Evaluation of mechanical properties of fiber-reinforced syntactic foam thermoset composites: A robust artificial intelligence modeling approach for improved accuracy with little datasets

1 Introduction

The term composite materials refer to a multi-material system that is made up of two or more constituents that are placed together in a single structure without undergoing any kind of chemical reaction. It is important to note that various materials are combined in order to produce a new material that possesses unique properties that are not identical to those of any of the components when they are taken separately. The resulting material can be applied in a wide range of applications, such as aerospace, wind turbine blades, and aircraft [1,2,3,4]. The components of any composite may be broken down into a reinforcing phase and a matrix phase since they are insoluble in one another. The reinforcing phase, which can take the shape of fibers, particles, or flakes, is embedded within the matrix phase. The matrix phase, which geometrically establishes the component, provides material cohesiveness and transmits forces from one fiber to another. On the other hand, the substance that makes up the reinforcing phase is hard and may be incorporated into the matrix in order to make it more robust. Nowadays, a wide variety of composite materials are employed for a variety of applications as in composite material reinforced with carbon fiber. Also, polymer matrix composites, metal matrix composites, natural composites like wood, sandwich panels composites, and ceramic matrix composites are the most frequent types of composites [5,6].

Syntactic foams are a form of material that has been characterized by researchers as being made up of spherical particles that are bonded together with a matrix resin that may be built of ceramics, polymers, or metals [2]. It is the most popular and attractive material used in a variety of applications due to its novel properties, which include high specific strength, a densely packed closed-cell structure, and a lower density than the majority of conventional materials. Some of its properties include good shock resistance, water permeability, floatation, and the durability to withstand hard water pressure [7–10].

Syntactic foam is created by introducing micro balloons or glass microspheres into the structural material. This creates a foam-like structure. It has been discovered that glass microspheres (3 M, K20) with a median diameter of 60 m and a density of 0.2 g/cm³ improve the viscosity of the resin material, in general, functioning as a rheology modifier. Moreover, the printed composites keep their form after being deposited on the substrate [11–13].

A comprehensive investigation of syntactic foam materials reveals that the mechanical characteristics can be customized by combining suitable matrix materials with hollow microspheres [14–16]. Such methods of combining glass microspheres with the matrix material are related to the additive manufacturing (AM) field as in the fused filament fabrications and the direct write technologies [11]. Glass microspheres, with a diameter of 10–300 m, are preferred by scientists due to their high compressive stresses, which much surpass those of other microsphere classes [17–19].

Injection molding or compression molding have often been cited as the manufacturing processes used for thermoplastic syntactic foam component production. However, additional research shows that AM of syntactic foams was successfully implemented – using the fused deposition modeling (FDM) process – to fabricate thermoplastic syntactic foam composites with reinforcements. This was accomplished by printing the syntactic foams in layers using the FDM process [20,21]. This approach, which is known as FDM, has the potential to boost productivity, save expenses, and boost the overall quality of the finished product. As a consequence, the high-speed technique and the flexibility in design scalability are being put to use in a diverse range of industries, including the medical, aerospace, automotive, and industrial sectors [22,23]. However, the FDM process experiences low assistance temperatures, unavoidable porosities, and problematically poor adhesion between layers, rendering this method infeasible when great mechanical performance is required [24–26]. On the other hand, it is possible to circumvent the limitations of the FDM process when working with polymer-based materials by using the AM technique known as direct writing (DW). It should be emphasized that in order to construct complex structures using direct ink writing, printing inks need to have high viscosity, high yield stress, and well-controlled rheological characteristics. It is feasible to improve the mechanical performance of AM thermoset syntactic foam composites using this method due to the versatility of the process, the extensive variety of materials that are appropriate for use, and the flexibility that it has [27,28].

While adding a high-volume ratio of reinforcements (like carbon fibers) to syntactic foam matrix resin can result in improved mechanical structures, there appears to be a limit to the fiber concentration levels being utilized in composite manufacturing, independent of the fiber type. As a result, measuring the mechanical performance of composites above the specific levels of reinforced content is challenging experimentally [13,29–32]. Because of this, the development of a practical method for evaluating the difficulty of predicting mechanical properties is an essential research priority that should be pursued with the goal of cutting down on the amount of labor that must be performed in the laboratory. Concerning the adaptive neuro-fuzzy inference system (ANFIS), it indicates a suitable strategy for predicting the mechanical characteristics of such composites across various engineering disciplines [33,34]. The hypothesized ANFIS models can generate an input–output appropriate algorithm based on experimental knowledge (in a state of fuzzy if-then rules), employing a hybrid learning approach [35–39]. ANFIS can also examine nonlinear functions and detects nonlinear components using nonlinear regression. In addition, they all have high prediction accuracy, enabling autonomous control, data categorization, and decision analysis [40–42].

In this regard, this proposed work has a novel ANFIS network architecture that can be used as a foundation for determining the fundamental mechanical properties of AM syntactic foam thermoset composites without the need for any experimental investigation. Indeed, there are different schemes to solve this type of problem, such as decision-making tools, stochastic models, and finite element analysis. However, these models need a large number of data sets and complicated computational analysis to predict the output. In this work, the inclination toward artificial intelligence techniques is due to the accuracy they have, even when they are applied to a small data set (as in this case), and this is the first added value of this work. Besides, this work aims to accurately predict the impact of short carbon fiber reinforcements on the mechanical characteristics of syntactic foams and thermoset composites, independent of fiber concentration, without necessitating any experimental evaluation. That is, sometimes it is difficult to perform experiments under certain conditions due to nozzle clogging and fiber agglomeration. However, this work is capable of predicting the mechanical performance in such cases due to the developed prediction models offering accurate mechanical behavior predictions, regardless of the fiber concentration as predictions here have the basis of sampling data for the study under various conditions without requiring any experimental inquiry, and this is the second added value of this work in the field.

2 Methodology

A method of processing variables known as fuzzy logic enables different underlying principles of varying probabilities to be processed through the same variable at the same time. Fuzzy logic is an approach to problem solving that uses an open, uncertainty range of information and algorithms. This spectrum of data and these heuristics allow for the possibility of obtaining a variety of accurate estimates. Because of this, fuzzy logic really has a distinct advantage over other approaches.

A fuzzy learning algorithm has five functional units: a rule base among several fuzzy if-then rules; a data warehouse that clearly states the membership functions of the set theory used in the fuzzy rules; an outcome facility that accomplishes the rules’ inference activities; a fuzzification connection that transfers inputs into degrees of connection with linguistic variables; and a defuzzification interface that transforms the inference’s fuzzy inputs into linguistic values (i.e., crisp output). The initial stage in performing fuzzy inference operations is comparing the input variables with the membership functions on the hypothesis part to determine the membership degree values of each linguistic variable (i.e., fuzzification). The membership values in the premise section are then combined (using a special T-norm operator) to derive the set of weight values for each rule. What follows is the generation of the qualified resultant output of each rule based on the rule’s weight. Finally, because the qualifying consequences have been aggregated, a crisp output is produced (i.e., defuzzification).

Figure 1 shows the ANFIS model structure used in this study. As can be seen in this diagram, the fuzzy inference system used for each model takes into consideration two variables: the amount of glass microspheres (GB) and the amount of carbon fiber (F _i ). On the other hand, it has only a single output that consists of the flexural modulus (FE _i ), the flexural stress (FS _i ), the compressive modulus (CE _i ), and the compressive stress (CS _i ).

Figure 1

ANFIS structure of this work.

Following that, Sugeno’s fuzzy if-then rules, which are based on the fuzzy if-then rules that have previously been applied, are put into effect [43]. The defuzzification approach that is used by the Sugeno system is more appropriately efficient because, rather than determining the geometric center of a region in two dimensions, it puts emphasis on the average of the weights, or the sum of the weights of a very small number of diverse data points to arrive at its conclusions. According to these guidelines, the results of running each rule are represented as a linear function of the data that were input into it, which stands for the output of the whole system. This represents the overall output. A Sugeno-Fuzzy rules system may be represented by the following equations:

In the above equations, GB and F _i represent the amount of glass microspheres and the amount of carbon fiber, respectively. On the other hand, these formulas only have a single output that consists of the flexural modulus (FE _i ), the flexural stress (FS _i ), the compressive modulus (CE _i ), and the compressive stress (CS _i ). Finally, p _1i , p _2i ,…, p _12i represent the values of constant coefficients.

The entire characterizations are then generated by extracting these rules at the node level of the ANN and gathering them using a decomposition technique. Under this consideration, the fuzzy inference system, in this article, has two inputs and one output. For the sake of simplicity, the output of the node functions goes through five-layer stages. The mathematical operations are broken down and described in Figure 2, which offers a diagrammatic description of the procedure.

Figure 2

Summarizing the mathematical operations through the five-layer stages in the ANFIS structure.

The layer’s nodes i are all square nodes with node functions at layer 1. The Gaussian bell-shaped functions having premise parameters (a, b, c) are chosen to calculate the output. The function’s values ( μ x ( GB i ) , μ y ( F i ) ) are determined by

When the linguistic variables connected with this node function are x and y, the values of x(GB _i ) and y(F _i ) indicate how well the given inputs (GB _i , F _i ) meet the requirements of the linguistic variables. The second layer is associated with the receiving input signal magnification of each node, and it then transmits the results out (i.e., the output indicates a rule’s firing strength w i ). Consider, for instance,

Normalization of the firing strength happens at layer 3; this layer is responsible for measuring the sum of the ith rule’s firing strength to the total firing strength of all rules. The layer’s outputs (i.e., W ¯ i ) will be referred to as normalized firing strengths and are calculated as

Layers 4 and 5 can obtain the normalized inputs and deliver the defuzzified values to layer 5, which outputs the final value (i.e., R and R′ respectively). The mathematical processes in these levels are described in Eqs. (9) and (10), where Q(i) shows the computed mechanical property (flexural modulus, flexural stress, compressive modulus, and compressive stress):

3 Results and discussion

This research makes use of the data presented in the study of Nawafleh et al. [13] to highlight how effective our ANFIS model is. Training (70%) and testing (30%) of the obtained data make up these statistics. The parameters of the Sugeno fuzzy inference system were set with the use of training data, and the least-squares estimation coupled backpropagation gradient descent algorithm was utilized in order to create the values of each estimated model. Contrarily, the availability of the testing data set enables an evaluation of the generalizability of the fuzzy inference system that was produced. Figures 3 and 4 show, in contrast to the results of the experiments, the expected flexural and compressive stresses of syntactic foams. Unquestionably, a more in-depth account of the experimental results can be found in the study of Nawafleh et al. [13]. Specifically, the results of this work indicate quite clearly that the ANFIS prediction model is able to achieve a fairly truthful prediction of the values of the flexural stress and flexural modulus during the training period.

Figure 3

An examination of the observed values with the expected ones for the flexural stress: (a) an example of the data and (b) a 3D layout.

Figure 4

An examination of the observed values with the expected ones for the compressive stress: (a) an example of the data and (b) a 3D layout.

As the training data may be mapped between inputs and outputs, the fuzzy inference structure is a single-output Sugeno fuzzy system. The ANFIS models operate according to this guiding idea. In other words, there is only one output for every single group of the data set that serves as input. The ANFIS structure is now attempting to manage these data by using the output level of each rule to determine the firing strength of the rule. The last step involves obtaining the normalized inputs and supplying the defuzzified values to the final layer in order to generate a single output.

For example, note how precisely the red boxes (predicted values) are close to the blue circles (experimental work). All the same, Figures 5 and 6 notably demonstrate the accuracy of this model when used to predict the flexural and compressive modulus, respectively.

Figure 5

An examination of the observed values with the expected ones for the flexural modulus: (a) an example of the data and (b) a 3D layout.

Figure 6

An examination of the observed values with the expected ones for the compressive modulus: (a) an example of the data and (b) a 3D layout.

The tendency to predict the performance of the compression test rather than the tensile test is something that stands out in this body of work as being particularly notable. Under tensional loading, the compressive stresses of all composites were much greater than their tensile stresses. This was due to the presence of flaws in the composites, such as voids and glass microspheres, which caused stress intensification and served as crack initiation sites. In other words, the voids disappear as the material is compressed, yet this does not affect the fracture stress [13]. On the other hand, in flexure testing, specimens are subjected to both tension and compression; as a result, they fail with lower loads and at an earlier stage than those that are subjected to exclusively compressive stresses. Hence, these are the reasons for investigating these outputs. Although the trend of predicted values for all mechanical features was found to diverge somewhat from the observed (experimental) values, one should not, of course, disregard the examination of all probable types of errors. A closer look at Figure 7 reveals that the computed errors of the ANFIS-created models are relatively minimal. Tables 1 and 2 provide the numerical values of the calculated relative errors (REs) in both flexural and compressive tests. Accordingly, the RE is measured as

Figure 7

(a) Flexural stress, (b) compressive stress, (c) flexural modulus, and (d) compressive modulus RE for ANFIS model types.

The value of the expected mechanical properties is denoted by α p , while the measure of the experimental mechanical properties is denoted by α exp .

However, mean absolute percentage errors (MAPEs), R-squared correlation coefficient, and mean squared error (MSE) of mechanical tests described in this article are determined as follows:

The statistical overview of the projected values for the ANFIS mechanical tests is shown in Table 3; eventually, the observed statistical analysis results will offer information on the correctness of the developed ANFIS model.

Figure 8 provides a prediction for the correlation coefficient R ² curve fitting for each linear degree model. Figures 9 and 10, on the other hand, present more sophisticated 3D surface modeling of the data that were produced. According to the R ² values that are shown in these figures, the model can more reliably predict the mechanical performance of the composites that have been generated.

Figure 8

A linkage has been made between the ANFIS and numeric mechanical performance analysis. (a) Stress in the flexural direction, (b) stress in the compressive direction, (c) modulus in the flexural direction, and (d) modulus in the compressive direction.

Figure 9

Modeling of 3D curves for the ANFIS stresses values with stress due to (a) flexure and (b) compression.

Figure 10

Modeling of 3D curves for the ANFIS modulus values with modulus due to (a) flexure and (b) compression.

Table 4 presents the intercorrelations of the estimated equations for each ANFIS model among the predicted measures of the created composite; in this particular instance, these data provide support for the models that were projected using the expected ANFIS model.

Testing additional validation points, applied to this model, allows for an examination of the equations in Table 4, which checks the work’s dependability and makes it feasible to investigate. The study of these validation points demonstrates a positive connection between the experimental work and the ANFIS models, as indicated in Tables 5 and 6 (as numerals) and Figures 11 and 12. These results are consistent with the predictions of the proposed ANFIS model, despite the fact that there are discrepancies between the two sets of data. Indeed, this is because it is difficult to determine which results are more accurate due to the very small sample size (about 100 data points). The reason for this small sample size is the difficulty of conducting tests, as shown in the study of Nawafleh et al. [13], and this is the main benefit of employing ANFIS – in this work – in making reliable performance predictions while utilizing a small sample size.

Figure 11

The ANFIS model’s confirmation: (a) stress in the form of flexure and (b) stress in the form of compression.

Figure 12

The ANFIS model’s confirmation: (a) modulus in the form of flexure and (b) modulus in the form of compression.

4 Conclusions

Syntactic foams are unique material systems with high specific strength, low density, and impact resistance. Nevertheless, it is difficult to make syntactic foam composites, especially in AM techniques, making mechanical performance prediction challenging. Due to nozzle blockage and fiber aggregation, in this work, we investigated the mechanical effects of short carbon fiber reinforcement in syntactic foam thermoset composites without experimental assessment or carbon fiber level. Hence, the prediction models could accurately predict mechanical performance regardless of fiber concentration. More precisely, ANFIS models’ precision, even on limited data sets, allows this prediction. Compression and flexure testing on generated samples verified the ANFIS models and ensured mechanical performance. The strong agreement between the experimental observations and the ANFIS predictions suggests that they may accurately measure mechanical performance with high consistency. Precisely, flexural stress, flexural modulus, compressive stress, and compressive modulus showed low MAPE of 9.02, 3.65, 4.28, and 2.48%, verifying the accuracy of the proposed model. The mean square error, MAPE, and correlation coefficient were also used to evaluate the model. As another parametric demonstration, the correlation coefficient was tested to verify the ANFIS models’ reliability. Both tests yielded R-square values near 1, indicating the accuracy of the works. Consequently, by utilizing ANFIS models, AM systems may be made more reliable without exploratory experiments, enabling their use in interesting technical applications. Moreover, this approach is being scaled up to attain composite scale manufacturing, enhancing the full examination of such reinforced carbon fiber syntactic foam structures across a broader range of applications, for example, as in the maritime and aerospace sectors.