^{1}

^{1}

^{*}

Modeling and simulation allow methodical variation of material properties beyond the capacity of experimental methods. The polymers are one of the most commonly used matrices of choice for composites and have found applications in numerous fields. The stiff and fragile structure of monolithic polymers leads to the innate cracks to cause fracture and therefore the engineering applications of monolithic polymers, requiring robust damage tolerance and high fracture toughness, are not ubiquitous. In addition, when “many-parts” cling together to form polymers, a labyrinth of molecules results, which does not offer to electrons and phonons a smooth and continuous passageway. Therefore, the monolithic polymers are bad conductors of heat and electricity. However, it is well established that when polymers are embedded with suitable entities especially nano-fillers, such as metallic oxides, clays, carbon nanotubes, and other carbonaceous materials, their performance is propitiously improved. Among various additives, graphene has recently been employed as nano-filler to enhance mechanical, thermal, electrical, and functional properties of polymers. In this review, advances in the modeling and simulation of grapheme based polymer nanocomposites will be discussed in terms of graphene structure, topographical features, interfacial interactions, dispersion state, aspect ratio, weight fraction, and trade-off between variables and overall performance.

A comparison of the properties of polymers with other materials is shown in

Some of the most commonly used polymers include epoxy, polycarbonate, polyester, Polymethyl Methacrylate (PMMA), polyethylene, polypropylene, polyvinyl chloride, Polystyrene (PS), nylon, and teflon etc. The epoxies are most commonly used to produce PMCs. The polymers are reinforced with various kinds of reinforcements. The first manufactured PMCs used glass fibers as reinforcement. Besides glass fibers, some frequently used reinforcements include metallic oxides [_{2}) is on rise because of its applications in paints, varnishes, and paper industry [

There is an exponential rise in the use of graphene in the last decade. Novoselov et al. [

Various theoretical and computational approaches have been employed to explore the effect of graphene as reinforcement on the performance of polymer nanocomposites including but not limited to, quantum mechanical-based methods [

A schematic diagram of a hexagonal unit cell of graphene structure is shown in

are high energy sites and preferable localities for chemical reactants to attack. All these factors make graphene a very highly chemically reactive entity. The carbon atoms are connected through strong covalent bonds. There is sp^{2} orbital hybridization between P_{x} and P_{y} that forms σ-bond [_{z} forms π-bond with half-filled band that allows free motion of electrons. These interatomic forces play a crucial part in defining the proficiency of graphene as reinforcement in polymers.

As atoms keep oscillating with characteristic oscillating frequencies, the oscillatory period of carbon atoms for some bounded trajectories reaches terahertz frequencies [_{1} and potential

energy E is given in Equation (2), where m_{I}, r_{I}, and f_{I} are mass of, position of, and force on I^{th} atom, respectively, and

and non-bonded interactions among atoms and molecules. Polymer Consistent Force Field (PCFF) can be used to specify such interactions [

where P_{I} and M_{I} are momentum and mass of I^{th} atom, respectively. The energetics of the non-bonding interactions between all pairs of atoms can be defined using Lennard Jones potential (LJ-potential) as given in Equation (6) [

where r_{ij} is the separation distance between atoms i and j, and r_{cut} is the cut-off distance. For bonding interactions, the bond-stretch and bond-bending potentials are given by Equation (7) and Equation (8), respectively, and the dihedral potential is given by Equation (9) where Q_{ijkl} is the angle formed by i, j, k, and ℓ bonds [

In continuum-nanoscopic approach, also known as quasi-continuum method, every point in the continuum is considered as an extensive region wherein atomistic degrees of freedom are explicitly taken into account [

where

where x_{ij}, y_{ij}, and z_{ij} are the displacements of the local nodes in an arbitrary tetrahedral element. The bond length r_{ij} is given by Equation (12) [

where F^{T} is the transpose of F-matrix. The Green-Lagrange strain tensor E is given by Equation (13) [

where I is the identity tensor. From Equation (10) and Equation (12), the bond length in terms of Green-La- grange strain tensor is given by Equation (14) and Equation (15) [

where

where r_{L}, r_{0L}, and

where θ_{ijk} is the angle between triplet of atoms i, j, and k. The consine of dihedral can also be written as Equation (21) where r_{ij}, r_{jk}, and r_{kl} are the distances between atoms i and j, j and k, and k and ℓ, respectively, and Q_{ijkl} is the dihedral angle between the quartet of i, j, k, and ℓ atoms. Equation (21) can take the form of Equation (22) where the values of α, β, M, and N are defined in Equations (23)-(26), respectively [

The graphene has many other stable configurations apart from honeycomb hexagonal lattice structure. For example, different chemisorbed configurations of epitaxial graphene coexist on single crystal Ni (111) such as top-fcc, top-hcp, and top-bridge [

The exploitation of topographically modified geometries in synthetic and bio-inspired materials is a novel area of research [

for stable folding and bending energy at folds is compensated by intersheet adhesion (Van der Waals interactions) [

The topographical features of graphene can be observed experimentally and explored using computer simulations [^{2} hybridization which is stronger than sp^{3} hybridization found in diamond. Therefore, graphene shows superior mechanical properties. However, at the application of elevated temperatures, C-C bonds deteriorate and protrude out of the graphene plane. These misaligned bonds do not offer their intrinsic mechanical strength in the axial direction. The graphene sheets are held together by weak Van der Waals forces which can be subdued quite easily making sheet sliding inevitable in Multi-Layered Graphene (MLG). It was also observed that Young’s modulus significantly decreased due to layer sliding at different temperatures [

To model dynamics and energetics of graphene C-C covalent bonds, second-generation Brenner many-body interatomic potential can be used [

where

distance between pairs of nearest-neighbor atoms i and j, and _{o} is the initial length of RVE (nm), and ΔH is the elongation. The obtained value of axial Young’s modulus of graphene-PMMA nanocomposite was 59.536 GPa [

along with agglomerated graphene. In addition, the proposed model considers perfect structure of graphene. On the contrary, the actual graphene contains surface and structural defects which may deteriorate the mechanical properties of graphene and produced nanocomposites. An additional influential factor could be layer sliding which was not considered in the modeling approach but takes place in reality due to weak Van der Waals interactions between the graphene layers.

The schematic diagram illustrating the interfacial interactions between graphene and polymer matrix is shown in

According to ASM Handbook on composites (vol. 21 [

The improvement in mechanical properties of GBPNCs significantly depends on interfacial interactions as load must efficiently be transferred from the matrix to the reinforcement to alleviate the stress concentration in relatively weaker matrix phase. One of the factors influencing interfacial interactions is interfacial area. The interfacial interactions can significantly be improved with increasing interfacial area. Interfacial area can be increased by reducing reinforcement size and by altering the topography as shown in

increase in surface area, interfacial area with polymer matrix also increases which can significantly improve the mechanical properties of nanocomposites. In addition to that, mechanical interlocking between polymer and reinforcement will take place which can cause a further increase in the mechanical properties of nanocomposites as shown in

The graphene-polymer interactions can be defined via LJ-potential wherein spring elements with a nonlinear behavior can be used to couple the graphene and polymer matrix [_{1}, of the elastic member subjected to cooperative buckling is given by Equation (31), where b is sample width, D_{0} is bending rigidity,

In discrete buckling, also known as delaminated buckling, all layers buckle in disparate way. As indicated by

name, discrete buckling occurs when interfacial interactions are not strong enough to withstand the applied force. When compressive load exceeds the compressive strength and interfacial interactions are poor, delamination and discrete buckling take place as shown in _{2}, is mentioned in Equation (32), where D_{1} and D_{2} are bending rigidities of materials 1 and 2, and γ(u) is adhesion energy [

plied force is independent of buckling case, then change in potential energy (

tion (34). Delamination occurs when ΔΠ > 0, and does not occur when ΔΠ < 0 [

To determine whether cooperative buckling occurs or discrete buckling, Ritz method can be used [

where C is an undetermined coefficient and satisfies the boundary conditions for both the cases shown in ^{4}). From these values, ΔΠ is given by Equation (36) [

minimizing with respect to C and setting values as in Equation (37) and Equation (38) [

above equations give the critical condition for ΔD as given by Equation (39) [

If ΔD < kL^{4}/120, the change in stiffness from cooperative to discrete is not sufficient to overcome the adhesion. In this case, cooperative buckling is preferred mode when ΔΠ < 0. In case when ΔD > kL^{4}/120, the change in stiffness from cooperative to discrete is sufficient to overcome adhesion. In this case, discrete buckling occurs when ΔΠ > 0. The discrete buckling can be avoided by improving the interfacial interactions [

The graphene is unfortunately produced in entangled form and even if it is somehow dispersed, it tends to re- aggregate due to attractive Van der Waals forces [^{2}/g) and the fact that commercially available graphene is heavily entangled restricts uniform dispersion of graphene in polymer matrix. It is a great barricade in its research and commercial applications. Instead of using agglomerated graphene, it is beneficial for the composite properties to use non-agglomerated graphene, as inter-layer sliding and the fact that aggregates act as stress concentrators can deteriorate the composite strength. The issue of graphene dispersion in polymer matrix has yet not been entirely outwitted, particularly for graphene fraction above a few wt% [

There are two main methods for graphene dispersion. In the first method, the graphene can be separated by applying some external force, such as mechanical mixing and sonication, and is then stabilized by enclosure in a polymer or surfactant complex [

E.g., super-acids can dissolve graphene in large quantities [

The enhancement in the properties of GBPNCs by graphene is still not commensurate with the abilities inherited in the graphene structure [^{−4} under NVT ensemble at every 300 steps up to 1500 steps. The temperature was fixed at 0.1 K. The energy, temperature, and pressure trends obtained during MD equilibration are shown in

The schematic diagram of various aspect ratios of graphene is shown in

phene is an influential factor for the performance of GBPNCs. Rahman carried out modeling of graphene-epoxy nanocomposites with graphene of two different aspect ratios: length to width≥: (a) 5, and, (b) 10 [_{i}: i = 1, 2, 3., if x_{i} > L_{i}, then,

and if x_{i} < 0, then,

here, L_{i}: i = 1, 2, 3, represents the lengths of unit cell along x, y, z directions [

Rahman observed that uniformly dispersed graphene with high aspect ratio can significantly improve in-plane Young’s modulus [

However, graphene is not effective in increasing the out-of-plane Young’s modulus. It is because the out-of- plane stiffness depends on the modulus of polymer and Van der Waals interactions between graphene-graphene and graphene-polymer. It was observed that experimental Young’s modulus was higher for smaller aspect ratio than the larger one [_{1C} is shown in _{1C} is 18%. And at A.R = 2.8, the increase in G_{1C} becomes 24%. It shows that the G_{1C} and fracture toughness increase with increasing aspect ratio of graphene [

Cranford studied the mechanical properties of graphene with different longitudinal lengths and observed that maximum deviation was shown by smallest effective length of 50 Å [

increase the difference in predicted and measured values. Depending on effective stiffness and adhesion strength, either cooperative ordiscrete buckling occurs. A trade-off among critical length (L), stiffness of adhesion (k) and effective rigidity (D) is critical in initial designing of molecular composites. Buckling of graphene layers and interfacial delamination are most commonly observed failure modes in GBPNCs. To avoid such failures, minimum length scale and adhesion energy are necessary to avoid delamination under compressive forces.

The schematic diagram of varying weight fractions of graphene in polymer matrix is shown in

Rahman developed two different types of models: Fixed Graphene Model (FGrM), and Varying graphene model (VGrM), and studied three different weight fractions of graphene (1 wt%, 3 wt%, and 5 wt%) [^{3}. The weight fraction of graphene was varied by changing the number of epoxy molecules in the unit cell. Each unit cell contained one graphene layer. As graphene layer is fixed in each unit cell, this model was termed as “Fixed Graphene Model (FGrM)”. The alternative way of changing the weight fraction is by changing the number of graphene sheets in the unit cell while number of epoxy molecules is kept constant. In this case, there is no significant change in the total number of atoms in the unit cell. As graphene layer is varied in each unit cell, this model was termed as, “Varying Graphene Model (VGrM)”. The conventional molecular dynamics scheme can be used to equilibrate the model to get the final configuration. The schematic of the unit cell is shown in

Rahman produced epoxy-graphene nanocomposites to determine the validity of developed models [

The values of critical stress, strain, and Young’s modulus of SLG supported on perforated Si substrate were recorded using Atomic Force Microscopy (AFM) nanoindentation test and values obtained were 130 GPa, 25%, and 1 TPa, respectively [_{ij} can be calculated using virial expression as given in Equation (42) [

where Vol is volume of the simulation box, M_{I} is mass of I^{th} atom, ^{th} direction, N is total number of atoms, ^{th} and J^{th} atoms, and

force between I^{th} and J^{th} atoms. When uniaxial deformation is applied using uniaxial tension or compression, the total potential energy changes which can be used to determine elastic constants. The internal stress tensor can be calculated by taking the first derivative of potential energy with respect to strain. The second derivative of potential energy with respect to strain yields stiffness matrix and deformation in unit cell is given by Equation (43) where the corresponding stresses in the system (_{i} is i^{th} component of internal stress tensor, and _{11}, σ_{22}, and σ_{33} are the virial stresses in x, y, and z directions, respectively, and

Ebrahimi et al. used the conjugate pair of the Green-Lagrange strain and the second Piola-Kirchhoff stress tensor to study the stress-strain response of graphene-chitosan nanocomposites under large deformations [

where

VRVE is the volume of NRVE [

The ability of a material containing crack to resist fracture, known as fracture toughness, is a simple yet trustworthy indicator of the material’s damage tolerance and hindrance against fracture, and is considered as one of the most important mechanical properties of the engineering materials. The fracture toughness depends on how the pre-existing and newly formed cracks propagate within a material. The plane strain fracture toughness (critical stress intensity factor, K_{1C}) can be calculated using Equation (62), where P_{max} is maximum load of load-dis- placement curve (N), f(a/w) is constant related to geometry of the sample and is calculated using Equation (63), B is sample thickness (mm), W is sample width (mm), and a is crack length (should be kept between 0.45 W and 0.55 W according to ASTM D5045) [_{1C}) can be calculated using Equation (64), where E is the Young’s modulus obtained from the tensile tests (MPa), and ν is the Poisson’s ratio of the polymer. The geometric function f(a/W) strongly depends on the a/W ratio as shown in

Parashar and Mertiny developed three dimensional RVE based multiscale model in finite element environment to study the influence of graphene on the fracture characteristics of graphene-polymer nanocomposites [_{S} is bond-stretching component of potential energy (J), E_{B} is angle-bending component of potential energy (J), D_{e} is dissociation energy (Nm), β is constant controlling the width of the potential (m^{−1}), r is C-C bond length (m), r_{0} is C-C equilibrium bond length

(m), K_{Ө} is force constant for bond bending (N∙m/rad^{2}), Ө is current angle of the adjacent bond (rad), Ө_{0} is initial angle of the adjacent bond (rad), and K_{sextic} is constant in bending term of potential (rad^{−4}). The continuum polymer matrix, atomistic graphene structure, and interface were modeled using SOLID45, BEAM4, and LINK8, respectively, in ANSYS finite element software [

The stress-strain behavior of C-C bonds, configuration of multiscale model, and variation in G_{1C} with graphene volume fraction are shown in Figures 24-26, respectively [

can be attributed to variation in stress distribution by graphene in the polymer matrix. Because of high stiffness and space frame structure, graphene absorbs most of the applied load. The graphene also shields the crack tip from crack-driving forces and opening loads. The modeling results showed that when graphene is present on both sides of the crack plane, the crack tip shielding effect is more pronounced.

Because of superior thermal conductivity of graphene, GBPNCs are promising candidates for high-performance thermal interface materials. The dissipation of heat from electronic devices may also be barricaded when thehigh thermal conductivity of graphene is efficiently utilized. The graphene has shown higher efficiency in increasing the thermal conductivity of polymers than CNTs [_{eff}) of GBPNCs has a non-linear dependence on graphene weight fraction [

The theoretical calculations predict thermal conductivity values of polymer nanocomposites far greater than the experimental values [_{m}) and the time increment (Δt) as given in Equation (70) [

The TBR (R_{bd}) is introduced at graphene-polymer interface by a phonon transmission probability (f_{m-GA}) which can be measured according to acoustic mismatch theory as given in Equation (71) [

where

tively. The improvement in thermal conductivity can be theoretically described by the percolation theory using a power law relationship as given in Equation (72) [

where

threshold, and t is universal critical exponent. A comparative infrared microscopy technique can be used to measure thermal properties of nanocomposites [^{−8} m^{2}K/W [_{eff} values are obtained due to efficient heat transfer channel provided by graphene. However, when graphene is oriented

perpendicular to the direction of heat flux, lower K_{eff} values are obtained due to TBR. The modified effective medium theory to estimate K_{eff} is given in Equation (73) [

where

volume fraction of graphene. The results show that the values predicted by EMT are greater than experimental values. One main reason for this is the particle size distribution which EMT does not take into account. In general, a large particle size distribution can increase the filling ratio as the finer particles can fill in the interstitial sites created by larger particles. The higher filling ratio helps in making strong networks of graphene which can significantly increase the thermal conductivity values compared with narrow size distribution.

Chu et al. proposed an analytical model based on Differential-Effective-Medium (DEM) theory to determine the thermal properties of graphene based nanocomposites [

The solution for the temperature fields, T_{p} and T_{m}, within the dispersed particles and surrounding matrix, respectively, is given by Equation (76) [

where H_{j} is the depolarization factor of the ellipsoidal particles along j-axis, k_{m} is the thermal conductivity of matrix, and k_{j} is thermal conductivity of GNP along j-axis [_{x} + H_{z} = 1, where,

Equation (76) can be used to approximate the dependency of effective thermal conductivity of nanocomposites on the reinforcement volume fraction as given in Equation (78) [

when the initial condition is kept k^{*} = k_{m} at GNP volume fraction f = 0, Equation (78) can be written after integration as Equation (79) [

where K_{z} and K_{x} are the thermal conductivity values of GNP along longitudinal and transverse axes, respectively. At a very large aspect ratio (_{x} and H_{z} can be taken as zero and unity, respectively. Therefore, Equation (79) can be simplified as Equation (80) [

To consider interfacial thermal resistance, Chu et al. considered GNP as core-shell structure as shown in _{s} and the effective thermal conductivity can be written as Equations (81)-(83) [

Tailoring the electrical properties of graphene can unlock many potential electronic applications of graphene [

carriers per dopant in graphene sheet [

Due to graphitic composition, graphene exhibits excellent electrical conductivity. The charge carrier mobility of graphene at room temperature is as high as 15,000 cm^{2}/Vs [

The electrical conductivity of nanocomposites strongly depends on the microstructure. When polymer is filled with a conducting nano-filler, conduction of the produced nanocomposite depends on how the nano-fillers are aligned [

McCullough’s model is a structure-oriented model which assumes that the microstructure and orientation of reinforcement inside the matrix are variables and depend on the reinforcement content [_{f}, a), where a is the effective

aspect ratio and ν_{f} is the volume fraction of reinforcement. The average value of chain parameter

where λ(a) is chain parameter and is dependent on reinforcement shape. When aspect ratio a of reinforcement exceeds 1, the chain parameter can be calculated using Equation (90) and Equation (91) [

where,

where,

where σ_{m}, σ_{f}, and σ are electrical conductivities of matrix, reinforcement, and composite, respectively, and ν_{f} and ν_{m} are volume fractions of reinforcement and matrix, respectively. The experimental determination of η(ν_{f}, a) and

where <à^{2}> is mean-square value of conductive chains aspect ratio (a). With the help of AFM in CA-AFM model, the value of <à^{2}> can be calculated as in Equation (95) [

where n is the number of conductive chains, ℓ_{i} is length of i^{th} chain, d_{i} is diameter of i^{th} chain,

are average length and average diameter of chain, respectively. Therefore, Equation (92) can be reduced to Equation (96), and density of composite can be calculated using rule of mixture as given in Equation (97) [

where ρ, ρ_{f}, ρ_{m}, are specific densities of composite, reinforcement, and matrix, respectively. During production of nanocomposites, porosity is inevitable due to, (1) air entrapment, (2) evaporation of volatiles, (3) any shrinkage during curing, and (4) the relative movement of reinforcement and polymer chains. The porosity influences the conductivity which can be considered using an apparent density (ρ_{ap}) connected with specific density through the pore coefficient as given in Equation (98) [

When the nanocomposites are produced by a reproducible technology, such as latex technology, in which conductivity trends remain the same when samples are reproduced using the same parameters [_{ρ} can be taken as constant and needs to be measured only once. From Equation (98), Equation (97) can be written as Equation (99) where the coefficient K_{ρ} depends on the method of composite production and on geometry of the reinforcement [_{ap} can be defined experimentally if the reinforcement fraction (w), sample’s volume (V) and mass (m) are known as given in Equation (100), and the volume fraction of reinforcement can be calculated using Equation (101) where volume fraction can be converted to weight fraction using Equation (102), and G can be calculated by combining Equation (103) and Equation (104) [

percolation threshold [^{2}. The weight fractions of GNPs were 0, 0.6, 0.9, 1.5, and 2.0 wt%. Electrical conductivity measurements in Direct Current (DC) mode were performed in a direction parallel to the sample top surface using a 2-probe configuration and a Keithley 2602 system source meter. Conductivity was calculated from the obtained I/V characteristics using Equation (105) [

where V is applied voltage and I is current value through a cross-section (A) and between the distance (b). The values of

[

The relationship between

2-point DC measurements are in accord. The conductivity of polymer at low GNP loading is close to dielectric polymer matrix as no continuous pathways of GNPs are available for conduction. The percolation threshold of 0.9 wt% is suggested by DC measurements and is corroborated by Scanning Electron Microscopy (SEM) and CA-AFM. At the percolation threshold marked by DC measurements, the conductivity increases sharply by five orders of magnitude. This can be explained by the presence of microscopic conductive sub-networks of GNPs. However, these sub-networks do not create continuous network. Therefore, these sub-networks are not initially detected by DC measurements until a continuous network is established.

With the help of optical and electron microscopy, the dispersed reinforcement and polymer matrix can be observed in qualitative and quantitative manners, respectively. However, in this binary system integrated at nanoscale, there are myriad of factors that expound the overall performance of produced nanocomposites. Some of those defining features are summed up in

The properties of nanocomposites are also significantly dependent on filler shape and size. The graphene size, shape, and topography can be controlled simultaneously as shown in

It is well established that dispersion state is a crucial factor in defining the properties of nanocomposites and is discussed in detail in Section 6. In addition, the orientation of reinforcement and polymer chains and degree of crosslinking significantly influence the mechanical, thermal, and electrical properties of GBPNCs [_{13}P^{2}). Although various models have been presented to date regarding GBPNCs in which various aspects have been covered, however, it is just the beginning and myriad of factors are still to be incorporated in the modeling and simulation of GBPNCs.

Following conclusions can be drawn based on the critical review of the published literature related to modeling and simulation of GBPNCs:

Various theoretical and computational approaches have been employed to explore the influence of graphene as reinforcement on the performance of polymer nanocomposites including but not limited to, quantum mechanical-based methods [

The individual layers of graphene, under external loadings and thermal stresses, undergo out-of-plane wrapping [

Buckling of graphene layers and interfacial delamination are most commonly observed failure modes in GBPNCs [

MD simulation showed that thermal fluctuations produce ripples on graphene surface which influence the mechanical properties of graphene [

With the help of optical and electron microscopy, the dispersed reinforcement and polymer matrix can be observed in qualitative and quantitative manners, respectively. However, in this binary system integrated at nanoscale, there are a myriad of factors that expound the overall performance of produced nanocomposites.

If we consider the variables shown in _{13}P^{2}). Although various models have been presented to date regarding GBPNCs in which various aspects have been covered, however, it is just the beginning and myriad of factors which are still to be incorporated in the modeling and simulation of GBPNCs.

The authors would like to thank the Department of Mechanical and Construction Engineering, Northumbria University, UK for the provision of research facilities, without which the analysis of relevant data was not possible.

Rasheed Atif,Fawad Inam, (2016) Modeling and Simulation of Graphene Based Polymer Nanocomposites: Advances in the Last Decade. Graphene,05,96-142. doi: 10.4236/graphene.2016.52011

A.R: Aspect ratio, AFM: Atomic force microscopy, ASM: American society of metals, ASTM: American society for testing and materials, CA-AFM: Conductive atomic force microscopy, CM: Continuum mechanics, CNTs: Carbon nanotubes, CVD: Chemical vapor deposition, DC: Direct current, DEM: Differential effective medium, DFT: Density functional theory, EMA: Effective medium approximation, FGrM: Fixed graphene model, G1C: Critical strain energy release rate, GBPNCs: Graphene based polymer nanocomposites, GNPs: Graphene nanoplatelets, LAMMPS: Large-scale Atomic/Molecular Massively Parallel Simulator, K1C: Mode-I plane strain fracture toughness, LJ-potential: Lennard-Jones potential, MD: Molecular dynamics, MLG: Multi- layered graphene, MM: Molecular mechanics, m-rGO: Functionalized reduced graphene oxide, MTM: Mori- Tanaka Method, NRVE: Nanoscopic representative volume element, PBC: Periodic boundary condition, PCFF: Polymer consistent force field, PMCs: Polymer matrix nanocomposites, PMMA: Polymethyl methacrylate, PS: Polystyrene, PVA: Polyvinyl alcohol, RVE: Representative volume element, SEM: Scanning electron microscopy, SERR: Strain energy release rate, SLG: Single layer graphene, TBR: Thermal boundary resistance, UHMPE: Ultra-high modulus polyethylene, VGrM: Varying graphene model.