
select or enter the values of input variables. Lastly, the tensile strength of nanocomposites is displayed
directly by clicking the Start Predict button. The GUI is freely available at https://github.com/Tien-
ThinhLe/NanocompositeTensileStrengthPrediction.
18
Figure 12. Matlab’s GUI for the prediction of tensile strength of nanocomposites based on GPR
model.
Nonetheless, one of the major limitations of ML methods is the lack of physical constraints
as in conventional investigation theory (i.e. relationship between mechanical stress and deformation
through Hooke’s law). Most recently, Zhu et al. (Zhu et al., 2019) discussed physics-constrained deep
learning in a high-dimensional surrogate model. Stewart et al. (Stewart and Ermon, 2016) proposed
label-free supervision for supervising an Artificial Neural Network model under constraints derived
from prior known laws of physics. Berg et al. (Berg and Nyström, 2018) trained an Artificial Neural
Network to approximate the solution by minimizing the violation of the governing Partial Differential
Equations in complex geometries. In fact, in an ML algorithm, the models are generally treated as a
19
black box, a built-in model. This idea has, hitherto, presented many challenges for researchers and
In this work, an ML-based GPR model was developed and trained to predict the tensile
strength of polymer/CNTs nanocomposites. The database for training the GPR model was based on
relevant information of individual phase constituting the nanomaterials. For polymer matrix, polymer
matrix type and mechanical properties of the polymer were taken into account. For CNTs, the physical
characteristics of CNTs and mechanical properties of CNTs – the CNTs’ weight fraction, CNT
modification method and processing method were also investigated. The developed GPR model
shown by R = 0.991 and 0.993, RMSE = 5.982 MPa and 5.327 MPa, MAE = 3.447 MPa and 3.539
MPa, for training and testing parts, respectively. In addition, the performance of GPR as a function
of training set size and uncertainty quantification were studied. Moreover, the prediction capability
of the GPR model as a function of the input variables was investigated locally. Results showed a good
formulation for estimating the tensile strength of nanocomposites, and comparison with existing
formulae. In addition, more data should be collected, especially for cases with high error as identified.
In terms of practical application, a GUI based on Excel should be developed for wider applicability.
Such a GUI could allow us to quickly compute the mechanical properties of nanocomposites as a
function of various input variables. Besides, the application could be consolidated by updating the
20
data. Finally, ML-based models should also be investigated to predict other mechanical properties of
Acknowledgement: The author would like to thank Assoc. Prof. Johann Guilleminot, Department of
Civil and Environmental Engineering, Duke University, Durham, NC 27708, USA, for his helpful
Supplementary Materials: The GPR prediction model is appended to this paper, together with a
https://github.com/Tien-ThinhLe/NanocompositeTensileStrengthPrediction.
Data availability: The raw/processed data required to reproduce these findings will be made
available on request.
% of wt wt
Number Matrix
Reference prop- Polymer matrix min max
of data notation
ortion (%) (%)
21
Kuan et al. (Kuan et
8 4.0 Polyurethane PU 0.5 4
al., 2005)
Kanagarai et al.
2007)
2007)
Masuda et al.
Torkelson, 2008)
22
Liu et al. (Liu,
1 0.5 Poly(methyl methacrylate) PMMA 5 5
Tasis, et al., 2007)
Coleman et al.
2004)
McCullen et al.
2007)
Bokobza et al.
2007)
23
Ogasawara et al.
2004)
2007)
24
Hou et al. (Hou et
1 0.5 Polyvinyl alcohol PVA 0.2 0.2
al., 2009)
Srivastava et al.
2009)
Table A2. Coding for polymer matrix, processing method and CNT surface modification method,
respectively.
25
8 PC Mechanical blending NH2-modified
Solution mixing–injection
22 SBR
molding
23 WBPU
26
Table A3. Initial statistical analysis of the database.
Variable Notation Unit Role Min Q25 Average Q75 Max StD CV (%)
Density of matrix ρp g/cm3 Input 0.91 1.10 1.20 1.30 3.12 0.21 17.78
Young’s modulus of matrix Ep MPa Input 0.24 75.30 986.47 1477.00 3060.00 911.49 92.40
Tensile strength of matrix Np MPa Input 0.49 9.96 42.10 58.70 132.00 38.10 90.50
CNT weight fraction wt % Input 0.01 0.50 2.05 2.00 20.00 2.89 140.64
Density of CNTs ρt g/cm3 Input 1.30 1.90 1.94 2.00 2.16 0.15 7.94
Average CNT diameter ϕt nm Input 5.50 15.00 32.01 47.50 127.50 27.91 87.19
Average CNT length lt nm Input 850.00 10000.00 26609.85 30000.00 252500.00 37920.33 142.50
Young’s modulus of CNT Et GPa Input 450.00 850.00 848.59 850.00 1100.00 87.49 10.31
CNT surface modification method M - Input 1.00 10.00 12.94 16.00 21.00 4.92 38.03
Processing method P - Input 1.00 6.00 13.05 19.00 22.00 6.40 49.07
Tensile strength of nanocomposite Nc MPa Target 0.55 14.50 51.67 79.50 190.00 43.72 84.61
27
Table A4. Linear statistical correlation analysis of the database.
Correlation
M ρp Ep Np wt ρt ϕt lt Et M P Nc
coefficient
M 1 -0.002 -0.239 -0.252 -0.005 -0.079 -0.499 -0.274 -0.07 -0.028 -0.092 -0.193
ρp 1 0.202 0.273 0.083 -0.039 -0.003 0.1 0.074 -0.199 -0.111 0.283
M 1 -0.069 -0.204
P 1 0.116
Nc 1
28
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