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Prediction of tensile strength of polymer carbon nanotube composites using

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Article in Journal of Composite Materials · September 2020

DOI: 10.1177/0021998320953540

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 Prediction of Tensile Strength of Polymer Carbon Nanotube

Composites using Practical Machine Learning Method

Tien-Thinh Le 1,*

1
Université Paris-Est, Laboratoire Modélisation et Simulation Multi Echelle, MSME UMR 8208

CNRS, 5 bd Descartes, 77454 Marne-la-Vallée, France

* Correspondence: tien-thinh.le@univ-paris-est.fr (T.-T.L.)

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

environmental standpoints, there is a burgeoning need for materials to be developed. However,

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

mixture proportions for manufacturing new materials.

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

experiments (Jain et al., 2013).

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,

because of CNTs’ peculiar microstructure, an interphase region is created at the CNT/polymer

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

overall mechanical performance of the nanocomposites.

Despite the significant improvements that can be achieved in CNT/polymer nanocomposites,

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

dispersion of CNTs in elastomer poly(dimethylsiloxane) at a high volume fraction. Dondero et al.

(Dondero and Gorga, 2006) employed the melt fiber spinning processing method to investigate

CNT/polypropylene nanocomposite at different orientations and concentrations. The electrospinning

processing method has been used by McCullen et al. (McCullen et al., 2007) in fabricating

CNT/poly(ethylene oxide) nanocomposites. In short, overall mechanical properties of CNT

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.

In terms of predicting the overall mechanical properties of CNT nanocomposites, multiple

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.

(Kanagaraj et al., 2007) for estimating the modulus of high-density CNTs/polyethylene

nanocomposites. Most recently, Hassanzadeh-Aghdam et al. (Hassanzadeh-Aghdam and Jamali,

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

investigation strategies were applied to a specific case of a CNT nanocomposite.

As mentioned above, the literature review reveals that the overall mechanical properties of

CNT nanocomposites depend on various parameters: mechanical behavior of individual phase,

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

any laboratory experiments are carried out.

3. Materials and Methods

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:

 Type of polymer matrix and its density;

 Mechanical properties of polymer matrix: Young’s modulus, tensile strength;

 Physical characteristics of CNTs: density, average length, average diameter;

 Mechanical properties of CNTs: Young’s modulus;

 Incorporation parameters: CNT weight fraction, processing method, CNT surface

modification method.

3.2. Machine learning method: Gaussian Process Regression

Various successful applications of ML methods in materials science have been proposed – for

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 ') :

f ( x) = GP(m( x), k ( x, x ')) (1)

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

and a variance  n2 such that:

 = Ã(0,  n2 ) (3)

From Equation (2), the likelihood is given by:

L(y | f ) = Ã(y | f ,  n2I) (4)

where

y = [ y1 , y2 ,..., yn ]T (5)

f = [ f ( x1 ), f ( x2 ),..., f ( xn )]T (6)

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

matrix based on a Gramian matrix as follows:

L(f ) = Ã(f | 0, K) (7)

where K  k ( xi , x j ) is the covariance matrix corresponding to the covariance function k .

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

then defined as:

L(y) = Ã(f | 0, K y ) (8)

where K y  K   n2 I . Let us define f*  f (x* ) as the vector of function values corresponding

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

conditioning Gaussians (Bishop, 2006), the predictive distribution L ( y * y ) is a Gaussian distribution

with mean and covariance defined as follows:

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

of polymer nanocomposites (Dao, Adeli, et al., 2020):

-
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

(PARTAL and CIGIZOGLU, 2009; HQ Nguyen et al., 2020).

3.3. Monte Carlo method for random sampling

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

method could be explored using the following equation (Le, 2015):

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.

3.4. Error measurements

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,

Desceliers, et al., 2019; Le et al., 2019; HQ Nguyen et al., 2020):

 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,

et al., 2019; Montavon et al., 2013):

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

3.5. Research methodology

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)

identification of weights of input variables, and (5) post-analysis.

 Step 1, Preparation of data: In this step, the data were collected from the available literature

involving 11 input variables, as introduced in Section 3.1.

 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

the selection of such a model (Section 4.3).

 Step 4, Identification of weights of input variables (Section 4.4).

 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

was developed for practical application (Sections 4.5 and 4.6).

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.

4.1. Convergence of random samples

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.

4.2. Influence of training set size

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

over 2000 Monte Carlo random sampling runs with Nrate=80%.

19
20
Figure 3. Influence of Nrate using the training data for (a) R, (c) IA, (e) Slope; using the testing data

for (b) R, (d) IA, (f) Slope.

21
22
Figure 4. Influence of Nrate using the training data for (a) MAPE, (c) RMSE, (e) MAE; using the

testing data for (b) MAPE, (d) RMSE, (f) MAE.

23
 Figure 5. 2D histogram distributions between R and (a) RMSE, (b) MAE, (c) MAPE, (d) IA and

(e) Slope.

4.3. Comparison to other methods

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

cross-validation was chosen as 5.

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

Monte Carlo random sampling.

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)

is poor (i.e. out of range in these figures).

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.

 (  1)  (  1)   100 in case of: R, IA and Slope

 GPR others

%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

2000 runs, including a calculation of the gain obtained.

Method R Slope IA RMSE MAE MAPE

used (MPa) (MPa)

LN -0.13 -1.94 0.08 725.84 459.50 1479.67

SVM 0.59 0.78 0.66 112.45 33.42 324.80

FL 0.90 0.82 0.94 19.31 12.80 71.67

RT 0.94 0.90 0.96 15.30 9.48 27.52

ANN 0.94 0.93 0.96 15.49 10.42 59.16

EBT 0.95 0.86 0.97 14.30 9.00 30.83

5
 GPR 0.96 0.91 0.98 12.14 7.56 31.73

%Gain R Slope IA RMSE MAE MAPE

LN +108.8 +284.8 +89.8 +98.3 +98.4 +97.9

SVM +37.3 +13.4 +31.3 +89.2 +77.4 +90.2

FL +6.3 +9.2 +3.7 +37.1 +40.9 +55.7

RT +2.5 +0.7 +1.3 +20.6 +20.2 -15.3

ANN +2.5 -2.2 +1.3 +21.6 +27.4 +46.4

EBT +1.5 +5.1 +1.0 +15.1 +15.9 -2.9

4.4. Identification of relevant factors

This section presents the identification of weights of input parameters for predicting the tensile

strength of nanocomposites, by removing statistical behavior (i.e. probability density function) of

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

representative probability density function of input variables.

9
 Table 2. Details of weighting: difference in average value (%).
Input parameter R IA Slope RMSE MAE

Processing method 0.02 0.00 0.04 0.26 0.00

Young’s modulus of CNT 0.03 0.02 0.23 0.27 0.03

Average CNT diameter 0.06 0.04 0.23 0.54 0.12

Polymer matrix 0.06 0.04 0.30 0.56 0.46

Average CNT length 0.09 0.05 0.31 0.97 1.10

Young’s modulus of matrix 0.10 0.06 0.31 1.18 1.25

Density of CNTs 0.17 0.09 0.54 1.86 1.35

Density of matrix 0.24 0.15 0.78 2.74 2.42

Tensile strength of matrix 0.89 0.56 1.28 10.01 10.43

Weight fraction of CNTs 1.47 0.79 3.14 16.45 16.25

CNT surface modification 3.37 1.89 5.96 32.49 28.55

method

4.5. Regression and uncertainty analysis

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

polymer nanocomposites is obtained.

11
 Figure 8. Regression graphs between actual and predicted tensile strength: (a) training data,

(b) testing data and (c) all data.

Table 3. Summary of prediction performance of GPR.

Data used R Slope IA RMSE MAE MAPE

Unit - - - MPa MPa -

Training 0.991 0.960 0.995 5.982 3.447 10.589

Testing 0.993 0.954 0.996 5.327 3.539 33.394

All 0.991 0.959 0.995 5.856 3.466 15.196

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

(c) CNT average length.

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

(c) processing method.

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

before conducting any experiments.

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

engineers in terms of understanding and application.

5. Conclusion and outlook

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

exhibited a strong performance in predicting tensile strength of polymer/CNTs nanocomposites,

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

performance of prediction in general.

Certainly, further investigations should be carried out in order to derive an empirical

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

nanocomposites and different types of reinforcement.

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

advices and comments on this paper.

Supplementary Materials: The GPR prediction model is appended to this paper, together with a

Matlab’s GUI prediction function. All supplementary materials are presented at

https://github.com/Tien-ThinhLe/NanocompositeTensileStrengthPrediction.

Funding: This research received no external funding.

Conflicts of Interest: The authors declare no conflict of interest.

Data availability: The raw/processed data required to reproduce these findings will be made

available on request.

Appendix A: Details of the database

Table A1. Organization of the database.

% of wt wt
Number Matrix
Reference prop- Polymer matrix min max
of data notation
ortion (%) (%)

Wu et al. (Wu et al., PLA-g-

4 2.0 Acrylic acid grafted polylactide 0.5 3
2009) AA

Yang et al. (Yang,

5 2.5 Poly(ethylene oxide) PEO 0.1 2
Shi, et al., 2007)

21
Kuan et al. (Kuan et
8 4.0 Polyurethane PU 0.5 4
al., 2005)

Shi et al. (Shi et al.,

4 2.0 Poly(vinyl chloride) PVC 0.1 0.5
2007)

Tseng et al. (Tseng

10 5.1 Epoxy Epoxy 0.1 1
et al., 2007)

Blond et al. (Blond Poly(methyl methacrylate)

6 3.0 PMMA 0.019 0.6
et al., 2006) (PMMA)‐functionalized

Safadi et al. (Safadi

3 1.5 Polystyrene PS 1 5
et al., 2002)

Xia et al. (Xia et al.,

2 1.0 Polypropylene PP 1 3
2004)

Qian et al. (Qian et

2 1.0 Polystyrene PS 1 1
al., 2000)

Kanagarai et al.

(Kanagaraj et al., 4 2.0 High density polyethylene HDPE 0.11 0.44

2007)

Xiao et al. (Xiao et

4 2.0 Low density polyethylene LDPE 1 10
al., 2007)

Yang et al. (Yang,

Pramoda, et al., 8 4.0 Low density polyethylene LDPE 0.5 2

2007)

Jose et al. (Jose et

2 1.0 Polypropylene PP 0.5 1
al., 2007)

Masuda et al.

(Masuda and 6 3.0 Polypropylene PP 0.92 1.1

Torkelson, 2008)

22
Liu et al. (Liu,
1 0.5 Poly(methyl methacrylate) PMMA 5 5
Tasis, et al., 2007)

Liu et al. (Liu, Eder,

2 1.0 Poly(methyl methacrylate) PMMA 2.1 5
et al., 2007)

Kim et al. (Kim and

3 1.5 Poly(methyl methacrylate) PMMA 0.01 0.1
Jo, 2008)

Coleman et al.

(Coleman et al., 7 3.5 Polyvinyl alcohol PVA 0.11 1

2004)

McCullen et al.

(McCullen et al., 2 1.0 Poly(ethylene oxide) PEO 1 3

2007)

Hou et al. (Hou et

5 2.5 Polyacrylonitrile nanofibers PAN 2 20
al., 2005)

Li et al. (Li and Poly(styrene-b-butadiene-co-

1 0.5 SBBS 3 3
Shimizu, 2007) butylene-b-styrene)

Bokobza et al.

(Bokobza and Bilin, 7 3.5 Styrene–butadiene rubber SBR 1 10

2007)

Zhao et al. (Zhao et

2 1.0 Polyamide-6 Nylon 6 0.5 0.5
al., 2005)

Shao et al. (Shao et

2 1.0 Polyamide-6 Nylon 6 0.5 1
al., 2006)

Kang et al. (Kang et Nylon

1 0.5 Poly(hexamethylene sebacamide) 1.5 1.5
al., 2006) 610

23
Ogasawara et al.

(Ogasawara et al., 3 1.5 Polyimide Triple A PI 3.3 14.3

2004)

Zhu et al. (Zhu et

3 1.5 Poly(amic acid) PI 2 9
al., 2006)

Liu et al. (Liu,

2 1.0 Polyetherimide PEI 0.5 1
Tong, et al., 2007)

Yuen et al. (Yuen,

Ma, Lin, et al., 4 2.0 Polyimide PI 1 7

2007)

Kwon et al. (Kwon

14 7.1 Waterborne polyurethane WBPU 0.01 1.5
and Kim, 2005)

Xia et al. (Xia and

3 1.5 Polyurethane PU 0.5 2
Song, 2005)

Xu et al. (Xu et al.,

5 2.5 Polyurethane PU 0.1 0.5
2006)

Jin et al. (Jin et al.,

9 4.5 Poly(ethylene terephthalate) PET 0.5 2
2007)

Bai et al. (Bai, Hard

2 1.0 Hard epoxy 0.5 1
2003) epoxy

Bai et al. (Bai and

9 4.5 Epoxy Epoxy 0.5 4
Allaoui, 2003)

Guo et al. (Guo et

4 2.0 Epoxy Epoxy 2 8
al., 2007)

Kim et al. (Kim and

9 4.5 Polycarbonate PC 0.1 1
Jo, 2009)

24
 Hou et al. (Hou et
1 0.5 Polyvinyl alcohol PVA 0.2 0.2
al., 2009)

Isayev et al. (Isayev

4 2.0 Polyetherimide PEI 1 10
et al., 2009)

Li et al. (Li and Poly(styrene-b-butadiene-co-

5 2.5 SBBS 1.25 15
Shimizu, 2009) butylene-b-styrene)

Yan et al. (Yan and

3 1.5 Polyamide 6 PCL 0.5 1.5
Yang, 2009)

Zhang et al. (Zhang

5 2.5 Polyimide PI 1 10
et al., 2009)

Srivastava et al.

(Srivastava et al., 12 6.1 Polyimide PI 0.5 3

2009)

Total 198 100

Table A2. Coding for polymer matrix, processing method and CNT surface modification method,

respectively.

Coding Polymer matrix Processing method CNTs surface modification method

1 Epoxy Ball milling Amine-modified

2 HDPE Bulk mixing C18-alkylated

3 Hard epoxy Electrospinning COOH-modified

4 LDPE Electrospinning–yarn twisting Diisocyanate functionalized

5 Nylon 6 Hot casting Gum Arabic-modified

6 Nylon 610 In situ condensation Hydroxy-modified

7 PAN In situ polymerization MA-modified

25
8 PC Mechanical blending NH2-modified

9 PCL Melt blending Octyl-modified

10 PEI Melt extrusion Oxidized

11 PEO Melt fiber spinning PBMA-grafted

12 PET Melt mixing PE-grafted

13 PI Pan milling–melt mixing PHT-g-PMMA modified

14 PLA-g-AA Simple mixing PMMA-grafted

15 PMMA Solid state shear milling Phenoxy-grafted

16 PP Solid state shear pulverization Pristine

Pristine with P3HT-g-PCL

17 PS Solution blending
compatibilizer

18 PU Solution casting Purified

19 PVA Solution mixing Acid

20 PVC Solution mixing–casting Diamine

21 SBBS Solution mixing–casting–curing

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 (%)

Matrix M - Input 2.00 8.00 13.28 19.00 24.00 6.78 51.07

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

Ep 1 0.571 0.152 -0.14 0.15 0.227 -0.041 0.101 -0.098 0.6

Np 1 0.131 -0.129 0.037 0.111 0.069 -0.268 0.197 0.89

wt 1 -0.014 -0.038 0.125 -0.051 -0.054 -0.147 0.094

ρt 1 -0.061 -0.106 -0.277 0.115 -0.021 -0.04

ϕt 1 0.05 -0.003 0.13 0.155 0.006

lt sym: 1 -0.059 -0.051 -0.271 0.129

Et 1 -0.137 -0.022 0.091

M 1 -0.069 -0.204

P 1 0.116

Nc 1

28
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