In the recent few years, nanocomposite materials have been largely involved in various evolving industrial and medical fields [1]. This was further motivated by the emergence and recent advancement of nanoscale technology, which is intended primarily for various sensing applications [2–4]. Nanoscale devices are manufactured based on nanocomposites that possess mechanical properties [5–7], thermal properties [8], ruggedness [9], resiliency [10], and electrical properties [11] and are cost effective. These factors fit the various requirements of the dedicated applications [12]. Conversely, bionanocomposites offer suitable ecofriendly properties to preserve the environment and diminish pollution [13–15]. The aforementioned outstanding properties of nanocomposites render them highly promising candidates for numerous critical applications across various industrial sectors [16–19]. Therefore, to underscore the positive impact of bionanocomposite materials in our daily environment, researchers have conducted various studies and experiments. Bionanocomposites intended for medical fields are often employed in tissue engineering [20–22], cancer therapy [23,24], infection therapy [25], biosensing [26–28], and other critical needs. Furthermore, they are employed for water and wastewater treatment [29–31]. In addition to medical applications, bionanocomposites are utilized in water and wastewater treatment [29–31]. In the oil and gas industry, nanocomposites play a vital role in preventing asphaltene dispersion inhibitors [32], and serve as coating materials to protect carbon/steel-based pipelines [33] and other well operations [34,35]. Additionally, nanocomposite materials are commonly utilized in various other applications, including food packaging [36,37], electronic devices [38,39], telecommunications [40,41], energy sustainability [42–44], biotechnology [45,46], and smart textiles [47–49].

To this end, the present work aims to investigate the properties of carbon nanotubes (CNTs), which are one of the most extensively studied nanocomposites owing to their excellent mechanical properties, tensile strength, thermal properties, ruggedness, ductility, availability, and cost-effectiveness [50–54]. Several recent studies have demonstrated the benefits of incorporating CNTs into other polymers. Prasad et al. [55] proposed a composition based on red mud with silicon rubber associated with multiwalled carbon nanotubes (MWCNTs) to produce a high electrical conductivity material of 24 S/cm with an effective electromagnetic shielding within the X band of 8–12 GHz, intended for hazardous areas to eliminate electromagnetic interferences. Similarly, Mao et al. [56] incorporated CNTs into an epoxy matrix to produce conductive polymers for electromagnetic interference applications in high frequency bands. Chen et al. [57] investigated the impact of incorporating MWCNTs and grapheme nanoplatelets into thermoplastic polyurethane to produce smart sensors for use in smart wearable electronics. The proposed sensor had the form of a spider web and demonstrated high flexibility strain, good linearity, and efficient response time (R2 = 0.98 at 50 % strain, 40 ms). Wu et al. [58] proposed a composite material consisting of CNTs, graphene (GR), and copper calcium titanate (CTCO) to produce an epsilon-negative metamaterial, with the primary focus on regulating the negative permittivity, which is necessary for controlling the low frequency dispersion characteristics. The CNTs were used to interconnect the GR sheets, whereas CTCO was added to enhance the permittivity of the material. In a similar study, described in [59], researchers used percolation of CNTs with epoxy polymer to achieve a negative permittivity and low frequency dispersion. They found that this occurred when the percolation threshold was exceeded by the CNTs content. Zhang et al. [60] investigated the potential of CNTs for producing smart devices for the internet of things. Czarny et al. [61] showed that the use of CNTs in nanotoxicology and nanoecotoxicology represents both a societal priority and a scientific challenge, with experimental approaches and concepts still emerging.

In the same context, the present work provides an extensive analysis on the micromechanical properties of the composite materials based on carbon fibers, glass fibers, and CNTs with epoxy-matrix, using an analytical method based on a generic crossing model.

2.1. Micromechanical model of Cox

Various analytical and numerical methods have been developed for a representative elementary volume. One of the micromechanical models proposed for this purpose is Cox’s approach [62], whose formula is given in Eq. (1). This approach was developed by Cox [62] and provides information regarding the shape of the shear stress along the length of the fiber. Additionally, this model allows for the calculation of the interfacial shear strength, which can be determined using Eq. (1), as follows.

Where

· ($G_m$) : shear modulus of the matrix;

· ($E_f$) : Young's modulus of the fiber;

· ($\epsilon$) : deformation;

· (a) : radius of the fiber;

· ($R$) : distance betweenfibers;

· ($\tau$) : shear stress of the interface;

· ($r_f$):the distance between fiber and the matrix.

2.2. Probabilistic approach of Weibull

Matrix damage in composite materials is assumed to follow a uniform stress law, as developed by Weibull. It is calculated as shown in Eq. (2) [63].

Where

· ${\sigma }_f$: applied stress;

· $V_{eff}$: matrix volume;

· m and ${\sigma }_0$: Weibull parameters.

· $V_0$: Initial volume of the matrix.

Weibull [63] developed a method to calculate the probability of fiber breakage in composite materials. He assumed that the fiber was part of a set of bonds, each with its own breaking strength. The probability of fiber breakage can be calculated using Eq. (3) [63]

Where

· ${\sigma }_{max}^f$: maximum stress applied to the fiber;

· ${\sigma }_{0f}$: initial stress applied to the fiber;

· $m_f$: Weibull parameters;

· $A_f={\pi }^{\ast }{a}^2;$

· $L_{equi}$: length of the fiber at equilibrium.

This section introduces the main reinforcing fibers utilized herein, namely carbon, CNT, and glass fibers. A fiber can be described as a cylindrical filament with a diameter much smaller than its length. Typically, the diameter of fibers is only a few micrometers, whereas their length can extend for several hundred meters without interruption.

The exceptional fineness and perfection of fibers allows for the possibility of properties that are not present in the same solid material. For instance, solid glass is typically rigid and extremely brittle, whereas in fibrous form, it retains its rigidity but also has an elongation at break greater than 3 %. The pursuit of high-performance composite materials requires the use of reinforcing fibers with high rigidity, tensile breaking stress, and lightness. Glass, CNT, and carbon fibers are particularly sought-after owing to these properties [64].

3.1. Fibers

Glass fiber: Glass fibers have been a popular choice for composite material reinforcement since the 1950s and are used more frequently than carbon fibers owing to their exceptional performance-to-cost ratio. Compared with solid glass, the fibrous shape of glass significantly reduces its fragility, thereby reducing the effect of crack propagation caused by defects. Glass of varying qualities can be produced depending on its formulation. E-glass fibers are the most commonly utilized glass fibers for reinforcement purposes. They offer the best performance-to-price ratio and are widely used in various applications [64,65]. Glass fibers exhibit excellent thermal and electrical insulation properties and high thermomechanical properties, such as Young’s Modulus, tensile and compressive breaking stress, even at temperatures above 300 °C. Additionally, their coefficient of thermal expansion is very low [64,66]. The E-glass fibers were treated by silane coupling agent. Silane coupling agents play an important role in the preparation of composites from organic polymers and inorganic fillers, such as glass, minerals, and metals. Evidently, E-glass fibers became smoother without any agglomerates after treatment with silane coupling agent [67].

Carbon fiber: Carbon fibers are known for their thermal and electrical conductivity, and high thermomechanical properties, including Young’s modulus, tensile and compressive breaking stress, even at temperatures above 300 °C in air. Additionally, they have a very low or even negative coefficient of thermal expansion. When exposed to temperatures above 400 °C in air, carbon fibers begin oxidizing and gradually lose their mechanical properties. Furthermore, they exhibit brittle behavior, and their impact strength and elongation at break decrease proportionally with the Young’s modulus. Carbon fibers have various advantages, including availability in several grades, high Young’s modulus and tensile strength, low density, thermal stability in the absence of O₂, good fatigue properties, good electrical and thermal conductivity, low or even negative coefficient of linear thermal expansion, excellent creep behavior, biocompatibility, and nontoxicity [64,68–76].

CNTs: Single-layer CNTs are composed of a single GR sheet and have a diameter of approximately 1 nm, with lengths that can reach a few micrometers. The GR sheet that forms the basis of a CNT has two types of edges: armchair and zigzag edges. The type of nanotube obtained depends on the direction in which the GR sheet is wound, thus resulting in either an armchair or zigzag type nanotube. Another way to form a CNT is by translating one edge of the GR sheet relative to the other, parallel to the axis of the tube. This results in a chiral nanotube. The most immediate and industrializable application consists of using these nanoobjects as additives in polymers, thermoplastics, thermosets, or elastomers to modify its mechanical, electrical, or thermal properties. The potential for using composite materials that incorporate CNTs is promising in the field of nanotechnology [77–82]. CNT-based nanocomposites have potential applications in various industries, including construction and microelectronics. For instance, companies such as Infineon, Samsung, and Maxwell Technologies have been exploring the use of CNT-based composites to enhance memory performance [78–82].

Table I presents the different mechanical and physical properties used in the simulation program.

Table 1 . Mechanical and physical properties used in the simulation program..

Fibers	E (GPa)	ρ (Kg/m3)	σrupt (MPa)	εrupt (%)	Ref.
Carbon (High strength)	20	1780	91	1.5	[74]
E-Glass	73	2540	33	4.8	[74]
CNTs	128	1380	62	6	[83]

3.2. Matrix polymer

Epoxy matrix: Epoxy resins are a class of thermosetting polymers known for their high mechanical properties, such as strength and modulus in both tension and compression. Additionally, they have excellent resistance to degradation when exposed to solvents in the environment [83]. Epoxy resins have the ability to react with the hydroxyl groups of natural fibers, such as flax, through a cross-linking process. This process involves the curing of the resin, which results in the formation of a rigid and durable three-dimensional network of polymers. At high temperatures (above 100 °C), the viscosity of the resin decreases, and polymerization occurs, thus resulting in the formation of a strong and rigid composite material. Thermosetting resins, such as epoxy resins, differ from thermoplastic resins in that they can only be shaped once, and cannot be reshaped once they are cured. Although thermoplastic resins have weaker mechanical properties compared to thermosets, they are not commonly used in high-tech sectors. The characteristics of the epoxy resin used in the genetic simulation are listed in Table II.

Table 2 . Characteristics of the epoxy resin used in the genetic simulation..

Matrix	E (GPa)	ρ (Kg/m3)	σrupt (MPa)	εrupt (%)	Ref.
Epoxy	3.8	1200	130	3.5	[85]

3.3. Simulation genetic model

To determine the shear damage at the interface, the genetic operator crossing of the two damages of the fiber and the matrix, as defined by Eqs. (2) and (3) respectively, was employed. The values of the interface damage were enhanced through the use of genetic operators such as selection, crossover, and mutation.

The present study calculated the shear damage at the fiber–matrix interface for three distinct hybrid nanocomposite materials, namely carbon–CNT/epoxy, CNT/epoxy, and glass–CNT/epoxy, using the genetic operator crossing approach. The fiber and matrix damage were calculated using Eqs. (2) and (3), respectively [86–92]. The values of the materials presented in Tables I and II were utilized in the calculation of the objective function. Subsequently, the interface damage was determined based on the length of each fiber for the studied materials, under the influence of various applied mechanical stresses, ranging from σ = 1,550 to 1,700 MPa.

According to Cox’s model [62], load transfer results in the generation of interfacial shear stresses, with the maximum amplitude at the ends of the fiber, and very small values in the middle. Additionally, Cox noted that these stresses exhibit symmetry at the midpoint of the fiber. To facilitate this analysis, he presented the two tensile and shear stresses, which helps justify our selection of specific points used in our genetic modeling, namely (–L, –L/2, 0, L/2, and L).

Figures 1–3 illustrate the damage at the interface for the three different types of hybrid nanocomposite materials. Figure 1 shows the initiation of shear damage at the interface (denoted as ‘D’) for the carbon–CNT/epoxy hybrid nanocomposite material. The damage begins at a threshold of 0.041 when σ = 1,550 MPa and increases to a maximum value of 0.118 at σ = 1,700 MPa. Additionally, Figures 1- 3 indicate the symmetry of weak damage in the middle of the fiber and strong damage at the ends of the fiber. These results suggest that the increase in mechanical stress leads to an increase in the degree of damage. Furthermore, in the case of this material, the degradation of the interface is relatively small. Figure 2 shows also the amount of damage at the interface of the CNT/epoxy nanocomposite material. The graph demonstrates that the shear damage at the interface, represented by D, initiates at a value of 0.175 for a mechanical stress of σ = 1,550 MPa and reaches a maximum value of 0.315 at σ = 1,700 MPa. As with the previous material, a symmetric pattern of weak damage in the middle and strong damage at the ends of the fiber was observed for the CNT/epoxy nanocomposite material. This is depicted by different colors, which are blue and green, as shown in Fig. 2. The increase in damage level in the CNT/epoxy nanocomposite material indicates that mechanical stresses are concentrated at the interface, thus resulting in more severe degradation of the interface compared with the carbon– CNT/epoxy hybrid nanocomposite material.

Figure 1. Fiber–matrix interface shear damage of carbon–CNTs/epoxy hybrid nanocomposite material.

Figure 2. Fiber–matrix interface shear damage of CNTs/epoxy nanocomposite material.

Figure 3. Fiber–matrix interface shear damage of glass-CNTs/epoxy hybrid nanocomposite material.

Figure 3 shows that the damage at the interface labeled as ‘D’ in the glass–CNT/epoxy hybrid nanocomposite material initiates at a value of 0.346 under a stress of 1,550 MPa and increases to a maximum value of 0.515 when subjected to a stress of 1,700 MPa. Additionally, the symmetry of the damage level is relatively low in the central region and significantly higher toward the ends of the fiber, as indicated by the contrasting the colors of blue and black in Fig. 3. The results suggest that the build-up of mechanical stresses leads to an increase in the damage level, which results in more severe degradation at the interface of the glass–CNT/epoxy hybrid nanocomposite material compared with the carbon–CNT/epoxy and CNT/epoxy materials.

The aforementioned findings were corroborated by a recent study (published in 2022) conducted by [93], which reported that the presence of CNTs greatly enhances the interfacial adhesion between the carbon fiber and the epoxy matrix.

This study performed a detailed analysis of the shear damage that occurs at the interface between the fiber and matrix in three types of nanocomposite materials, including CNT/epoxy, carbon–CNT/epoxy, and glass–CNT/epoxy. The damage caused to the fiber and matrix components was evaluated individually by employing a genetic operator crossing approach. As per the genetic simulation results, the carbon–CNT/epoxy hybrid nanocomposite demonstrated higher resistance to mechanical stress compared with the CNT/epoxy nanocomposite and glass–CNT/epoxy hybrid nanocomposite. These findings were supported by those of Kamae et al. [93] in their recent study published in 2022, where they demonstrated that the incorporation of CNT material significantly enhances the interfacial bonding between the carbon fiber and epoxy matrix. In conclusion, this comparative analysis underscore that the aforementioned materials under investigation exhibit significant promise for diverse engineering applications, including but not limited to electronic and telecommunications industries, wearable sensors with intelligent features, and biomedical applications. In addition, future scope of this study is to use its nanocomposite hybrids materials for the design of new, more efficient antennas for telecommunications applications.

The authors acknowledge the financial support received from the General Direction of Scientific Research and Technological Development of the Ministry of Higher Education and Scientific Research of Algeria. This work was supported by the General Direction of Scientific Research and Technological Development of the Ministry of Higher Education and Scientific Research of Algeria (PRFU: A25N01CU320120230001).

The authors declare no conflicts of interest.