Finite element micromechanical modeling of elastic and thermoelastic constants in graphene nanoplatelet-aluminum nanocomposites

Abstract

A numerical micromechanics approach based on the finite element method (FEM) is employed to evaluate the elastic modulus and coefficient of thermal expansion (CTE) of graphene nanoplatelet (GNP)-reinforced aluminum (Al) matrix nanocomposites. The modeling framework integrates representative volume elements (RVEs) with detailed consideration of GNP morphology in the nanocomposite structure. The critical role of GNP waviness and the formation of aluminum carbide (Al₄C₃) interphase, resulting from the interaction between graphene and the metal matrix, is examined in relation to nanocomposite properties. Variations in the volume fraction, geometry, and alignment of GNPs, along with the interphase thickness and its material properties, are considered to capture the microstructural influence on the elastic modulus and CTE of GNP/Al nanocomposites. The study reveals that the presence and growth of the Al₄C₃ interphase contribute positively to the mechanical and thermal elastic response of the nanocomposite. Although increasing graphene content, aspect ratio, and alignment enhances the elastic modulus and reduces the CTE, the presence of waviness in graphene nanofillers diminishes these benefits. Model validation is carried out through a comparison between micromechanics-based FEM outcomes and experimental data documented in the literature.

Keywords

aluminum nanocomposites graphene nanoplatelets elastic and thermoelastic constants micromechanics-based finite element approach graphene morphology

Introduction

Aluminum (Al) is an extremely versatile non-ferrous metal that possesses important physical and mechanical properties, making it suitable for various structural applications, including the fabrication of Al-based metal matrix composites (MMCs).^1–3 High specific strength, excellent corrosion resistance, reflectivity, ductility, and low density are among the key attributes that enable widespread applications of Al and its alloys in aerospace, marine, rail transport, and construction industries.^4–8 However, the growing demand for MMCs with acceptable mechanical and thermo-mechanical performance, driven by the requirements of emerging engineering applications, has led to the incorporation of secondary reinforcement phases such as ceramics and nanofillers.^9,10 The advancement of nanotechnologies demonstrates that nanoscale reinforcements can effectively enhance the mechanical and thermal performance of composites by increasing the surface-to-volume ratio and interfacial reactivity between the matrix and nanofillers.^11,12 Graphene nanoplatelets (GNPs), a class of carbon nanostructures formed by stacking a few to several layers of graphene, possess a very high elastic modulus and a low coefficient of thermal expansion (CTE), thereby imparting superior properties to nanocomposites. GNPs have garnered particular attention due to their exceptional stiffness, high aspect ratio, low density, and two-dimensional planar structure. Compared to other carbon nanostructures, such as single-walled and multi-walled carbon nanotubes (CNTs), GNPs exhibit a rough laminated architecture with a larger surface area, making them a promising alternative reinforcement for metal matrix nanocomposites (MMNCs) and an effective means of addressing the limitations of conventional MMCs.^13,14

Experimental and analytical studies available in the literature^15–20 highlight the significant impact of carbonaceous nanofillers on the mechanical and physical behavior of MMNCs. In an experimental investigation on the mechanical properties of Al-based nanocomposites reinforced with varying concentrations of CNTs, Bradbury et al.¹⁵ reported that the addition of 3% multi-walled CNTs resulted in approximately 160%, 262.7%, and 41.1% increases in hardness, tensile strength, and Young’s modulus, respectively. In addition to mechanical properties, the thermal expansion behavior of MMNCs must be investigated, as the structures made from such advanced materials are often exposed to elevated temperatures during manufacturing and service conditions. Such evaluation is essential for assessing the long-term reliability of MMNCs.¹⁶ Using a micromechanics-based simplified unit cell model, Hassanzadeh-Aghdam et al.¹⁶ investigated the influence of various microstructural parameters on the thermal expansion behavior of CNT/Al nanocomposites. These parameters include CNT volume fraction, aspect ratio, cross-sectional shape, waviness, orientation (either unidirectionally aligned or randomly dispersed), agglomeration state, and the formation of aluminum carbide (Al₄C₃) interphase resulting from the interaction between CNTs and the Al matrix. Their findings revealed that the cross-sectional shape of CNTs have a negligible effect on the CTE, whereas other factors, particularly agglomeration induced by strong van der Waals forces and CNT content, exert a significant influence. Empirical results reported by Tang et al.¹⁷ indicate that by raising the CNT content to 15 vol%, the CTE of the composite is reduced by 65%. of CNT-reinforced Al-MMNC. Li and Xiong¹⁸ investigated the influence of graphene nanosheet (GNS) content on the tensile properties and microstructure of GNS-reinforced Al-MMNC. Their study revealed that the Al₄C₃ phase forms at the GNS/Al interface, with its concentration directly correlated to the GNS content. The results demonstrated that incorporating 0.25 wt% GNS led to improvements of 38.27% in yield strength and 56.19% in tensile strength compared to pristine aluminum. In an experimental investigation into the effects of GNS volume fraction and alignment on the anisotropic thermal properties of MMNCs, Chu et al.¹⁹ examined the thermal conductivity and CTE of copper (Cu)-MMNCs reinforced with highly aligned GNSs. Their findings revealed that increasing the GNS volume fraction up to 30% resulted in a 35% enhancement in thermal conductivity and a 64% reduction in CTE, indicating high in-plane thermal conductivity and low through-plane thermal expansion. Zhou et al.²⁰ fabricated Al-MMNCs reinforced with few-layered graphene (FLG) using spark plasma sintering followed by hot extrusion, and investigated the direct contribution of FLG to the thermal expansion behavior of FLG/Al nanocomposites. Their experimental results showed that the addition of 0.4 vol% FLG led to a ∼4.45% reduction in the CTE. It was observed that Schneider’s analytical model slightly underestimated the experimental values, likely due to interfacial effects. Furthermore, through a comprehensive comparison of conventional fillers, including multi-walled and single-walled CNTs, carbon fiber, graphite, silicon carbide, and titanium carbide, FLG was identified as the most effective reinforcement for improving the CTE of Al-based nanocomposites.

One of the key challenges associated with MMNCs is the extensive knowledge required to investigate microstructural effects at the nanoscale. Unlike polymer-based composites, the dispersion of nanoscale reinforcements within MMNCs, along with the evaluation of their geometrical characteristics, is highly complex and often not cost-effective in terms of time and financial resources.^21,22 Hence, numerical schemes based on micromechanics-driven finite element method (FEM) serve as powerful and widely adopted tools for investigating the effective properties of MMNCs. Regarding the thermal expansion behavior of metal-based composites, Nam et al.²³ predicted the CTE of Al-MMCs containing a high fraction of SiC particles using analytical thermoelastic models and finite element analysis. Rezzig et al.²⁴ examined how Al₄C₃ interphase development and CNT clustering influence the overall properties of CNT-filled Al composites through a numerical micromechanics-based approach. It was observed that CNT agglomeration and interphase formation strongly affect the elastic behavior of Al nanocomposites, with the impact intensifying at greater CNT volume fractions.

Based on the preceding overview, graphene-based nanofillers demonstrate significant potential for enhancing the effective properties of MMNCs. However, research on the mechanical and thermal expansion behavior of graphene-reinforced MMNCs remains limited, largely due to manufacturing challenges that impede reliable experimental characterization. In this regard, simulations that incorporate critical microstructural phenomena such as morphological features and interfacial interactions are indispensable for improving the accuracy of nanocomposite modeling and aligning it more closely with physical reality. Notably, investigations addressing the influence of GNP waviness are particularly scarce. The present study aims to develop a micromechanics-based finite element model for evaluating the elastic modulus and CTE of GNP-reinforced Al-MMNCs. The structure of the paper is organized as follows: The second section introduces the governing equations and material properties of the MMNC constituents. The third section details the simulation framework of the three-phase representative volume element (RVE), including its constituent phases, modeling assumptions, and applied boundary conditions. In the fourth section, the micromechanics-based FEM is validated against available experimental data, followed by parametric studies examining the influence of microstructural factors on the effective properties. Finally, the fifth section summarizes the key findings and implications of the research.

Constitutive equations and material properties

The mechanical and thermal expansion properties of nanocomposite materials are extracted using finite element software packages. Nevertheless, the constitutive equations governing the computation of these properties are explicitly formulated and presented. Although heat generation typically results in non-uniform temperature distributions, the present study assumes that thermal diffusion leads to a uniform temperature field throughout the body volume. By neglecting inertial effects, the governing thermoelastic equation can be expressed as follows^25,26:

\nabla . σ = 0

(1)

where

\nabla .

and

σ

represent the divergence operator and the Cauchy stress tensor, respectively. In classical continuum mechanics, the constitutive stress-strain relationship at a point

x

is expressed as follows:

σ (x) = C (x) : (ε (x) - Δ T α (x))

(2)

where

C

denotes the fourth-order elasticity tensor,

α

stands for the second-order CTE tensor, and

ε

is the strain tensor. In addition,

Δ T

represents the temperature change throughout the body. Disregarding the thermal term, this equation simplifies to

σ (x) = C (x) : ε (x)

, based on which Young’s modulus is calculated. Since Al matrix, GNP, and Al₄C₃ interphase considered isotropic materials,

C

is in the following form,

C = \frac{E}{1 + ν} I + \frac{3 K}{ν (1 + ν)} 1 ⨂ 1

(3)

where

K = E / 3 (1 - 2 ν)

is the bulk modulus (

E

and

ν

refer to Young’s modulus and Poisson’s ratio, respectively),

I

and 1 stand for the fourth- and second-order identity tensors, respectively, and

⨂

is the dyadic product. Further, the kinematic equation for

ε

is given via the following relation under the small deformation assumption,

ε (x) = (\nabla u + {(\nabla u)}^{T}) / 2

(4)

where

\nabla

is the gradient operator and

u

represents displacement vector. Given that the volumetric expansion from temperature differences (

Δ T

) does not induce stress within the system, equation (2) transforms into the following form, serving as the criterion for determining the CTE,

α_{p} = \frac{1}{Δ T} ε_{p} = \frac{1}{Δ T} \frac{U_{p}}{l}, p = 1, 2, 3

(5)

where the subscript

p

denotes the direction,

l

can be the length, width, and thickness of RVE, and

U_{p}

indicates the displacement in the direction

p

The MMNC material selected in the present study comprises an Al matrix reinforced with either straight or wavy GNPs. Commercially available GNPs are typically produced with inherent morphological irregularities, including waviness, arising from synthesis and exfoliation processes. Achieving perfectly flat GNPs on a large scale remains challenging, as their morphology is strongly influenced by production methods, processing conditions, and dispersion techniques. Although advanced fabrication routes such as chemical vapor deposition or controlled exfoliation can yield relatively flatter graphene sheets, these approaches are not widely adopted for bulk production of GNPs intended for composite reinforcement. Consequently, waviness is generally regarded as an intrinsic characteristic of commercially available GNPs, and its complete elimination is difficult. Partial control may be achieved through optimized dispersion, alignment, or surface functionalization strategies, which can mitigate agglomeration and improve filler orientation. Accordingly, waviness is incorporated into the present modeling framework as a realistic microstructural feature to more accurately capture the behavior of GNP-reinforced MMNCs.^20,26 Additionally, the contribution of the Al₄C₃ interphase, produced through GNP-Al interaction, is also included by modeling it with a prescribed thickness. To evaluate the elastic modulus and CTE of the Al-MMNC, isotropic behavior is assumed for the Al matrix, GNP, and the Al₄C₃ interphase. The Al matrix has a Young’s modulus of 76.5 $G P a$ , Poisson’s ratio of 0.33, and a CTE of 26.72 $\times 1 0^{- 6} / K$ .²⁰ GNPs with a sheet-like form has a thickness of 2.5 $n m$ and an average lateral size of 50 $n m$ . Young’s modulus, Poisson’s ratio, and CTE of GNP are taken as 1000 $G P a$ , 0.22, and −8 $\times 1 0^{- 6} / K$ , respectively.^11,20 As well, the contribution of the Al₄C₃ interphase with a Young’s modulus of 309 $G P a$ , Poisson’s ratio of 0.2, and a CTE of 5 $\times 1 0^{- 6} / K$ is specified with a thickness of 0.25 $n m$ .¹⁶ It should be noted that, since this research is purely computational-numerical, certain geometric parameters of the components are defined hypothetically.

Finite element micromechanical modeling

In analyses aimed at determining engineering constants of nanocomposites, bottom-up multiscale simulations are conducted across the nano-, micro-, meso-, and macro-scales. Among these, micromechanical modeling, used to extract various effective properties of nanocomposites while accounting for microstructural effects, offers enhanced flexibility in capturing the geometry of constituent phases at the meso-scale. Assuming that the macroscopic nanocomposite exhibits a periodic structure composed of numerous RVEs, the effective properties of the bulk material can be generalized from the RVE-level results by accurately specifying the dimensions and distribution of nano-additives within each RVE. The demonstrated proficiency of RVEs in evaluating various behaviors of nanocomposites, as reported in numerous studies, further supports this assertion.^26–30 The three-dimensional RVEs considered in the present study are modeled as homogeneous cuboids, each having a length, width, and thickness equal to $l = 0.2 μ m$ . According to the available reports,²⁰ sheet-like straight GNP considered in the form of a cubic domain with square cross-section, where both length and width equal $L_{G} = 50 n m$ and a thickness equal $t_{G} = 2.5 n m$ (Figure 1(a)). Additionally, as illustrated in Figure 1(b), parameters $β$ and $λ$ , which represent the vertical distance from the center of the wavy GNP to its upper surface and the imaged length of the wavy GNP, respectively, are included in the simulation process of wavy GNP ( $β / λ =$ 0.2 and $λ / t_{G} =$ 20). Figure 1(c) shows a schematic view of the modeling of the Al₄C₃ interphase region with a thickness of $t_{i} = 0.25 n m$ and a total surface coating of the GNP.

Figure 1.

Dimensional specifications of the simulated samples: (a) straight GNPs, (b) wavy GNPs, and (c) the GNP/Al interphase region.

Since the study of microstructure-level features including orientation and waviness, volume fraction, aspect ratio, GNP/Al interphase, and other geometrical influences of GNPs is intended by employing a micromechanical finite element model formulated on the basis of the RVE concept, based on the shape of the simulated GNPs and their dimensions, we consider the volume fraction ( $V_{G}$ ) and the aspect ratio ( $A R$ ) of GNP respectively as follows:

V_{G} = \frac{n {L_{G}}^{2} t_{G}}{l^{3}}

(6a)

A R = \frac{L_{G}}{t_{G}}

(6b)

where

n

denotes the total number of GNPs dispersed within the RVE. The RVE of composites reinforced with micro-scale fibers usually consists of a few fibers due to the equal size unit of RVE and fibers. However, micromechanical modeling of the effects of nanofillers requires the distribution of a large number of nanofillers due to the very small dimensions of nanoscale fillers. Throughout this analysis, for the sake of satisfying the convergence condition, the number of GNPs is considered 40. In order to generate RVEs, a code based on Python software has been built with the inputs of RVE size, volume content and number of GNPs, and the thickness of the Al₄C₃ interphase.

The RVE is generated under the assumptions of perfect bonding between components, avoidance of GNP intersection through defined minimum spacing, and periodicity in geometry. The minimum relative distance is defined to ensure that the distribution of GNPs within the RVEs approximates a uniform arrangement. If this value is set too large, the target volume fraction may not be achieved; conversely, if it is too small, overlap between particles may occur or a condition resembling agglomeration may arise. Moreover, based on the periodic geometry conditions, the parts of the fillers that are in the side volume elements of the RVE are transferred to the corresponding facing surface. Figure 2 displays a simplified view of the applied periodic geometry conditions.

Figure 2.

Simplified view of the periodic geometry conditions.

The initial step in the RVE generation process is to script a GNP (with interphase or without interphase) in a random origin with a random orientation (according to aligned or random distribution of GNPs). Afterward, other GNPs are transferred or rotated with the mentioned assumptions in relation to the initial GNP, so that the inputs of the problem, including the volume fraction and the number of GNPs, are satisfied. By establishing the volume fraction of GNPs embedded in the Al matrix, the RVE generation process is stopped. Figure 3(a) and (b) illustrate the generated RVEs with 2 vol% straight and wavy GNP, respectively, with an aligned distribution. Also, the random distribution of the straight and wavy GNP is displayed in Figure 3(c) and (d), respectively. It should be noted that all the RVE components, including the Al matrix, straight and wavy GNP, and GNP/Al interphase, are meshed with C3D10 elements. The C3D10 element is a three-dimensional, 10-node quadratic tetrahedral element available in the finite element software, in which each node possesses three translational degrees of freedom. This element type is particularly suitable for meshing complex geometries and curved boundaries, and provides higher accuracy compared to linear tetrahedral elements (Figure 3(e)).

Figure 3.

Aligned dispersion of 2 vol% (a) straight and (b) wavy GNPs in Al-MMNC, random dispersion of 2 vol% (c) straight and (d) wavy GNPs in Al-MMNC, (e) C3D10 element.

Additionally, optimizing the number of elements is necessary to avoid increasing the calculation time and extract accurate results, which will be discussed later. In order to calculate the elastic modulus, a uniaxial tensile test was performed. The RVEs have one side that is restricted, and the other side is subjected to the uniaxial displacement of ${\bar{u}}_{p} = 0.01 l$ . Then, Hooke’s law is used to determine the equivalent elastic modulus ( $E$ ) of the nanocomposite through evaluation of the reaction force at the fixed boundary and subsequent stress calculation. By discretizing RVEs into appropriate elements, the conditions for loading and extracting CTE of MMNC reinforced with straight or wavy GNPs are provided in accordance with the structural relationships mentioned in Section 2. According to equation (5), the CTE determination of GNP-reinforced Al-MMNC requires the application of a temperature difference $Δ T$ in the entire volume of RVE. The temperature difference $Δ T$ leads to an increase in the volume of RVE. This increase in volume shows the variation in length in three main directions, the extraction of which leads to establishing equation (5). For this purpose, it is necessary to apply periodic boundary conditions in determining the properties in which the RVE should be deformed.

Periodic boundary conditions applied to RVE surfaces state that all points on a surface have the same displacement. It is important to mention that since the basis of this boundary condition is the establishment of connections between the nodes on the free surfaces of the RVE, a higher mesh density at the interface between nano-additives and the matrix produces a large node count on the RVE surfaces, which can significantly extend computation time. To overcome this problem, the application of periodic boundary conditions with the equation constraints, which has been shown to be accurate in estimating the solutions,²⁶ has been utilized in present research. The boundary conditions implemented in this research are listed in Table 1. It should be noted that in the case where the GNPs are distributed randomly in the Al matrix (Figure 3(c) and (d)), the displacement

U_{p}

caused by the temperature difference

Δ T

is almost equal in all three directions. In this case, the reported CTE is the average of the results in three principal directions for the sake of convergence. Also, in the case of unidirectional dispersion of GNPs in the Al matrix (Figure 3(a) and (b)), the reported longitudinal CTE is in the

x

direction. It should be emphasized that, to ensure the accurate and reliable reporting of properties, three representative building blocks are simulated for each case study, and the results are subsequently averaged.

Table 1.

Boundary conditions applied to the RVE surfaces for the characterization of GNP/Al MMNC elastic modulus and CTE.

RVE surface	Young’s modulus			CTE
RVE surface	$E_{1}$	$E_{2}$	$E_{3}$	$α_{1}$	$α_{2}$	$α_{3}$
$x = 0$	$u_{1} = 0$	$u_{1} = 0$	$u_{1} = 0$	$u_{1} = 0$	$u_{1} = 0$	$u_{1} = 0$
$y = 0$	$u_{2} = 0$	$u_{2} = 0$	$u_{2} = 0$	$u_{2} = 0$	$u_{2} = 0$	$u_{2} = 0$
$z = 0$	$u_{3} = 0$	$u_{3} = 0$	$u_{3} = 0$	$u_{3} = 0$	$u_{3} = 0$	$u_{3} = 0$
$x = l$ ^a	${\bar{u}}_{1}$	$u_{1}$	$u_{1}$	$U_{1}$	$U_{1}$	$U_{1}$
$y = l$ ^a	$u_{2}$	${\bar{u}}_{2}$	$u_{2}$	$U_{2}$	$U_{2}$	$U_{2}$
$z = l$ ^a	$u_{3}$	$u_{3}$	${\bar{u}}_{3}$	$U_{3}$	$U_{3}$	$U_{3}$

^aPeriodic boundary conditions (all nodes on the surfaces $x, y, z = l$ have the unique identical displacementby the utilization of equation constraints).

Results and discussion

Validation

It is necessary to demonstrate that the simulated RVEs perform correctly in predicting the engineering constants of MMNCs. Comparison between finite element outcomes with experimental tests can be effective for this purpose. The GNP-reinforced Al-MMNC system tested by Zhou et al.²⁰ has been selected to this aim. Based on the available information, the GNPs have a thickness and length of

2.5 n m

and

830 n m

, respectively (

A R = 332

). The Al matrix has a Young’s modulus of 76.5

G P a

, Poisson’s ratio of 0.33, and a CTE of 26.72

\times 1 0^{- 6} / K

. In addition, Young’s modulus, Poisson’s ratio, and CTE of GNP are assumed to be 1000

G P a

, 0.22, and −8

\times 1 0^{- 6} / K

, respectively. In order to generate suitable RVEs to investigate the influences of rising the content of GNPs on the CTE of GNP-reinforced Al-MMNC, straight GNPs are modeled according to Figure 1(a) and the reported dimensions. Then, RVEs with different volume fractions of GNP, which were modeled based on the algorithm mentioned in Section 3, are subjected to a temperature difference of

Δ T = 1

°C. The distribution of GNPs within the RVE is considered to be random and the periodic boundary conditions with the equation constraints are established. The exerted temperature difference leads to an increase in the volume of RVE and a change in length in different directions, which provides the basis for the calculation of CTE (Equation (5)). Since GNPs distribution in the matrix is random, the reported CTE represents the average of the results obtained across the three principal directions and three RVEs. The results presented in Table 2 show that the finite element predictions can well determine the CTE of GNP/Al MMNCs. The CTE was reduced by 2.24% as the GNP volume fraction increased to 0.2%. This reduction in the experimental test is reported as 2.13%, which indicates an acceptable compliance.

Table 2.

Comparison between finite element predictions and experimental test results for GNP/Al MMNC CTE.

GNP volume content (%)	CTE ( $1 0^{- 6} / K$ )
GNP volume content (%)	FEM	Experiment²⁰
0	26.72	26.72
0.05	26.62	26.58
0.1	26.51	26.37
0.2	26.12	26.15

Convergence study

Finite element analysis results are sensitive to both the distribution of nanofillers in the RVE and the mesh size, which can compromise the clarity of the findings. It is necessary to select the number of nanofillers and elements in such a way as to optimize the calculation time in addition to achieving accurate results. For this reason, the convergence studies have been conducted on the number of GNPs and elements. For each case, multiple random realizations of the RVE were simulated. The variation across different realizations was found to be negligible, so the mean values are reported in the figures without explicitly plotting the standard deviations. With the GNP volume fraction maintained at 2%, Figure 4(a) demonstrates how the number of randomly dispersed GNPs without considering the interphase region affects the ultimate CTE of GNP/Al MMNC. It is clear that the determined CTE is saturated for values exceeding 35 numbers, and without changing the results, it is accompanied by an increase in the calculation time. For this reason, during this analysis, the number of GNPs distributed in the RVE is considered to be 40. As illustrated in Figure 4(b), RVE discretization with a coarse element size (a smaller number of elements) does not have high accuracy in determining CTE. With the increase in the number of elements, the CTE of GNP/Al MMNC does not change much compared to the number of elements beyond 70,000. In this research, taking into account the effects of different microstructure levels, RVEs have been discretized in such a way that the number of elements does not affect the results.

Figure 4.

Convergence study to optimize the number of (a) nanofillers and (b) elements.

Parametric study

The elastic modulus of GNP-reinforced Al-MMNCs is a crucial property reflecting the composite’s ability to resist deformation under stress. Using the finite element micromechanical model and RVE approach, the influences of GNP volume fraction, aspect ratio, waviness, and the Al₄C₃ interphase features (thickness and material properties) on the engineering constants are investigated.

Figure 5 shows the variation of GNP/Al MMNC elastic modulus with the graphene volume fraction. The elastic modulus is improved significantly with increasing GNP volume fraction. For both straight and wavy GNPs, the trend is almost nonlinear. The straight GNP configuration consistently provides a higher elastic modulus compared to the wavy GNPs at all volume fractions. This superiority of straight GNPs arises from their more effective load transfer capability. Conversely, the graphene waviness reduces the efficiency of reinforcement. The results demonstrate that straight nanofillers provide better stress transfer and higher stiffness than their wavy counterparts. As shown in Figure 6, with the increase of GNP volume fraction, the MMNC CTE decreases significantly, which is more noticeable for straight GNPs. The results show that the straight GNPs can be better for improving the thermal expanding constant of MMNCs as they give lower CTE values. It is found from Figures 5 and 6 that the formation of carbide interphase notably affects the engineering constants of MMNCs as mentioned in the previous studies.^31–35 Considering the Al₄C₃ interphase region with the assumed thickness, can be effective in improving the engineering constants of GNP/Al MMNCs.

Figure 5.

Effects of GNP volume fraction on the elastic modulus of GNP/Al MMNC.

Figure 6.

Effects of GNP volume fraction on the CTE of GNP/Al MMNC.

Figure 7(a) shows the variation of elastic modulus of GNP/Al MMNCs with the value of $β / λ$ in the presence and absence of interphase. Wavy GNPs with the higher $β / λ$ more reduces the elastic modulus. This finding is consistent with prior studies, which demonstrated that straight nanofillers offer superior reinforcement compared to wavy structures due to the better stress transfer.^36–39 As can be seen in Figure 7(b), with the increase of parameter $β / λ$ , the CTE shows an increasing trend.

Figure 7.

Effects of wavy GNP geometry on the (a) elastic modulus and (b) CTE of GNP/Al MMNC.

The thickness of the Al₄C₃ interphase significantly influences the elastic modulus of the MMNC, as shown in Figure 8(a). For both straight and wavy GNPs, increasing the interphase thickness results in a higher elastic modulus for GNP/Al MMNCs. The increase is more pronounced in the straight GNP configuration. The elastic modulus of the interphase also contributes to the overall stiffness of the GNP/Al MMNCs as can be seen in Figure 8(b). As the elastic modulus of the interphase increases, the Al nanocomposite becomes stiffer due to improved load transfer across the matrix-filler interface.

Figure 8.

(a) Effects of interphase thickness on the elastic modulus of GNP/Al MMNC; (b) effects of interphase elastic modulus on the elastic modulus of GNP/Al MMNC.

In addition, as seen in Figure 9(a), the increase in the thickness of the Al₄C₃ interphase is associated with the decrease in CTE. The decrease slope of CTE for MMNC reinforced with 2 vol% straight GNP is higher than that of the wavy GNP. It was clarified that the Al₄C₃ interphase region formed by the interaction between graphene nanofillers and metal matrix can be effective for the improvement of CTE of MMMCs. The importance of interphase led to the investigation of the effects of its Young’s modulus and CTE of on the thermal expansion constant of the GNP/Al MMNC. In Figure 9(b), the CTE of the present MMNC is plotted as a function of the elastic modulus of the interphase. It is known that with the increase of the elastic modulus of the interphase, the CTE of the MMNC decreases, which is very small due to the very small volume fraction of interphase compared to nanofillers. The effect of increasing the CTE of the interphase on the CTE of the GNP/Al MMNC is also shown in Figure 9(c).

Figure 9.

(a) Effects of interphase thickness on the CTE of GNP/Al MMNC; (b) effects of interphase elastic modulus on the CTE of GNP/Al MMNC; (c) effects of interphase CTE on the CTE of GNP/Al MMNC.

The effect of GNP aspect ratio on the engineering constants of GNP/Al MMNC is explored using the developed finite element technique by considering a fixed nanofiller volume fraction of 2% into the RVE. Figure 10(a) shows that the Al MMNC elastic modulus increases with the rise of GNP aspect ratio. It is obvious from Figure 10(b) that CTE decreases slightly by increasing the aspect ratio of randomly dispersed GNPs.

Figure 10.

(a) Effects of GNP aspect ratio on the (a) elastic modulus and (b) CTE of GNP/Al MMNC.

Table 3 illustrates effects of some critical microstructures including interphase, wavy shape and dispersion type of GNPs on the elastic modulus and CTE of GNP/Al MMNCs. It is clear that the aligned dispersion of graphene nanofillers within the nanocomposite performs better in improving the engineering constants. Generally, the alignment of straight GNPs with an interphase region can give the maximum value of elastic modulus and minimum value of CTE for the GNP/Al MMNCs.

Table 3.

Effects of GNP dispersion type, waviness, and interphase on the elastic modulus and CTE of GNP/Al MMNC.

Sample	Aligned dispersion				Random dispersion
	Straight		Wavy		Straight		Wavy
	Without interphase	With interphase	Without interphase	With interphase	Without interphase	With interphase	Without interphase	With interphase
CTE (10^-6/K)	22.68	22.4	24.4	24.22	23.88	23.6	24.75	24.52
E (GPa)	86.17	87.36	83.73	84.21	85.33	86.96	82.7	83.3

The observed reduction in modulus with increasing waviness can be attributed to the loss of effective load transfer efficiency between the GNPs and the Al matrix. As the waviness increases, the alignment of GNPs with the loading direction deteriorates, thereby reducing their reinforcing capability and diminishing the stiffness of the composite. Similarly, the diminishing advantage in the CTE with increasing waviness arises from the reduced constraint imposed by the GNPs on the thermal expansion of the matrix. Flatter GNPs provide stronger restriction against matrix expansion, whereas wavy morphologies weaken this effect, leading to higher effective CTE values. Although the presence of an Al₄C₃ interfacial layer can enhance bonding between GNPs and the Al matrix, thereby improving stress transfer and reducing thermal mismatch, it may also introduce potential drawbacks. Specifically, the brittle nature of the carbide phase can act as a preferential site for crack initiation and propagation under mechanical or thermal loading. This dual effect suggests that while the Al₄C₃ layer contributes positively to modulus and CTE control, its presence must be carefully considered, as it may compromise the long-term durability of the nanocomposite.^2,40–44

Conclusion

Using a micromechanical model based on the FEM, the elastic modulus and CTE of GNP/Al MMNCs were investigated. The effects of microstructural contributors were evaluated through the RVE approach. Each RVE, comprising an Al matrix, straight or wavy GNPs, and an Al₄C₃ interphase, was subjected to a prescribed displacement and temperature gradient under the assumptions of perfect interfacial bonding and non-intersection between randomly dispersed nanofillers and their coatings. The results clearly demonstrated that the incorporation of graphene nanofillers into the Al matrix enhances the engineering constants of MMNCs. Furthermore, it was found that aligned dispersion of GNPs within the matrix yields superior thermo-mechanical properties compared to random dispersion. A reduction in CTE and an increase in elastic modulus were observed with increasing volume fraction and aspect ratio of GNPs. The detrimental effect of GNP waviness on the engineering constants was also evident. Additionally, the contribution of the interphase region was found to be significant; increasing its thickness led to a decrease in CTE and an increase in elastic modulus. These findings highlight that optimizing GNP geometry, specifically volume fraction, dispersion pattern, aspect ratio, and waviness, along with precise control of interphase characteristics, is critical for enhancing the engineering performance of GNP-reinforced Al-MMNCs.

Footnotes

ORCID iDs

Reza Ansari

Saeid Sahmani

Author contributions

Maedeh Saberi: Writing—original draft, Software, Data curation, Validation, Visualization. Alireza Moradi: Writing—review and editing, Methodology, Formal analysis, Data curation. Reza Ansari: Supervision, Resources, Methodology, Investigation, Formal analysis, Conceptualization. Mohammad Kazem Hassanzadeh-Aghdam: Writing—original draft, Visualization, Project administration, Conceptualization. Saeid Sahmani: Writing—review and editing, Project administration, Investigation, Conceptualization.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

Data will be made available on request.*

References

Bae

Joo

Shin

, et al. A new strategy for developing interfacial bonding using non-metallic atoms in carbon nanotube-reinforced aluminum composites. Mater Sci Eng, A 2024; 893: 146128.

Malaki

Kasar

, et al. Advanced metal matrix nanocomposites. Metals 2019; 9(3): 330.

Gangolu

Rao

Prabhu

, et al. Effects of temperature and strain rate on compressive flow behavior of aluminum-boron carbide composites. J Compos Mater 2014; 48(11): 1313–1321.

Khalili

SMR

Daghigh

Eslami Farsani

. Mechanical behavior of basalt fiber-reinforced and basalt fiber metal laminate composites under tensile and bending loads. J Reinforc Plast Compos 2011; 30(8): 647–659.

Feng

Malingam

Ping

. Mechanical characterisation of kenaf/PALF reinforced composite-metal laminates: effects of hybridisation and weaving architectures. J Reinforc Plast Compos 2021; 40(5-6): 193–205.

Huang

Ouyang

Guo

, et al. Graphite film/aluminum laminate composites with ultrahigh thermal conductivity for thermal management applications. Mater Des 2016; 90: 508–515.

Suthar

Patel

. Processing issues, machining, and applications of aluminum metal matrix composites. Mater Manuf Process 2018; 33(5): 499–527.

Guo

Qiu

, et al. Mechanical response behavior, constitutive modeling and microstructural evolution of a 7003-T6 Al alloy rolled plate under high-speed impact loading. J Mater Res Technol 2025; 36: 2731–2746.

Liu

Fan

Tan

, et al. Reinforcement with intragranular dispersion of carbon nanotubes in aluminum matrix composites. Compos B Eng 2021; 217: 108915.

10.

Gao

Zhan

Chen

, et al. A micromechanical model of graphene-reinforced metal matrix nanocomposites with consideration of graphene orientations. Compos Sci Technol 2017; 152: 120–128.

11.

Keramati

Moradi

Ansari

, et al. Enhancing thermo-electro-mechanical properties of piezoelectric fiber-polymer composites via boron nitride nanotube addition: a predictive micromechanical model. Results Eng 2025; 28: 107289.

12.

Park

Ruoff

. Chemical methods for the production of graphenes. Nat Nanotechnol 2009; 4(4): 217–224.

13.

Kumar

HGP

Xavior

. Graphene reinforced metal matrix composite (GRMMC): a review. Procedia Eng 2014; 97: 1033–1040.

14.

Tabandeh-Khorshid

Kumar

Omrani

, et al. Synthesis, characterization, and properties of graphene reinforced metal-matrix nanocomposites. Compos B Eng 2020; 183: 107664.

15.

Bradbury

Gomon

Kollo

, et al. Hardness of multi wall carbon nanotubes reinforced aluminium matrix composites. J Alloys Compd 2014; 585: 362–367.

16.

Hassanzadeh-Aghdam

Ansari

Mahmoodi

. Thermal expanding behavior of carbon nanotube-reinforced metal matrix nanocomposites-A micromechanical modeling. J Alloys Compd 2018; 744: 637–650.

17.

Tang

Cong

Zhong

, et al. Thermal expansion of a composite of single-walled carbon nanotubes and nanocrystalline aluminum. Carbon 2004; 42(15): 3260–3262.

18.

Xiong

. Effects of graphene content on microstructures and tensile property of graphene-nanosheets/aluminum composites. J Alloys Compd 2017; 697: 31–36.

19.

Chu

Wang

, et al. Thermal properties of graphene/metal composites with aligned graphene. Mater Des 2018; 140: 85–94.

20.

Zhou

Chen

Mikulova

, et al. Thermal expansion behaviors of few-layered graphene-reinforced Al matrix composites. J Alloys Compd 2019; 792: 988–993.

21.

Bhoi

Singh

Pratap

. Developments in the aluminum metal matrix composites reinforced by micro/nano Particles–A review. J Compos Mater 2020; 54(6): 813–833.

22.

Moradi

Ansari

Hassanzadeh-Aghdam

, et al. Evaluating the role of agglomerated carbon nanotubes in the effective properties of polymer nanocomposites: an efficient micromechanics-based finite element framework. Comput Mater Sci 2025; 246: 113337.

23.

Nam

Requena

Degischer

. Thermal expansion behaviour of aluminum matrix composites with densely packed SiC particles. Compos Appl Sci Manuf 2008; 39(5): 856–865.

24.

Rezzig

Tarfaoui

Abdeslam

. The effect of interphase and agglomeration on the mechanical properties of aluminum matrix nanocomposites reinforced with carbon nanotubes. Mater Today Commun 2023; 37: 107103.

25.

Abueidda

Dalaq

Abu Al-Rub

, et al. Micromechanical finite element predictions of a reduced coefficient of thermal expansion for 3D periodic architectured interpenetrating phase composites. Compos Struct 2015; 133: 85–97.

26.

Moradi

Ansari

Hassanzadeh-Aghdam

, et al. Morphology-centric computational simulation for predicting coupled-field responses of graphene nanoplatelet/barium titanate nanowire/polydimethylsiloxane nanocomposites. Compos Struct 2025; 370: 119385.

27.

Moradi

Ansari

Hassanzadeh-Aghdam

, et al. Modeling-based mechanistic insights into the role of barium titanate shape and microstructural defects in coupled-field responses of piezoelectric nanocomposites. Compos B Eng 2025; 305: 112755.

28.

Kundalwal

Ray

. Effective properties of a novel continuous fuzzy-fiber reinforced composite using the method of cells and the finite element method. Eur J Mech Solid 2012; 36: 191–203.

29.

Zhu

Jeong

Lim

, et al. Probabilistic multiscale modeling of 3D randomly oriented and aligned wavy CNT nanocomposites and RVE size determination. Compos Struct 2018; 195: 265–275.

30.

Dong

Bhattacharyya

. A simple micromechanical approach to predict mechanical behaviour of polypropylene/organoclay nanocomposites based on representative volume element (RVE). Comput Mater Sci 2010; 49(1): 1–8.

31.

Turan

. Investigation of mechanical properties of carbonaceous (MWCNT, GNPs and C60) reinforced hot-extruded aluminum matrix composites. J Alloys Compd 2019; 788: 352–360.

32.

Mourdjeva

Karashanova

Nihtianova

, et al. Microstructural characteristics of Al4C3 phase and the interfaces in Al/graphene nanoplatelet composites and their effect on the mechanical properties. J Mater Eng Perform 2024; 33(21): 11607–11616.

33.

Zhao

Bai

, et al. The interfacial structure of Al/Al4C3 in graphene/Al composites prepared by selective laser melting: first-Principles and experimental. Mater Lett 2019; 255: 126559.

34.

Kolev

Lazarova

Petkov

, et al. Investigating the effects of graphene nanoplatelets and Al4C3 on the tribological performance of aluminum-based nanocomposites. Metals 2023; 13(5): 943.

35.

Teng

. Recent progress in graphene-reinforced aluminum matrix composites. Front Mater Sci 2021; 15(1): 79–97.

36.

Joshi

Sharma

Harsha

. Effect of waviness on the mechanical properties of carbon nanotube based composites. Phys E Low-dimens Syst Nanostruct 2011; 43(8): 1453–1460.

37.

Chou

. Failure of carbon nanotube/polymer composites and the effect of nanotube waviness. Compos Appl Sci Manuf 2009; 40(10): 1580–1586.

38.

Moradi

Ansari

Hassanzadeh-Aghdam

, et al. Finite element modeling of the effective creep compliance of carbon nanotube-polymer nanocomposites: a critical microstructure-level investigation. Mech Adv Mater Struct 2024; 31(24): 6109–6125.

39.

Yanase

Moriyama

. Effects of CNT waviness on the effective elastic responses of CNT-Reinforced polymer composites. Acta Mech 2013; 224(7): 1351–1364.

40.

Sun

Meng

Xie

, et al. Wire-based friction stir additive manufacturing enables enhanced interlayer bonding in aluminum-matrix composites. J Manuf Process 2025; 153: 1–15.

41.

Zhang

Wan

Wen

, et al. Wire-based friction stir additive manufacturing of TiC reinforced Al-Cu-Mg composite: particle refinement and dispersion. Compos Appl Sci Manuf 2025; 196: 109009.

42.

Cao

Killips

. Advances in the science and engineering of metal matrix nanocomposites: a review. Adv Eng Mater 2024; 26(20): 2400217.

43.

Zeinedini

Shokrieh

. Agglomeration phenomenon in graphene/polymer nanocomposites: reasons, roles, and remedies. Appl Phys Rev 2024; 11(4): 041301.

44.

Baig

Mamat

Mustapha

. Recent progress on the dispersion and the strengthening effect of carbon nanotubes and graphene-reinforced metal nanocomposites: a review. Crit Rev Solid State Mater Sci 2018; 43(1): 1–46.