Journal of Materials Science & Technology  2019 , 35 (6): 1147-1152 https://doi.org/10.1016/j.jmst.2018.12.012

Finite element analysis of the effect of interlayer on interfacial stress transfer in layered graphene nanocomposites

C.C. Roach, Y.C. Lu*

Department of Mechanical Engineering, University of Kentucky, Lexington, KY, 40506-0503, USA

Corresponding authors:   * Corresponding author.E-mail address: ycharles.lu@uky.edu (Y.C. Lu).

Received: 2018-03-19

Revised:  2018-05-24

Accepted:  2018-10-10

Online:  2019-06-20

Copyright:  2019 Editorial board of Journal of Materials Science & Technology Copyright reserved, Editorial board of Journal of Materials Science & Technology

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Abstract

Understanding the roles of interlayers in reinforcement efficiencies by layered graphene is very important in order to produce strong and light graphene based nanocomposites. The present paper uses the finite element method to evaluate the interfacial strain transfers and reinforcement efficiencies in layered graphene-polymer composites. Results indicate that the presence of compliant interlayers in layered graphene plays significant roles in the transfers of strain/stress from matrix to graphene and subsequently the reinforcement effectiveness of layered graphene. In general, the magnitude of shear strain transferred onto the rigid graphene decreases as the thickness of the interlayer increases. This trend becomes insignificant as the graphene becomes sufficiently large (s>25,000). The shear strain at the interface of graphene-matrix is also greatly influenced by the interlayer modulus. A stiffer interlayer would result in a higher shear strain transferred on the graphene. The performance of the interlayers is further affected by the property of the composite and the architecture of the layered graphene stack. If a composite contains more graphene phase, the efficiency of reinforcement by a layered graphene becomes improved. If a graphene stack contains more interlayers, the effectiveness of reinforcement at the edges of the graphene becomes negatively affected.

Keywords: Interlayer ; Layered graphene ; Nanocomposites ; Interfacial stress transfer ; Finite element method

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C.C. Roach, Y.C. Lu. Finite element analysis of the effect of interlayer on interfacial stress transfer in layered graphene nanocomposites[J]. Journal of Materials Science & Technology, 2019, 35(6): 1147-1152 https://doi.org/10.1016/j.jmst.2018.12.012

1. Introduction

Due to their exceptional mechanical properties and unique geometric characteristics, the two-dimensional (2D), layered graphene (also known as graphene flakes or graphene sheets) have been increasingly used as reinforcement fillers for fabricating high-performance nanocomposites. However, there is one important issue that has yet to be fully explored: the roles of interlayers embedded in those layered graphene (see Fig. 1 in Reference [1] for SEM image and model of a layered graphene). The interlayers that lie between adjacent graphene layers are rather complicated materials: either the chemicals and/or solvents that are used to intercalate the graphene, or the polymer chains that penetrate into the spaces between graphene during the preparations of composites, or a mixture of both [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. Due to the complicity of the material composition, the mechanical properties of the interlayers are often unknown or difficult to characterize. In addition, the presence of weak van der Waals force between graphene layers may further complicate the properties at the interlayers [12,13]. The other unknown is about the sizes of the interlayers (spacing between adjacent graphene layers), which could vary drastically due to the processing conditions used to intercalate the graphene sheets and/or to fabricate the composites. We have previously studied the effects of interlayers on elastic moduli of layered graphene nanocomposites by using combined Arridge’s lamellar model and Tandon-Weng composite model [1]. Results show that the properties of the interlayer in layered graphene stacks have noticeable impact on elastic properties of the nanocomposites, particularly the out-of-plane properties (E2 and G12). Further, as the size of the interlayer increases, all elastic properties of the composites have been greatly weakened.

Fig. 1.   Finite element model for a composite with a three-layered graphene stack: (a) the shell element model of a layered graphene nanocomposite, (b) the cross-sectional view of the shell element.

The present work is to investigate the effects of interlayers on interfacial stress transfer in layered graphene composites. The knowledge on the transfer of stress from the soft matrix to a reinforcement fiber and subsequently the variation of stress along the fiber is critical in the design of a composite. Historically, the transfer of stress (or strain) between a matrix and a reinforcement fiber has been modeled by the well-known shear-lag model. The original shear-lag model, proposed by Cox [14], was composed of a single, one-dimensional fiber embedded in an elastic matrix. The model has been subsequently improved by considering the axisymmetric geometry of fiber-matrix system [15], [16], [17], [18], [19]. The shear-lag model was further extended to model the two-dimensional planar fillers [20,21]. Recently, a new shear-lag model has been developed by Young et al. for analyzing the stress/strain transfers in the monolayer graphene polymer composites [22]. The analytical predictions have been verified by the experimental observations conducted by Raman spectroscopy [23,24].

Unlike the monolayer graphene, a layered graphene is composed of rigid graphene layers and soft, mysterious interlayers sandwiched between adjacent graphene. Such a structure represents a much complicated problem for analyzing the transfer of stress/strain between matrix and graphene, which is beyond the scope of analytical shear-lag model. The finite element method (FEM) was used in the present work to model the layered graphene embedded in a polymer matrix. Composites with different interlayer properties (modulus, Poisson’s ratio) and interlayer spacing were modeled and the shear stress/strain distributions at the interfaces of graphene/matrix were analyzed. The effect of the interlayers was further investigated by considering volume fraction and architecture of the layered graphene in the composites.

2. Finite element model and validation

2.1. Finite element model

Composites reinforced with layered graphene were modeled by finite element method using commercial program ANSYS. Since the composite with a layered graphene was essentially a plate-like structure, the mid-plane of such structure was extracted to represent the composite. The mid-plane was subsequently modeled with the continuum shell elements (S4) (Fig. 1). The detailed structure of the composite was defined by using the “shell lay-up” commend in ANSYS.

The properties of pure graphene layer were well established, with the modulus and Poisson’s ratio of 1050 GPa and 0.19, respectively [2], [3], [4], [5], [6]. The commonly used epoxy resin was chosen as the matrix material, with the modulus and Poisson’s ratio of 2.1 GPa and 0.35, respectively [25]. The properties of the interlayer in layered graphene were comprehensively studied by varying parameters such as interlayer thickness (ti), interlayer modulus (Ei), and interlayer Poisson’s ratio (νi). The effects of interlayers were further investigated by varying the aspect ratio of layered graphene flakes and the volume fraction of layered graphene in the composites. Table 1 details the properties of the interlayers used in finite element analysis.

Table 1   Summary of the properties of layered graphene used in the finite element modeling.

Interlayer thickness, ti (nm)0.34, 1, 2
Interlayer modulus, Ei (GPa)0.01, 0.1, 1, 2.1, 5, 10
Interlayer Passion’s ratio, νi0.2, 0.3, 0.4, 0.49
Numbers of layers in graphene sheet1, 3, 5, 7, 9
Volume fraction of layered graphene, Vg (%)0.25, 0.5, 1, 2, 5, 10
Aspect ratio of layered graphene sheet, L/t74, 198, 333, 740, 2613, 7002, 11764

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2.2. Finite element model validation

To validate the finite element model, a composite with a monolayer graphene (with no presence of interlayers) was first modeled and the result was compared with the one obtained from analytical shear-lag model proposed by Young et al [22]. Fig. 2 shows a model of a monolayer graphene sheet imbedded within a polymer resin. It is assumed that the forces due to the shear stress at the interface (τi) are balanced by the force due to the applied axial stress (σ), such that [22]

τidx=-tdσ or dx=- τit(1)

Fig. 2.   Sketch showing the model of monolayer graphene within a polymer resin used to drive the shear-lag model.

Solving Eq. (1) eventually led to the distributions of shear stress (τi) and shear strain (εi) across the graphene (x) as follows [22]

τi=nEgεm$\frac{sinh(ns\frac{x}{L})}{cosh{(\frac{ns}{2}})}$ (2a)

εim[1-$\frac{sinh(ns\frac{x}{L})}{cosh{(\frac{ns}{2}})}$] (2b)

where εi is the strain on the graphene, εm is the strain in the matrix, x is the position across the graphene, L is the length of the graphene and t is the thickness of the graphene. The constant n is computed by n=2GmEg(tT), where Eg is the Young’s modulus for the graphene, Gm is the shear modulus for the matrix, and Tis the thickness of the matrix.

It is noticed that the shear stress (τi) and shear strain (εi), defined in Eqs. 2(a) and 2(b), can be readily converted from each other. Therefore, either of these two quantities may be used for evaluating the performance of the reinforcements in the composites. Recently, experimental techniques have been developed to measure the transfers of the stresses or strain at the interfaces of graphene-polymer matrices [23,24,[26], [27], [28]. In those experiments, a monolayer graphene was placed onto the top of a thin polymer plate. The plate was then stressed and the resultant axial deformation on the graphene could be directly monitored by using 2D Raman spectroscopy, which led to the estimations of the axial strain (εi) on the graphene. For this reason, the interfacial shear strain (εi), rather than the interfacial shear stress (τi), is often reported in the experiments [22], [23], [24]. Following the same practice, the present paper also reports the shear strain (εi) for evaluating the transfers of stress in layered graphene polymer nanocomposites.

In the monolayer graphene composite, the graphene had a length of 3400 nm and a thickness of 0.34 nm, which corresponded to a length-to-thickness aspect ratio s=L/t = 10,000. The matrix material used was epoxy, with properties defined earlier. The graphene volume fraction in the composite was 1%. In the finite element model (Fig. 1), the matrix was strained to 0.2% and the strain distributions across the graphene were computed. Fig. 3 shows the profiles of εi on the monolayer graphene obtained from both finite element method and shear-lag model. It is observed that the results from the two methods are fairly consistent, which essentially validates the model and procedures used in the finite element method.

Fig. 3.   Comparison of interfacial shear strain in the graphene-epoxy composite computed from the finite element method and the shear-lag model.

3. Results and discussion

3.1. Effects of interlayer thickness

Unlike a monolayer graphene, the layered graphene is made of alternating layers of monolayer graphene and interlayers. The geometry and properties of a monolayer graphene have been extensively studied and the nominal thickness and Young’s modulus of the graphene are known as 0.34 nm and of 1050 GPa, respectively [2], [3], [4], [5], [6]. In contrast, the size and properties of the interlayer may vary drastically. It is often concluded that the minimum spacing between adjacent graphene layers has to be greater than the thickness of the graphene itself, i.e., t ≥ 0.34 nm [22]. During the process of graphene intercalation and/or the process of composite fabrication, the solvents and/or the polymer coils will penetrate into the spaces. Therefore, the spacing between graphene layers will likely expand, by up to several nanometers. With these considerations, the spacing of the interlayer was varied at 0.34 nm, 1 nm, and 2 nm in the present study.

Fig. 4 shows the effect of interlayer thickness on the interfacial shear strain of a 3-layer graphene stack (i.e., graphene/interlayer/graphene) in a polymer matrix. The graphene volume fraction was 5% and the modulus of the interlayer was 0.01 GPa. The graphene was assumed to be a square-shaped flake with the edge of 3400 nm. For layered graphene, the calculation for the length-to-thickness aspect ratio would have to take into account of the interlayer thickness (ti). For example, for the 3-layer graphene stack, the aspect ratio is computed as: s=L/(t+ti+t). So, with the interlayer spacing of 0.34 nm, 1 nm, and 2 nm, the corresponding aspect ratios for the layered graphene are: s = 3333, 1,984, 740, respectively. It is observed that the presence of “soft” interlayer in the layered graphene has had significant influence on the interfacial shear strain. As the composite is subjected to a uniaxial strain of 0.2%, the maximum strain on the rigid graphene is less than 0.15%. This indicates that the elastic stress applied on the composite is not fully transferred to the rigid graphene layer and thus the graphene has not fully reached its full reinforcement capacity. The results also show that the size (thickness) of the interlayer greatly affects the interfacial shear strain. As the thickness increases (from 0.34 nm to 2 nm), the maximum shear strain on the graphene decreases and the overall effectiveness of the graphene (the width of horizontal portion of εi) is reduced.

Fig. 4.   Effect of interlayer thickness on interfacial shear strain transfer. The graphene volume fraction is 5% and the interlayer modulus is 0.01 GPa. The graphene length L = 3400 nm.

Fig. 5 shows the interfacial shear strain for layered graphene at two distinct dimensions: L = 340 nm and L = 12,000 nm. Depending upon the size of the interlayers (0.34 nm, 1 nm, 2 nm), the corresponding length-to-thickness aspect ratios for these two layered graphene are: s = 333, 198, 74 and s = 11,764, 7,002, 2,613, respectively. At the smaller aspect ratios (s<350), the effectiveness of the layered graphene is completely diminished: under a 0.2% applied strain, the shear strain at the graphene is only about 0.002%, approximately 1/100th of the applied strain. The shear strain is seen to further decrease as the thickness of the interlayer between graphene increases (from 0.34 nm to 2 nm). However, at larger aspect ratios (s>2500), the layered graphene is seen to be highly effective: the elastic strain (0.2%) applied at the matrix has been fully transferred to the graphene stack. Furthermore, as the graphene becomes sufficiently large, the magnitude of the interfacial shear strain is seen to be independent upon the interlayer thickness. Those findings are consistent with the experimental results reported by Gong et al [23,24]. They performed the shear-lag analysis on monolayer graphene nanocomposites and concluded that lateral dimensions of the graphene sheet have a significant influence on the reinforcement efficiency of the graphene [23].

Fig. 5.   Effect of interlayer size on interfacial shear strain transfer. The graphene volume fraction is 5% and the interlayer modulus is 0.01 GPa. (a) L = 340 nm and (b) L = 12,000 nm.

3.2. Effects of interlayer properties

Layered graphene (with alternating layers of graphene and interlayer) has drawn great interests due to its wide range of applications [12,13,[29], [30], [31], [32], [33]. However, the precise characterization for the properties of the interlayers in layered graphene has remained to be challenging. Studies show that the shear modulus of the interlayer in the raw multilayered graphene is typically in the range of 0.19 GPa to 0.34 GPa (due to primarily the presence of van de Waals force) [13]. When used for composite applications, the layered graphene sheets or flakes would have to be treated with solvents or dispensed into polymer matrices. Consequently, the spacing between neighboring graphene layers is expanded and the solvents and/or polymer coils may penetrate into those spaces, which can significantly change the properties of the interlayers. In this study, the modulus of interlayer was varied from 0.01 GPa (corresponding to the properties of solvents or soft polymers) to 10 GPa (corresponding to the properties of stiff polymer matrices).

Fig. 6 and 7 show the distributions of interfacial shear strain of a 3-layer graphene stack with various interlayer modulus. The aspect ratio of the graphene was 3333 and the graphene volume fraction in the composite was 5%. It is seen that as the interlayer modulus increases, the shear stain at the interface of graphene-matrix increases (Fig. 7). When the interlayer modulus is low, the interfacial shear strain on the graphene only reaches 0.1%, approximately 50% of the applied strain (0.2%). As the interlayer modulus is greater than $\widetilde{2}$ GPa, the graphene stack is seen to reach its full potential: the elastic strain applied at the composite has been fully transferred to the rigid graphene.

Fig. 6.   Effect of interlayer modulus on interfacial shear strain transfer. The graphene volume fraction is 5% and the graphene aspect ratio of 10,000. (a) The interface thickness is 0.34 nm and (b) the interface thickness is 2 nm.

Fig. 7.   Relation of interfacial shear strain and interlayer modulus.

Poisson’s ratio is another important interlayer characteristic that can affect the overall composite strength [13]. Fig. 8 shows the effect of interlayer Poisson’s ratio on interfacial shear strain in graphene-polymer composites. The interlayer Poisson’s ratio was varied as 0.2, 0.3 and 0.49. The thicknesses of the interlayer was 0.34 nm and the volume fraction was 5%. The graphene length-to-thickness aspect ratio was s = 3333. Overall, the Poisson’s ratio of the interlayer is seen to have a negligible effect on the shear strain at the interface of graphene-matrix.

Fig. 8.   Effect of interlayer Poisson’s ratio on interfacial shear strain transfer. The graphene volume fraction is 5% and the graphene aspect ratio is 3333.

3.3. Effect of graphene volume fraction

The mechanical properties of a composite are highly dependent upon the volume fraction of the reinforcement phase. For monolayer graphene based composites, due to the exceptionally high aspect ratios of the graphene, the modulus of the composite may be approximated by the “rule-of-mixture” model, E=EgVg+EmVm, that was developed primarily for continuous, long fiber composites [25]. Based on such simple model, the modulus (E) of the composite would be linearly dependent upon the fraction of the reinforcement phase (Vg).

The interfacial stress transfer or the reinforcement efficiency of a 2D graphene is also related to the fraction of the graphene phase in the composite. As illustrated in the simple shear-lag model (Equation 2), the shear strain (εi) transferred on the graphene is related to the parameter of “t/T”, (t is the thickness of the graphene and T is the thickness of the matrix), which is essentially a direct measure of graphene volume fraction (Vg). The effect of graphene volume fraction on interlayer performance was evaluated through a series of finite element models, in which the volumetric fraction of the layered graphene in the composites was varied, from 0.25% to 10% (Table 1). The graphene stack was kept as the same three-layer structure (graphene/interlayer/graphene) and the aspect ratio was set as 3333. The interlayer had a thickness of 0.34 nm and a modulus of 2.1 GPa. As shown in Fig. 9, the reinforcement efficiency of the graphene, i.e., the width of the horizontal portion in εg, has been affected by the content of graphene phase. As the fraction of the matrix phase increases, the effectiveness of the graphene reinforcement decreases, even though the properties of the interlayer are the same for all composites.

Fig. 9.   Effect of graphene volume fraction on interfacial shear strain transfer of graphene-matrix composites. The graphene volume fraction is 5% and the graphene length is 3400 nm.

3.4. Effect of number of layers

Results shown in previous sections have demonstrated the effects of interlayer spacing and interlayer material characteristics on interfacial shear strain transfer in graphene-matrix composites. For those studies, it was mostly assumed that the graphene stack is a three-layer structure: graphene/interlayer/graphene. In practice, the graphene can be at various architectures: monolayer, bi-layers, tri-layers, and few-layers. So, one critical fact about the interlayer is the number of layers (or interlayers) presented in a graphene stack.

Fig. 10 shows the effects of the number of interlayers on the shear strain at graphene-matrix interface. Overall, the effect is seen to occur mostly at the edges of the graphene sheets. As the number of the interlayer in a graphene stack increases, the shear strain at the edges of the graphene decreases, from 0.14% to almost 0%. The result is in agreement with the force balance in the shear-lag model. For a multi-layer graphene stack, as shown in the insert in Fig. 10, the original force balance equation (Eq. (1)) becomes:

τidx=-( Nt+Mt1) dσ or dx=- τiNt+Mti(3)

where t and ti are the thickness of the graphene and interlayer, respectively. N and M are the numbers of graphene layer and interlayer presented in a graphene stack. As the numbers (N, M) of the graphene/interlayer decreases, the shear stress (τi) increases. The magnitude of τi reaches the maximum for a ultra-thin, single-layer graphene sheet, where N = 1 and M = 0 and thus Eq. (3) reduces to Eq. (1). The extremely high shear stress occurred in a monolayer graphene can sometimes be transmitted into the normal stress on the graphene itself, especially at the edges, a phenomenon known as the “edge effect” [34,35]. The present study shows that as the number of interlayer increases, the edge effect in a layered graphene may be weakened or even completely diminished. However, away from edges, all graphene exhibit similar reinforcement efficiency regardless of the number of layers (interlayers) in the graphene sheets.

Fig. 10.   Effect of the number of interlayer in a graphene stack on interfacial shear strain transfer. The graphene volume fraction is 5% and the graphene length is 3333. The inset shows the force balance in a multi-layered graphene stack.

4. Conclusion

The analytical shear-lag theory is acceptable for predicting the reinforcement efficiency of monolayer graphene nanocomposites but not applicable to composite with layered graphene. The finite element method has been used to model the layered graphene nanocomposites and to compute the interfacial strain distributions and reinforcement efficiencies of the layered graphene. Results indicate that the presence of “soft” interlayers in the layered graphene has had significant influence on the transfers of strain/stress from matrix to graphene. At small length-to-thickness aspect ratios (s<350), the magnitude of shear strain on the graphene decreases as the thickness of the interlayer increases. As the graphene becomes sufficiently large (s>2500), the magnitude of the interfacial shear strain is no longer affected by the interlayer thickness and the layered graphene becomes highly effective regardless the interlayer properties. FEM results also show that the shear strain at the interface of graphene-matrix increases with the increase of interlayer modulus, but is not affected by the values of interlayer Poisson’s ratio. Further, the performance of the interlayers is also affected by the overall volumetric fraction of the graphene in the composites and by the architecture of the layered graphene. As the volumetric fraction of graphene decreases, the effectiveness of the layered graphene becomes inversely affected, even though the properties and geometry of the interlayer remain the same. As the layered graphene changes from monolayer to few-layers, it will lose the effectiveness in reinforcement at its edges.

Acknowledgments

The material is based upon the work supported by NASA Kentucky under NASA award No.: NNX15AR69H.

The authors have declared that no competing interests exist.


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