Artificial Intelligence Method for Predicting Mechanical Properties of Sand/Glass Reinforced Polymer: a New Model

1. Heshmati^a, S. Hayati ^a*, S. Javanmiri^a, M. Javadian^b

^a Department of Mechanical Engineering, Kermanshah University of Technology, Kermanshah, 67156-85420, Iran

^b Department of Computer Engineering, Kermanshah University of Technology, Kermanshah, 67156-85420, Iran

1. Introduction

In the last decades, polymeric concrete (PC) has been widely used for new constructions and repairing old constructions due to its good properties, such as rapid setting, high strength, corrosion, and water resistance. Polymer concrete (PC) is fabricated generally by combining the polymers and the fillers. The resin, (e.g. epoxy or polyester) plays the role of a binder instead of cement binders in plain concretes [1]. The unsaturated isophthalic and orthophthalic polyesters can be used as a binder in PC at special conditions such as harsh environments like acid or alkaline media or water [2]. Moreover, epoxy resins are widely used for the manufacturing of polymer concretes due to their suitable mechanical properties and especially their high binding resistance.

Because of the adhesion of the polymeric concrete, repairing both polymer and conventional cement-based concretes are possible. Polymer concrete is an ideal material for underground constructions because of its neutral chemical structure and water impermeability. Conducted studies about mechanical behaviors of such materials under chemical attack situations certify their performance in such situations [3, 4]. While cement-bound mortars cannot resist to chlorine-based acids and the effects of sulfate, polymer-based mortars show resistance both as repair mortar or coating. Polymer concrete also shows good resistance to water and has a high hydraulic capacity thanks to its smoothness [5]. Their adhesion property is the most important property of these materials as they are widely used to reinforce concrete structures [6-12].

Several research works have been conducted for determining the material characteristics of different types of polymer concrete [13-16]. Abdel-Fattah and El-Hawary [17] conducted experiments to study the flexural behavior of polymer concrete (PC) made with epoxy resin and polyester with varying percentages. The results show that the modulus of rupture and ultimate compressive strain for PC were much higher. Thus, the ductility was improved in comparison with the ordinary Portland cement concrete. Komendant et al. [18] investigated the effect of testing temperature on the compressive strength as well as the influence of thermal cycling between 23 and 71 0C on the strength and elastic properties of concrete. They observed that the compressive strength of the concrete is reduced by 3–11% at 43 0C and 11–21% at 71 0C. However, only a few articles have been published in the literature which consider the effect of environment conditions on the PC properties. In some other works [19-22] a numerical analysis approach was used to investigate the mechanical properties of concrete.

Since the PC exhibits brittle failure behavior, improving its post-peak stress-strain behavior is an important aspect for the application of PC. Hence, developing better PC systems and also characterizing the fracture properties and flexural strength in terms of constituents are essential for the efficient use of PC [23, 24]. Chopped strand glass fiber has been applied to polymer composites for improving the strength and controlling the cracking [25, 26]. Thus, before being used in practical and industrial applications, the study of their mechanical properties is necessary. To characterize the failure behavior of the polymer composites with respect to the constituents, some attempts have been made for efficient use [27] or optimizing the mechanical properties [28, 29].

However, since these materials are newly developed, the study of their mechanical properties is much necessary before using them in practical and industrial applications. Analytical methods such as numerical homogenization are used to determine composite material properties [30]. The microstructure interpolation method is also applied for multiple length scale structural optimization [31]. In addition, Akbari et al [32] applied a multi-scale method based on the homogenization technique to investigate the influence of microscopic parameters on the macroscopic behavior of polycrystalline materials under different loading configurations.

Since there is still no mathematical model to obtain the mechanical properties of PCs in general, most of the research work on the mechanical properties are limited to experimental studies [33-41]. Thus, the only way for estimation of the mechanical properties of a PC is through a time consuming and expensive experimental process.

Therefore, in this study, the artificial intelligence (AI) soft computation modeling based on experimental data is chosen to construct a simple and reliable model for the estimation of mechanical properties of the PCs. As an example, Shabani and Mazaheri used the artificial neural network models in numerical modeling of nano-sized ceramic particulates reinforced metal matrix composites [42]. They studied the accuracy of various artificial neural network training algorithms in FEM modeling of Al2O3 nanoparticles reinforced A356 matrix composites. A hybrid artificial intelligence-based model is also used to study the bond strength of CFRP-lightweight concrete composites [43].

The tensile and bending test specimens of sand/glass reinforced polymer concrete with different weight percentages of sand/glass are fabricated. Modulus of elasticity and the ultimate tensile stress are obtained via the tensile and bending tests. The extracted results are then used for training and testing of the neural network models. Different types of neural networks including feed-forward neural network (FFNN), radial basis neural network (RBNN) beside the support vector machine (SVM), and active learning method (ALM) are employed for connecting the properties of the sand/glass reinforced PC to the properties of the resin and weight percentages of sand and chopped glass. All the models are fitted to the problem properly with acceptable accuracy. Finally, a model is presented as a simple formula based on the FFNN structure to obtain the mechanical properties of the sand/glass reinforced PC.

Instead of using expensive experimental studies for the estimation of sand/glass reinforced polymers properties, the presented model can easily be used via any programming software. The presented scheme can also be used for the study of any other complicated material.

2. Methodology

In this section, the utilized methods for obtaining the experimental results and constructing the artificial intelligence (AI) models are discussed. At first, each value of modulus of elasticity for the composite (E_c) or ultimate stress for the composite (σ_uc) is experimentally obtained with respect to three parameters including the modulus of elasticity for the resin (E_r) or ultimate stress for the resin (σ_ur), the weight percentage of sand (%sand), and weight percentage of chopped glass fiber (%glass) in the composite structure. In order to obtain stiffness and ultimate compressive strength of the composite, E_c or σ_uc for each certain value of E_r or σ_ur, %sand, and %glass, ten specimens with similar configurations are made for tensile and bending tests. Therefore, every result is derived as an average result of 5 tensile tests and 5 bending tests. The results for E_c and σ_uc are obtained for 240 different configurations (as tabulated in Tables A-1, 2, 3, and 4), which means there are over 2400 tests performed to obtain the presented results.

In the next step, the obtained results are used to construct the neural network models. Two separate networks are founded in every section, one for estimation of E_c with respect to E_r, wt% of sand, and wt% of glass fiber, and another for estimation of σ_uc with respect to σ_ur, wt% of sand, and wt% of glass fiber. The modulus of elasticity for both resin and composite (E_r and E_c) are in GPa and the ultimate tensile stresses for both resin and composite (σ_ur and σ_uc) are in MPa. The datasets are divided into two categories consisting of train and test datasets. For the below explained structures of the utilized neural networks, the components of the networks, including weight and bias terms, are acquired via an optimization process to fit the training dataset results in the training procedure. After the training procedure, the network should be tested over the test datasets. The testing observation datasets are not used in training and preserved for testing the generality of the neural network model. The same procedure of training and testing is performed for the ALM model as well. The generality of a method means that the method should give proper results for any other data other than the training data. In this study, 80% of the results are taken as the training datasets (192 observations) and the rest are left for testing of the networks (48 observations). A flow chart of the computational procedure is shown in Fig. 1.

Fig. 1. A flow chart for prediction of mechanical properties of polymer concrete

Fig. 2. The fabricated metallic mould

 (a)

(b)

Fig. 3. Dimensions of the (a) tensile and (b) bending samples

2.1. Experimental Setup

Different types of resins including EPON 828, EPON 862, Epoxy L135i, LY564, and PVA are used to fabricate the test specimens. Sand particles and chopped glass fibers are also used for reinforcement of test specimens. The sand is sieved by two different sieves so all the grains of sand are about 2-4 mm in diameter and also all strands of chopped glass fiber are about 6mm in length.

The preparation of specimens is the most important stage of any testing method. The American Society for Testing and Materials (ASTM) has a tensile test standard designed to determine the tensile properties of unreinforced and reinforced plastics in the form of standard dumbbell (dog-bone) shaped test specimens. The tensile test specimen has the basic shape of a tensile dog bone according to ASTM D 638 (Type I). The dimensions of the specimen are 168mm in length, 13mm in width, and 5mm in thickness (Figs. 2-3).

After preparing the materials, the mould should be prepared, so the process of preparing the mould is as follow:

Fig. 4. Removal of specimens from the stainless steel mould

• Assembling the mould and making sure that both upper and lower parts are well attached.
• Using a suitable lubricant (machine oil, grease, or any commercially available mould release agent) to avoid the sticking of samples on the mould.
• Setting the fabricated mould on the flat surface so that the mould does not move during casting the epoxy mixture.
• Adding sand and chopped glass fiber into the cavities of the mould if necessary.

The next step is preparing the matrix, as mentioned; the matrix is a mixture of resin epoxy, and hardener, so at the beginning, these two materials are mixed well and stirred for more than 5min then the mixture is sonicated in the ultrasonic bath for more than 15min. this process helps the removal of almost all of the bubbles from the matrix so a uniform mixture will be achieved. When a uniform mixture of the resin and hardener is provided, the epoxy mixture must be poured into the cavities on the stainless steel mould. Then the mould should be placed in the oven to cure the samples for 5h at 55℃. Finally, the specimens are ready to be removed from the stainless steel mould, but there might be some unwanted extensions on the contour of the specimens that need to be removed before testing (Fig.4).

When it comes to the testing set up and execution, according to conducted tensile and bending tests, two methods, one for tensile tests and the other one for bending tests, are necessary to follow. Figs. 5 and 6 illustrate the experimental setup for both tensile and bending tests.

For the tensile tests, dog-bone specimens are placed in the top and bottom grips and tightened while one visually observes alignment of the long axis of the specimen with the direction of the pull and for bending tests, the specimen is placed on the two lower edges and the third edge on the top moves down until the specimen is broken. Tensile and bending test specimens are fabricated with different sand and chopped glass fiber weight fractions: 0% to 55% with increments of 5% for sand, and 0% to 15% with increments of 5% for chopped glass fiber. Both tensile and bending tests are performed at room temperature under a crosshead speed of 5mm/min. In addition, it is necessary to be mentioned that for each weight percentage, five tensile and one bending test samples are prepared to achieve more reliable results.

As seen in Appendix A, the specimens with 55% sand have a higher stiffness than other samples. According to this appendix, the ultimate tensile strength for the sample that contains 20% of sand is 5.1637MPa. By increasing the weight percentage of sand, the ultimate tensile strength increases significantly and for specimens containing 40% and 55%, the ultimate tensile strength is 5.6995MPa and 7.1863MPa, respectively. However, for samples that contain 20% and 40% of sand, the Young’s modulus doesn’t change widely but for 55% sand specimen, young’s modulus is about 2.5 times higher than 20% and 40% sand specimens.

To study the effect of chopped glass fiber on the ultimate tensile strength of polymers and also to find out about the influence of increasing the amount of chopped glass fiber on the ultimate tensile strength of specimens, two different amounts of chopped glass fiber are mixed with pure polymer. Tables A-1 to A-4 also include information about the samples having specific amount of chopped glass fiber. As illustrated, by adding 5% extra chopped glass fiber to the polymer, its ultimate tensile strength increases about 30% and as expected in comparison with sand contained specimens, the chopped glass fiber samples have a higher ultimate tensile strength.

The simultaneous influence of adding chopped glass fiber and sand particles is also investigated. Two different percentages are presented in the table; the first specimen contains 40% sand and the second one contains a mixture of 40% sand and 5% chopped glass fiber. According to the presented results in Appendix A, adding 5% of chopped glass fiber can increase the ultimate tensile strength up to 60%. Also, the amount of young’s modulus increases from 1.82GPa to 3.50GPa by adding 5% glass fiber into specimens.

2.2. Feed-Forward Neural Network (FFNN)

Feed-forward neural networks (FFNN) are the most primary neural networks. The FFNNs are successfully utilized to model nonlinear problems or estimate complicated functions. The FFNN is usually used as a simple numerical instrument to estimate the results of complicated phenomena after being trained. The structure may have multiple hidden layers but usually, it consists of one hidden layer and one output layer. Multiple layers may lead to extra complication of the network that causes problems in the training procedure and convergence. The structure of an FFNN with one hidden layer is depicted in the Fig. 5. Note that the shown inputs and output of the system belong to the estimation of the mechanical properties of the sand/glass polymer composite.

The input data should be pre-processed before it is entered into the hidden layer. The pre-processing is a linear transform that maps the minimum and maximum of input data ([x_min, x_max]) into the domain between -1 and 1 ([-1, 1]). In the hidden layer, the mapped input data is multiplied by a weight matrix (W_h) and added to a bias vector (b_h). This summation is then applied by a tangent sigmoid (tansig) transfer function. Note that the weight matrix of W_h in the hidden layer is a N_n×N_i matrix of real numbers where N_n is called the number of neurons in the hidden layer and N_i is the dimension of the input vector or the number of input parameters that is equal to 3 for this problem. As mentioned above, the three input parameters of the network are E_r or σ_ur, %sand, and %glass. The bias vector of b_h in the hidden layer is also an N_n×1 vector of real numbers. The number of neurons is an important factor in a neural network that significantly affects both the accuracy and complexity of that network and it will be discussed later.

In the output layer, a similar process is applied to the output of the hidden layer. The data is multiplied by a weight matrix (W_o) and then added to a bias vector (b_o). The transfer function of the output layer is a pure linear (purelin) function that gives the same value of its input as its output. The weight matrix of W_o in the output layer is a N_o×N_n matrix of real numbers and the bias vector of b_o in the output layer is a N_o×1 vector of real numbers where N_o is the number of outputs. This problem has one output (E_c or σ_uc), so N_o=1. Since the output of the output layer is within the domain of [-1, 1], in order to obtain the real output values a post-process is applied to its outputs. Similar to the pre-process, the post-process is a linear transform but it maps the data from [-1, 1] into the domain between minimum output value and maximum output value ([y_min, y_max]).

After the construction of the neural network structure, the network must be trained. The training process is generally an optimization process for tuning the network parameters including W_h, b_h, W_o, and b_o.

Fig. 5. The structure of a single layer FFNN model proposed for estimation of mechanical properties of sand/glass polymer composites

Fig. 6. The structure of an RBNN model proposed for estimation of mechanical properties of sand/glass polymer composites

During the training optimization process, the goal is to find a set of network components which minimizes the mean square error (MSE) between the network results and the real results of the training datasets. The MSE value is strictly relevant to the root mean square (RMSE). Training is done with semi-analytical backpropagation approaches such as Levenberg-Marquardt (LM) and Bayesian regularization (BR) or numerical approaches such as genetic algorithm (GA) and particle swarm optimization (PSO).

2.3. Radial Basis Neural Network (RBNN)

The structure of a radial basis neural network (RBNN) is shown in Fig. 6 which looks so similar to a single layer FFNN. The first difference is in the type of the hidden layer transfer function that is a radial basis function. In addition, in the hidden layer, an element by element multiplication operator (.*) is applied to the output of the weight matrix and the bias vector. Moreover, in the structure of RBNNs the pre-process and post-process functions return their input values.

The RBNNs take a lot of neurons, but they are easily designed and trained. RBNN gives excellent results when a lot of training data are available. The training of an RBNN is processed by adding neurons. In an exact RBNN, the number of neurons is equal to the number of input data vectors.

2.4. Support Vector Machine (SVM)

Support vector machine (SVM) is an artificial intelligence method that is widely used for classification and regression problems and it is also known as support vector regression (SVR) method. In this method, an approximate function (f(x)) is trained to fit the training dataset using a minimization method.

Fig. 7. The structure of an SVM model proposed for estimation of mechanical properties of sand/glass polymer composites

 The structure of the constructed support vector machine in this study is illustrated in Fig. 7 which takes the resin properties (E_r or σ_ur) and weight percentages of sand and glass as inputs and gives the mechanical properties of the sand/glass polymer resin composites (E_c or σ_uc) as the output. In Fig. 7, the parameters  and  are the Lagrangian multipliers, N is the number of observations and K(x_i, x) is the kernel function.

2.5. Active Learning Method (ALM (

ALM is a fuzzy regression algorithm, which works well in uncertain environments [44]. The basic idea of this algorithm is breaking a Multiple Input-Multiple Output (MIMO) system into several simpler Single Input-Single Output (SISO) subsystems as shown in Fig. 8.(a). Afterward, the algorithm combines these subsystems by a fuzzy inference engine in order to achieve the overall behavior of the system. Fig. 8.(b) shows a SISO subsystem of the ALM algorithm called Ink-Drop-Spread (IDS), where two valuable information (Narrow-Path and Spread)are extracted from it. Narrow-Path (NP) and Spread (SP) extracted from each SISO subsystem are then combined by a fuzzy inference unit. Equation (1) shows how these pieces of information are combined. Parameter  is the NP of each SISO subsystem and is the confidence degree of the NP and can be computed by Eq.(2). The ALM algorithm also considers the uncertainty for each data point by using a fuzzy membership function called an ink, as shown in fig. 8.(c). Fig. 8.(d) shows the ink drop spread of 7 data points in an IDS unit. It also shows NP and SP resulted from the IDS unit.

Fig. 8. (a) ALM algorithm breaks a multi-input-single-output function into simpler single-input-single output subsystems and combines the results by a fuzzy inference engine. (b) Each single-input-single-output subsystem consists of a plan called an IDS plan and a feature extractor unit which extract two useful pieces of information, Narrow Path (NP) and Spread (SP). (c) The Gaussian membership function which is called an ink. This membership function is considered for each data point in every IDS plans. (d) The inks of 7 data points are spread in an IDS plan which forms a pattern. The NP and SP are extracted from the IDS plan.

where

where is the Spread inverse and  is the membership degree of the data point to each SISO subsystem.

The Narrow path could be obtained by the weighted-average method, as in equation (3).

where d(x_i, y) is the darkness value of coordinate (x_i, y). The Spread can be computed by equation (4).

where Th is the threshold of the IDS plane which is set by the user (usually Th=0 for modeling purpose).

Until now, numerous successful applications of ALM have been reported in function approximation[45, 46], classification [47-49], clustering [50, 51] and control [45, 52-54]. However, in [55] they show that ALM shows its best advantage when a high level of uncertainty existed in the system.

3. Model Construction and Evaluation

In this section, the neural network models are constructed and evaluated. The models including FFNN, RBNN, SVM, and ALM are trained and tested separately for estimation of E_c and σ_uc. The models are compared with each other in respect to regression plots and statistical indices such as R², RMSE, and VAF that are introduced below.

3.1. FFNN Results

The FFNN model is founded and trained twice, once for estimation of E_c in respect to E_r, %sand, and %glass and once for estimation of σ_uc in respect to σ_ur, %sand, and %glass. The models are trained and tested, using 192 datasets for training and 48 datasets for testing, as follows.

3.1.1. FFNN Model for Estimation of E_c

To estimate the modulus of elasticity of the sand/glass polymer composite (E_c), the FFNN model with 8 neurons in the hidden layer (N_n=8) is founded. The number of neurons is attained through a try and error procedure for finding the most exact network with the simplest structure. Hence, different numbers of neurons are applied to the network several times and the convergence of the networks is investigated noting the training and testing datasets. Finally, a single layer FFNN with 8 neurons in the hidden layer was revealed to be the most convergent network. This model also satisfies the simplicity factor with an 8×3 matrix of W_h, an 8×1 vector of b_h, a 1×8 matrix of W_o, and a 1×1 vector of b_o that generally means 26 components.

The Bayesian-regularization (BR) algorithm is used for training the network which is a fast and exact algorithm. The BR method is essentially a gradient-based method that chooses the first set of network components vector by random. Therefore, every time the BR method solves the problem, it gives different results for the network components. In order to achieve the best possible structure, the training process with the BR method is performed multiple times.

The statistical convergence indices of the finally achieved FFNN for training and testing datasets are shown in Table 1. The regression plots in Fig. 11 show the convergence between the experimental values of E_c and the FFNN results. Noting Fig. 11, the FFNN model gives proper results for both train and test datasets. Therefore, the constructed FFNN model seems to be an exact and general model for the problem. Further model evaluation is presented in the following sections.

3.1.2. FFNN Model for Estimation of σ_uc

To estimate the ultimate tensile stress of the sand/glass polymer composite (σ_uc), a similar FFNN with 11 neurons in the hidden layer is founded and trained. The statistical indices for comparison of the experimental results with the network results are given in Table 2. The same try and error procedure is applied for training the FFNN with the BR method. The indices show a proper accuracy for the network while the accuracy has a fall in comparison to the previous network.

3.2. RBNN Results

Similar to the previous section, the RBNN structure is constructed and trained once for estimation of the E_c with respect to E_r, %sand, and %glass and once for estimation of σ_uc in respect to σ_ur, %sand, and %glass. The training and testing results are primarily investigated with respect to the mentioned statistical indices.

3.2.1. RBNN Model for Estimation of E_c

For estimation of the modulus of elasticity of the composite, an RBNN with 56 neurons in the hidden layer is created. In order to reach the best accuracy, the spread value of the radial basis layer is taken equal to 10 and the goal mean squared error is taken equal to 0.01. These values for spread and goal are obtained through a try and error process.

The resulted indices for the RBNN results in comparison to the experimentally achieved composite modulus of elasticity are depicted in Table 1. The results show an excellent convergence and generality for the RBNN model.

3.2.2. RBNN Model for Estimation of σ_uc

A similar RBNN structure for estimation of the ultimate tensile stress of the composite is constructed with 111 neurons in the hidden layer. In order to reach the best accuracy, the spread value of the radial basis layer is taken equal to 8 and the goal mean square error is taken equal to 0.6. The values are obtained after a try and error process.

The statistical indices comparing the RBNN results with the experimentally achieved composite ultimate tensile stress are shown in Table 2. The results show an excellent convergence and a good generality for the RBNN model.

3.3. SVM Results

In this study, the Gaussian kernel function is utilized for the constructed SVM model. The parameter b is the threshold of the SVM system known as the bias term. This structure estimates the problem through an f(x) function with a deviation of ε that is a predefined parameter for accuracy and it is set to be equal to 0.001 in this study. The L1QP solver is used to solve the minimization problem that gives an SVM structure with a set of 192×3 support vectors. All the mentioned SVM settings are achieved through a try and error process to give the best possible results.

3.3.1. SVM Model for Estimation of E_c

The achieved indices for comparison of SVM results with the experimentally resulted values for composite modulus of elasticity are shown in Table 1. The results show proper accuracy for both testing and training procedures.

3.3.2. SVM Model for Estimation of σ_uc

The indices comparing the SVM results with the experimental values of the composite ultimate tensile stress are shown in Table 2. The results show a poor convergence for the model in training and testing states despite the previous SVM model. Therefore, the SVM model seems not to be proper for this problem.

3.4. ALM Results

In this section, the ALM structure is used for estimation of the E_c with respect to E_r, %sand, and %glass, and for estimation of σ_uc with respect to σ_ur, %sand, and %glass. A primary evaluation is possible noting the regression plots.

3.4.1. ALM Model for Estimation of E_c

The achieved statistical indices for comparison of ALM results with the experimentally resulted values for composite modulus of elasticity are shown in Table 1. The results show an acceptable convergence for the model in the training domain and testing results. The number of partitions in the ALM algorithm is 4,7 and 1 for E_r, %sand, and %glass respectively. The Ink radius is also 0.085 and the threshold value is 0.01.

3.4.2. ALM Model for Estimation of σ_uc

The convergence indices comparing the ALM results with the experimental values of the composite ultimate tensile stress are shown in Table 2. The results show proper accuracy for both testing and training procedures. The number of partitions in the ALM algorithm is 5, 11, and 2 for E_r, %sand, and %glass respectively. The Ink radius is also 0.005 and the threshold value is 0.01.

3.5. Model Evaluation

Evaluation of the obtained models for estimation of the mechanical properties of sand/glass polymer composites is performed in this section.  In this regard, the selected performance indices are R², RMSE, and variance account for (VAF) which their equations can be written as follows:

where y and y′ are the predicted and measured values, respectively, ỹ is the mean of the y′ values and N is the total number of data. The model will be excellent if R² = 1, VAF =100 and RMSE = 0.

In the following subsections, the different constructed neural network models are evaluated with respect to the above-mentioned indices. With respect to the statistical indices, the most accurate models are chosen for the problem.

3.5.1. Model Evaluation for Estimation of E_c

The statistical performance indices including R², RMSE, and VAF for the developed neural network models for estimation of E_c are presented in Table 1. It is observed that the resulted indices for each model are presented for both training and testing datasets.

Since the testing process is much important for generality and even convergence analysis, here it is recommended to consider the testing results as the decisive factor to ascertain the most accurate models. Comparing the resulted indices show that the most accurate model is the RBNN model for estimation of E_c. This model gives an excellent convergence and generality due to both train and test results. The results of the FFNN model are in the next grade with a slight difference. However, the FFNN model still gives excellent accuracy and generality. The poorest results belong to the RBNN model which has acceptable performance for training datasets but it doesn’t give proper results for testing datasets.

Table 1. R², RMSE, and VAF results of the developed models for estimation of E_c

3.5.2. Model Evaluation for Estimation of σ_uc

The statistical performance indices including R², RMSE, and VAF for the developed neural network models for estimation of σ_uc are presented in Table 2. The aforementioned statistical indices for each model are presented for both training and testing datasets.

Noting the performance indices for the testing process in Table 2, the FFNN model gives the best results for testing datasets. Whilst the RBNN model gives better results for the training process, the FFNN model seems to be the best choice for estimation of σ_uc and the RBNN model is in the next grade, because, as mentioned above, the testing results have higher importance. The SVM model can also be specified as the poorest model for estimation of σ_uc.

3.5.3. FFNN Model 5-Folds Cross Validation

The initial evaluation of models shows that the best accuracy and generality are achieved with an FFNN model. To ensure the generality of this model over all observations, a 5-fold cross validation is applied to the structure that is shown in Table 3 and 4 respectively for estimation of E_c and σ_uc based on mean R² value.

In the 5-fold cross-validation process, the 240 observations are divided into 5 independent parts including 48 observations. Afterward, the model is trained and tested for 5 times (5 folds). In each fold, one of the separated parts is taken as the test data and the other 4 parts are taken as the training data. In this way, 100% of the dataset is used for both testing and training.

Table 2.R², RMSE and VAF results of developed models for estimation of σ_uc

Table 3. The mean R² results for 5-fold cross-validation of FFNN for E_c estimation

Table 4. The mean R² results for 5-fold cross-validation of FFNN model for σ_uc estimation

The 5-folds cross validation results are in a closed range. Therefore, the results certify the generality of the FFNN model.

4. Model Presentation

In the previous sections, the models are constructed and investigated in terms of convergence and generality. The best models are chosen and it is time to represent proper models for obtaining the mechanical properties of sand/glass polymer composites. Since in addition to the accuracy the simplicity is an important factor in model presentation, for both estimation problems the FFNN model is presented that has a simple structure with excellent accuracy.

4.1. Model Presentation for E_c

An FFNN model proposed for estimation of the E_c is presented in this section. As mentioned above, in an FFNN model the input vector (x) at first should be mapped from [x_min, x_max] into the [-1, 1] domain through a linear function.

Due to the observations of the problem, the domain of [x_min, x_max] can be stated as 1.636≤E_r ≤4 GPa, 0≤%sand≤55 percent, and 0≤%glass≤15 percent. Therefore, the mapped inputs (x_p) can be obtained as follows.

Noting Fig. 5, the pre mapping output (y_p) that is a value between -1 and 1 is obtained as follows.

The pre mapping output should then be mapped from [-1,1] into [y_min, y_max] domain where y_min=E_cmin= 1.5944 GPa and y_max=E_cmax = 12.4596 GPa. Then the model output (E_c in GPa) will be obtained through a linear mapping as follows.

Having the weight and bias matrices, this model can be applied to estimate the modulus of elasticity for sand/glass polymer composites (E_c) for any in-range datasets using any programming software. The weight and bias matrices are presented as follows.

4.2. Model Presentation for σ_uc

An FFNN model proposed for estimation of the σ_uc is presented in this section. Similar to the previous model, at first, the input vector (x) should be mapped from [x_min, x_max] into the [-1, 1] domain through a linear function.

Due to the observations of the problem, the domain of [x_min, x_max] can be stated as 54.48≤σ_ur≤93.540 MPa, 0≤%sand≤55 percent, and 0≤%glass≤15 percent. Therefore, the mapped inputs (x_p) can be obtained as follows.

Again, noting Fig. 5, the pre mapping output (y_p) that is a value between -1 and 1 is obtained via Eq. 9.

The pre mapping output should then be mapped from [-1, 1] into [y_min, y_max] domain where y_min=σ_ucmin = 7.7242 MPa and y_max=σ_ucmax = 93.540 MPa. Then the model output (σ_uc in MPa) will be obtained through a linear mapping as follows.

Having the weight and bias matrices, this model can be applied to estimate the modulus of elasticity for sand/glass polymer composites (σ_uc) for any in-range datasets using any programming software. The weight and bias matrices are presented as follows.

5. Conclusions

In this study, the neural network soft computation modeling based on experimental datasets is used to construct a realistic model for the prediction of mechanical properties of sand/glass polymer composites. The tensile and bending tests are conducted to obtain the modulus of elasticity and the ultimate tensile stress of sand/glass reinforced polymer composite specimens. The extracted results are then used for training and testing of the neural network models. The model is supposed to give the mechanical properties of the sand/glass polymer composite including the modulus of elasticity and ultimate tensile stress in respect to the modulus of elasticity and ultimate tensile stress of the resin and weight percentages of sand and glass in the composite. The ALM and SVM models and two different types of neural networks including FFNN and RBNN are employed for generating a realistic model. All of the models are trained to fit the problem datasets properly through a try and error procedure. The try and error process is performed to minimize the resulted RMSE value as much as possible to obtain the most acceptable configuration of each model. Then, for both training and testing data, the extracted results of ALM, FFNN, RBNN, and SVM models are compared together in terms of accuracy using the statistical indices including R², RMSE, and VAF. Noting the obtained statistical indices, although all the models are excellent over the training process, the FFNN model is selected as the reference model because of its accuracy over the test data and simple structure. Since the FFNN model gives the best coincidence over the test data, it has the best generality among the obtained models. Finally, the models are presented as a simple formula based on the FFNN structure to obtain the mechanical properties of the sand/glass polymer composite with an excellent agreement with the experimental results.

Appendix A

The experimentally obtained results for modulus of elasticity (E_c) and ultimate tensile stress of the sand glass resin composites (σ_uc) are tabulated in this section. The neural network results are also added to the tables to perform a comparison between the experimental results and the utilized neural networks. Tables A-1 and 2 give the obtained E_c from experiments and neural networks for training and testing observations.

Tables A-1. Obtained E_c from experiments and neural networks for training observations. (Continued)

Tables A-1. Obtained E_c from experiments and neural networks for training observations. (Continued)

Tables A-1. Obtained E_c from experiments and neural networks for training observations. (Continued)

Tables A-1. Obtained E_c from experiments and neural networks for training observations. (Continued)

Tables A-1. Obtained E_c from experiments and neural networks for training observations.

Table A-2. Obtained E_c from experiments and neural networks for testing observations. (Continued)

Table A-2. Obtained E_c from experiments and neural networks for testing observations.

In addition, tables A-3 and 4 give the obtained σ_uc from experiments and neural networks for training and testing observations.

Tables A-3. Obtained σ_uc from experiments and neural networks for training observations. (Continued)

Tables A-3. Obtained σ_uc from experiments and neural networks for training observations. (Continued)

Tables A-3. Obtained σ_uc from experiments and neural networks for training observations. (Continued)

Tables A-3. Obtained σ_uc from experiments and neural networks for training observations. (Continued)