Novel Hybrid Polymer Composites with Graphene and MXene Nano-Reinforcements: Computational Analysis

This paper presents a computational analysis on the mechanical and damage behavior of novel hybrid polymer composites with graphene and MXene nano-reinforcements targeted for flexible electronics and advanced high-strength structural applications with additional functions, such as real-time monitoring of structural integrity. Geometrical models of three-dimensional representative volume elements of various configurations were generated, and a computational model based on the micromechanical finite element method was developed and solved using an explicit dynamic solver. The influence of the geometrical orientation, aspect ratio, and volume fractions of the inclusions, as well as the interface properties between the nano-reinforcements and the matrix on the mechanical behavior, was determined. The results of the presented research give initial insights about the mechanical and damage behavior of the proposed composites and provide insight for future design iterations of similar multifunctional materials.


Introduction
After the discovery of graphene, focus shifted to two-dimensional (2D) nanomaterials. These nanofillers drew attention due to their flexible properties, allowing us to create multi-functional composite materials that can be used for a range of applications such as aerospace, energy storage, and electromagnetic interference shielding [1][2][3]. Twodimensional nanomaterials such as graphene, due to their high aspect ratio (AR), offer increased fatigue resistance and fracture toughness through deflection of crack propagation and bridging mechanics [4][5][6]. However, such fillers are hydrophobic and do not form bonds with polymers. Such a pursuit led to the discovery of a new class of 2D materials in 2011. MXenes are ternary layered compounds, produced by selectively etching A-group layers from the MAX phases. They exhibit unique properties such as hydrophilic nature combined with high electrical and thermal conductivity, capability to intercalate ions, high electrical capacity, excellent electrochemical activity, and great mechanical properties [7][8][9][10][11][12][13][14][15]. These properties make them of great interest for many applications, and they can be utilized in developments of composite materials with polymer matrixes for flexible electronics and advanced high-strength structural applications with additional functions such as real-time monitoring of structural integrity. There are more than 30 different types reported, and hundreds computationally studied in silico [7,16]. Titanium carbide Ti 3 C 2 T z is the most widely researched MXene exhibiting hydrophilic properties, which makes it dispersible in a range of polar solvents [17,18]. Ti 3 C 2 T z of high aspect pores were distributed non-linearly in a functionally graded circular shape, and graphene nano-reinforcements compensated the decrease in the Young's modulus occurring due to linearly distributed air-voids.
The tensile response and the damage mechanism of MXene/epoxy composites MXene/polyvinyl alcohol composites were investigated [61] by developing a micromechanical finite element model, which was calibrated taking into account experimental results. The predictions based on this model demonstrated that MXene shows great promise for polymer matrix-based composites by significantly improving their mechanical properties, as well as high-strength multifunctional MXene-polymer film materials with high mechanical properties, which can be applied for real time monitoring of structural integrity by utilizing the electrical conductivity of MXene. Bioinspired MXene/polymer nanocomposites with nacre mimetic brick and mortar structures were modelled using classical analytical methods and numerical methods based on the finite element approach [62]. It was demonstrated that such structures result in an interlocking mechanism between MXene inclusions, leading to a significant increase in stiffness and strength. Orthotropic elastic properties of epoxy composites with MXene and graphene 2D nano-reinforcements were studied [63] by applying numerical methods. Recent works demonstrated that the mechanical and damage behavior of composites can be significantly improved by proper selection of reinforcements.
The aim of this research is to study the mechanical and damage behavior of hybrid polymer composites with MXene and graphene nano-reinforcements by developing a computational model based on the micromechanical finite element method, which would allow initial insights on the mechanical and damage behavior of such hybrid composites to be presented and estimations to be made about the influence of the geometrical orientation, aspect ratio and volume fractions of the inclusions as well as the interface properties between the nano-reinforcements and the matrix.

Materials and Methods
The investigated composite materials are composed of an epoxy matrix and 2D nanosheets of graphene and MXenes. The mechanical properties of the materials are presented in Table 1. The graphene-matrix interphase properties used in the presented computational analysis were based on the previous research [25,57], which studied these properties using an inverse modelling approach. To investigate the mechanical behavior of such hybrid polymer composites reinforced with graphene and MXene nanosheets, a computational model was developed on the basis of the micromechanical finite element method. Geometrical models of three-dimensional representative volume elements (RVEs) with various volume fractions (denoted as f G and f MX , respectively) of graphene and MXene inclusions, various aspect ratios (ρ G and ρ MX ), and different alignment configurations were created using Digimat-FE (Extreme Engineering, MSC.Software GmbH, Munich, Germany). In practice, the alignment of nano-reinforcements can be achieved through electrical methods [68,69]. A volume fraction for graphene inclusions was set to 0.1%. The MXene volume fractions were set in a range from 0.8% to 1.6% for the RVEs with randomly placed inclusions and in a range from 0.8% to 6.4% for the RVEs with aligned inclusions. An aspect ratio value of 500 was used for the graphene inclusions, while aspect ratio values of 200 and 400 were used for the MXene inclusions.
The RVEs with randomly placed inclusions were built as cubes with a size of 595 nm, while the RVEs with aligned inclusions were built as rectangular cuboids with dimensions of 595 × 290 × 595 nm, aligning the inclusions in the x-z plane. The graphene and MXene inclusions were generated as discs with a thickness of 0.335 nm and 1 nm [1,[70][71][72], respectively. As inclusions/polymer matrix interfaces make a significant influence on the mechanical behavior of composite materials reinforced with nanosheets, the approach of effective interface models was adopted where the thin layer, surrounding the inclusions, is generated with specific properties. Based on the experimental observations presented in the article [73], the thickness of the effective interface layers was set to 1 nm. Typical RVEs used in this research are presented in Figure 1.
Digimat-FE (Extreme Engineering, MSC.Software GmbH, Munich, Germany). In practice, the alignment of nano-reinforcements can be achieved through electrical methods [68,69]. A volume fraction for graphene inclusions was set to 0.1%. The MXene volume fractions were set in a range from 0.8% to 1.6% for the RVEs with randomly placed inclusions and in a range from 0.8% to 6.4% for the RVEs with aligned inclusions. An aspect ratio value of 500 was used for the graphene inclusions, while aspect ratio values of 200 and 400 were used for the MXene inclusions.
The RVEs with randomly placed inclusions were built as cubes with a size of 595 nm, while the RVEs with aligned inclusions were built as rectangular cuboids with dimensions of 595 × 290 × 595 nm, aligning the inclusions in the x-z plane. The graphene and MXene inclusions were generated as discs with a thickness of 0.335 nm and 1 nm [1,[70][71][72], respectively. As inclusions/polymer matrix interfaces make a significant influence on the mechanical behavior of composite materials reinforced with nanosheets, the approach of effective interface models was adopted where the thin layer, surrounding the inclusions, is generated with specific properties. Based on the experimental observations presented in the article [73], the thickness of the effective interface layers was set to 1 nm. Typical RVEs used in this research are presented in Figure 1. The created RVEs were imported to the commercial finite element software Abaqus FEA (Dassault Systemes, Vélizy-Villacoublay, France), which was used to develop computational model and carry out the simulation tasks. The periodic boundary conditions [74] were opted in Digimat-FE (Extreme Engineering, MSC.Software GmbH, Munich, Germany) and were imported along with the geometrical models to Abaqus FEA (Dassault Systemes, Vélizy-Villacoublay, France). The RVEs were subjected to uniaxial tensile loading along the x-axis direction [61,63] The RVEs were meshed using the three-dimensional 4-node linear tetrahedron element (C3D4) type. A minimum size of 8 nm was applied for the mesh, resulting in a total number of 1-2 million, depending on the RVE's configuration. The experimental stress-strain curve of epoxy [67] was inserted in the program and the multilinear hardening plasticity model was considered to define the response to the mechanical loading. To investigate the influence of the MXene/epoxy interface properties on the strength of the proposed composite, several values of the Young modulus EMX and strength of the interface was used (multiplying by 0.5, 0.75 and 1.5 to those of the matrix Em). These values were in the range determined in the previous research [61]. The maximum principal stress criterion was applied for the simulation of matrix and interfaces cracking using the values provided in Table 1 as it was demonstrated that these are reasonable value for the maximum principal strength [75,76]. The MXene and graphene inclusions were not damaged during the simulation as the obtained maximum principal stress did not exceed the strength limit of these materials. The developed computational model was solved using Abaqus explicit and converged, delivering reliable and stable results. The created RVEs were imported to the commercial finite element software Abaqus FEA (Dassault Systemes, Vélizy-Villacoublay, France), which was used to develop computational model and carry out the simulation tasks. The periodic boundary conditions [74] were opted in Digimat-FE (Extreme Engineering, MSC.Software GmbH, Munich, Germany) and were imported along with the geometrical models to Abaqus FEA (Dassault Systemes, Vélizy-Villacoublay, France). The RVEs were subjected to uniaxial tensile loading along the x-axis direction [61,63] The RVEs were meshed using the three-dimensional 4-node linear tetrahedron element (C3D4) type. A minimum size of 8 nm was applied for the mesh, resulting in a total number of 1-2 million, depending on the RVE's configuration. The experimental stress-strain curve of epoxy [67] was inserted in the program and the multilinear hardening plasticity model was considered to define the response to the mechanical loading. To investigate the influence of the MXene/epoxy interface properties on the strength of the proposed composite, several values of the Young modulus E MX and strength of the interface was used (multiplying by 0.5, 0.75 and 1.5 to those of the matrix E m ). These values were in the range determined in the previous research [61]. The maximum principal stress criterion was applied for the simulation of matrix and interfaces cracking using the values provided in Table 1 as it was demonstrated that these are reasonable value for the maximum principal strength [75,76]. The MXene and graphene inclusions were not damaged during the simulation as the obtained maximum principal stress did not exceed the strength limit of these materials. The developed computational model was solved using Abaqus explicit and converged, delivering reliable and stable results.

Results and Discussion
Stress distributions in RVEs subjected to tension along the x-axis direction are shown in Figure 2 and damage evolution is shown in Figure 3.

Results and Discussion
Stress distributions in RVEs subjected to tension along the x-axis direction shown in Figure 2 and damage evolution is shown in Figure 3.  both RVEs (Figure 3a,e). In the RVE with randomly placed inclusions, matrix damage was observed at a strain value of 0.016 ( Figure 3a). As the strain increases, the main crack starts to form (Figures 2b and 3b) and propagate (Figures 2c and 3c). A complete fracture of the RVE with randomly placed inclusions was observed at a strain of 0.038 (Figure 2d). In contrast, in the case of the RVE with aligned inclusions, it was observed at a strain of 0.045 (Figure 2h). After a complete fracture (Figure 3d,h), the stress dropped, and crack pinning and deflection of the epoxy matrix were observed in the fractured RVEs (Figure 2d,h). The influence of geometrical orientation of the inclusions is shown in Figure 4a. Both the effective Young's modulus and the tensile strength were higher in the case of aligned MXene and graphene inclusions. In the composites containing ρG = 500, ρMX = 125, fG = 0.1%, fMX = 1.6%, and EMX = 0.5Em, the effective Young's modulus was 3.4 GPa, and the tensile strength was 40.1 MPa with randomly placed inclusions, while the effective Young's modulus along the alignment (x-axis) direction was 4.6 GPa and the tensile strength was 58.4 MPa with aligned inclusions. The influence of the aspect ratio of aligned inclusions is shown in Figure 4b, at ρG = 500, fG = 0.1%, fMX = 3.2%, and EMX = 0.5Em. At MXene aspect ratio values of 60, 125, and 250, the effective Young's moduli were 4.7 GPa, 6.17 GPa, and 8.1 GPa, respectively. Additionally, higher values of elongation at break were observed at higher values of aspect ratio. At the beginning, the MXene/epoxy interfaces start to fail (Figure 3e). High-stress concentrations at the edges of the nano-reinforcements were observed (Figure 2a,e) resulting in the formation of localized cracking at the edges of the nano-reinforcements in both RVEs (Figure 3a,e). In the RVE with randomly placed inclusions, matrix damage was observed at a strain value of 0.016 ( Figure 3a). As the strain increases, the main crack starts to form (Figures 2b and 3b) and propagate (Figures 2c and 3c). A complete fracture of the RVE with randomly placed inclusions was observed at a strain of 0.038 (Figure 2d). In contrast, in the case of the RVE with aligned inclusions, it was observed at a strain of 0.045 (Figure 2h). After a complete fracture (Figure 3d,h), the stress dropped, and crack pinning and deflection of the epoxy matrix were observed in the fractured RVEs (Figure 2d,h).
The influence of geometrical orientation of the inclusions is shown in Figure 4a. Both the effective Young's modulus and the tensile strength were higher in the case of aligned MXene and graphene inclusions. In the composites containing ρ G = 500, ρ MX = 125, f G = 0.1%, f MX = 1.6%, and E MX = 0.5E m , the effective Young's modulus was 3.4 GPa, and the tensile strength was 40.1 MPa with randomly placed inclusions, while the effective Young's modulus along the alignment (x-axis) direction was 4.6 GPa and the tensile strength was 58.4 MPa with aligned inclusions. The influence of the aspect ratio of aligned inclusions is shown in Figure 4b, at ρ G = 500, f G = 0.1%, f MX = 3.2%, and E MX = 0.5E m . At MXene aspect ratio values of 60, 125, and 250, the effective Young's moduli were 4.7 GPa, 6.17 GPa, and 8.1 GPa, respectively. Additionally, higher values of elongation at break were observed at higher values of aspect ratio.  The MXene/epoxy interface did not have a significant influence on the effective Young's modulus of the composite; however, higher properties of the interface result in significantly higher values of the tensile strength ( Figure 5). In the RVEs with randomly placed MXenes having ρG = 500, ρMX = 125, fG = 0.1%, and fMX = 1.6%, the tensile strength was 40.1 MPa, 48.6 MPa, and 63.5 MPa, at EMX = 0.5Em, EMX = 0.75Em, and EMX = 1.5Em, respectively ( Figure 5a). In RVEs with ρG = 500, ρMX = 125, fG = 0.1%, and fMX = 6.4%, where MXenes aligned, the tensile strength was 69.2 MPa, 82.6 MPa, and 116.1 MPa, at EMX = 0.5Em, EMX = 0.75Em, and EMX = 1.5Em, respectively (Figure 5b). Moreover, elongation at break was almost two times higher when EMX = 1.5Em compared to the case when EMX = 0.5Em.
These results confirm the previous findings [63] that the elastic properties of this composite are not highly influenced by the MXene/epoxy interface properties. However, this research shows that the interface has an influence on the fracture response. The influence of the MXene volume fraction is shown in Figure 6. An increase in the volume fracture resulted in a significant increase in the effective Young's modulus of the composite. With randomly placed inclusions as ρG = 500, ρMX = 125, fG = 0.1%, and EMX = 0.5Em, the obtained effective Young's moduli were 2.8 GPa and 3.4 GPa at fMX = 0.8% and The MXene/epoxy interface did not have a significant influence on the effective Young's modulus of the composite; however, higher properties of the interface result in significantly higher values of the tensile strength ( Figure 5). In the RVEs with randomly placed MXenes having ρG = 500, ρMX = 125, fG = 0.1%, and fMX = 1.6%, the tensile strength was 40.1 MPa, 48.6 MPa, and 63.5 MPa, at EMX = 0.5Em, EMX = 0.75Em, and EMX = 1.5Em, respectively ( Figure 5a). In RVEs with ρG = 500, ρMX = 125, fG = 0.1%, and fMX = 6.4%, where MXenes aligned, the tensile strength was 69.2 MPa, 82.6 MPa, and 116.1 MPa, at EMX = 0.5Em, EMX = 0.75Em, and EMX = 1.5Em, respectively (Figure 5b). Moreover, elongation at break was almost two times higher when EMX = 1.5Em compared to the case when EMX = 0.5Em.
These results confirm the previous findings [63] that the elastic properties of this composite are not highly influenced by the MXene/epoxy interface properties. However, this research shows that the interface has an influence on the fracture response. The influence of the MXene volume fraction is shown in Figure 6. An increase in the volume fracture resulted in a significant increase in the effective Young's modulus of the composite. With randomly placed inclusions as ρG = 500, ρMX = 125, fG = 0.1%, and EMX = 0.5Em, the obtained effective Young's moduli were 2.8 GPa and 3.4 GPa at fMX = 0.8% and These results confirm the previous findings [63] that the elastic properties of this composite are not highly influenced by the MXene/epoxy interface properties. However, this research shows that the interface has an influence on the fracture response.
The influence of the MXene volume fraction is shown in Figure 6. An increase in the volume fracture resulted in a significant increase in the effective Young's modulus of the composite. With randomly placed inclusions as ρ G = 500, ρ MX = 125, f G = 0.1%, and E MX = 0.5E m , the obtained effective Young's moduli were 2. Summarizing the results in Figures 5 and 6, it should be noted that a stronger interface is able to improve both the maximum stress and the maximum strain, while the volume fraction of MXene is able to improve the maximum stress but the maximum strain is then decreased as the composite becomes more brittle. When developing compositions of new hybrid polymer composites with graphene and MXene nano-reinforcements, it is important to take these insights into account and control the manufacturing procedures in order to ensure a proper interface as well as optimize the MXene volume fraction in such cases where the degradation of structures occurs at small deformations.
Besides load carrying capabilities, the proposed composite can utilize the electrical properties of MXene. For example, when a structure made of such composite loses its structural integrity, the electrical conductivity is also deteriorated. This can be easily measured in real-time and give a reference about the structural integrity state.

Conclusions
Novel hybrid polymer composites with graphene and MXene nano-reinforcements were proposed, and a computational model based on the micromechanical finite element method was developed for studying the mechanical and damage behavior of hybrid polymer composites with MXene and graphene nano-reinforcements.
The influence of the geometrical orientation, aspect ratio and volume fractions of the inclusions, as well as the interface properties between the nano-reinforcements and the matrix, on the mechanical behavior was studied. The modelling demonstrated that both the effective Young's modulus and the tensile strength were higher in the composites with aligned MXene and graphene inclusions in comparison to the composites with randomly placed inclusions. An increase in the volume fracture of the nano-reinforcements results in a significant increase in the effective Young's modulus of the analyzed composites. The effective Young's modulus of the composite with aligned nano-reinforcements containing 0.1% and 6.4% volume fractions of graphene and MXene, respectively, was 3.65 times higher compared to the Young's modulus of the matrix. Moreover, the tensile strength increased as the volume fraction of MXene increased in the composites with aligned nano-reinforcements. However, in the composites Summarizing the results in Figures 5 and 6, it should be noted that a stronger interface is able to improve both the maximum stress and the maximum strain, while the volume fraction of MXene is able to improve the maximum stress but the maximum strain is then decreased as the composite becomes more brittle. When developing compositions of new hybrid polymer composites with graphene and MXene nano-reinforcements, it is important to take these insights into account and control the manufacturing procedures in order to ensure a proper interface as well as optimize the MXene volume fraction in such cases where the degradation of structures occurs at small deformations.
Besides load carrying capabilities, the proposed composite can utilize the electrical properties of MXene. For example, when a structure made of such composite loses its structural integrity, the electrical conductivity is also deteriorated. This can be easily measured in real-time and give a reference about the structural integrity state.

Conclusions
Novel hybrid polymer composites with graphene and MXene nano-reinforcements were proposed, and a computational model based on the micromechanical finite element method was developed for studying the mechanical and damage behavior of hybrid polymer composites with MXene and graphene nano-reinforcements.
The influence of the geometrical orientation, aspect ratio and volume fractions of the inclusions, as well as the interface properties between the nano-reinforcements and the matrix, on the mechanical behavior was studied. The modelling demonstrated that both the effective Young's modulus and the tensile strength were higher in the composites with aligned MXene and graphene inclusions in comparison to the composites with randomly placed inclusions. An increase in the volume fracture of the nano-reinforcements results in a significant increase in the effective Young's modulus of the analyzed composites. The effective Young's modulus of the composite with aligned nano-reinforcements containing 0.1% and 6.4% volume fractions of graphene and MXene, respectively, was 3.65 times higher compared to the Young's modulus of the matrix. Moreover, the tensile strength increased as the volume fraction of MXene increased in the composites with aligned nanoreinforcements. However, in the composites with randomly placed nano-reinforcements, an increase in the volume fraction resulted in only an increase in the effective Young's modulus. The MXene/epoxy interface did not have a significant influence on the effective Young's modulus of the composite; however, higher properties of the interface resulted in significantly higher values of the tensile strength. Further, higher mechanical properties Polymers 2021, 13, 1013 9 of 12 were observed at higher aspect ratios of nano-reinforcements. Crack pinning and deflection of the epoxy matrix were observed in the fractured RVEs.
Graphene nano-reinforcements may be used for additional strengthening of multifunctional composites with MXenes and further expand the scope of application of such materials by utilizing the great mechanical properties and electrical conductivity of MXene. The proposed novel hybrid polymer composites with graphene and MXene nanoreinforcements can be applied for flexible electronics and advanced high-strength structural applications with additional functions as real-time monitoring of structural integrity.
The results of the computational analysis revealed that MXene and graphene nanoreinforcements demonstrate considerable promise in the development of novel multifunctional composites, exhibiting excellent mechanical properties.