Effects of size and atomic defects on the buckling of single-layer and double-layer graphene nanoplates using AFEM

Abstract

This study investigates the effect of single atomic vacancy defects with different random distributions on the buckling behavior of single- and double-layer graphene nanoplates. A parametric analysis was conducted to assess the influence of nanoplate dimensions, aspect ratio, defect concentration, number of layers, and boundary conditions on the critical buckling load (). To capture the discrete nature of graphene, the nanoplates were modeled using a space frame approach, where covalent bonds between carbon atoms were represented as beam elements and interlayer van der Waals forces were simulated using linear elastic springs. The atomic positions were generated using MATLAB and imported into ANSYS, where neighboring atoms were connected based on an energy-equivalent mapping between interatomic and elastic beam energies. Springs were added between atoms located within a defined cutoff distance to represent long-range interlayer interactions. A linear finite element buckling analysis was performed by solving an eigenvalue problem using beam and spring elements with linear elastic behavior. Model validation was achieved by comparing simulation results to data available in the literature. The results show that vacancy defects significantly reduce —up to 40% at 10% defect concentration. Furthermore, three distinct distributions of defects were examined for each concentration to explore spatial effects on buckling resistance.

Introduction

Recent advances in the study of nanoplates, which feature two-dimensional structures at the nanoscale, have attracted growing interest. Single-layer and multilayer graphene sheets are among the most important nanostructures due to their extraordinary properties and broad range of potential applications^1–4. Graphene, recognized as the thinnest and strongest material known, has become a focal point due to its outstanding mechanical, electrical and thermal characteristics. The unique properties of graphene positions it as promising candidate for numerous applications in fields such as electromechanical devices⁵, catalytic oxidation⁶, nanosensors^7–10, nanoactuators¹¹, coatings and shields with electromagnetic microwave absorption^12,13, adsorptive materials for environmental remediation¹⁴, transistors¹⁵, solar energy systems¹⁶, lithium-ion batteries¹⁷, and biomedical technologies¹⁸. As another application case, graphene has been incorporated to reinforce magnetorheological elastomer nanocomposite^19–22. Owing to its transformative potential across the electronics, energy storage, and healthcare sectors, this nanostructure has become the subject of rigorous theoretical and experimental investigation.

In the above, some applications of the graphene have been reviewed, and to have a better view on the literature review of the current research, the rest of the reviewed references are categorized and presented here in terms of their methodology including: Molecular/atomic-based methods such as molecular dynamics (MD), Continuum-based methods and atomic finite element method (AFEM).

The second category refers to molecular dynamics (MD) as an approach that has been employed by many researchers to analyze small scale behavior of different materials including graphene sheets²³. Recently, Wang and Zhang used MD simulations to evaluate the Young’s modulus and fracture toughness of bilayer graphene, showing that temperature has a minimal effect on the elastic modulus but plays a key role in determining fracture toughness²⁴. In another study, Wang et al. investigated the failure of imperfect graphene with two types of defects: Stone-Wales (S-W) defects and vacancies²⁵. Fadaei Heydari et al.²⁶ explored the impact of graphene oxide nanoparticles (GO-NP) on the mechanical and thermal properties of PU/PCL nanocomposites using MD simulations. They found that a 2% GO-NP concentration provided the best balance, enhancing thermal conductivity, heat flux, and mechanical strength without adverse effects on the other material properties.

Another category of research pertains to the continuum-based approaches in which the elasticity theories incorporated with the beam or plate theories. Behfar and Naghdabadi utilized this method to examine the vibration characteristics of multilayer graphene sheets embedded in an elastic medium²⁷. As another study, they applied an analytical approach to predict the flexural modulus of multilayer graphene sheets²⁸. Kitipornchai et al.²⁹ developed a continuum-plate model to analyze multilayer graphene sheet vibrations. Jafari et al.³⁰ explored the free vibration behavior of nanoplates using classical and various non-classical theories, including first-order strain gradient, second-order strain gradient, non-local elasticity, and implicit gradient. They employed the Navier and Galerkin methods to numerically solve the governing partial differential equations and obtained parametric responses for natural frequencies, assessing the impact of parameters such as classical and nonlocal properties, boundary conditions, and plate dimensions. Namin and Pilafkan³¹ applied nonlocal elasticity theory to analyze the vibrational behavior of defective graphene sheets, demonstrating that defects within the atomic structure reduce the natural frequencies of these sheets. Pradhan³² employed nonlocal elasticity theory to examine the influence of nanoscale effects on the critical buckling load of single-layer graphene, revealing that buckling capacity increased with nonlocal parameter values and decreased with increasing graphene sheet length. Samaei et al.³³ utilized nonlocal Mindlin plate theory to model single-layer graphene sheets, assessing their buckling behavior in an elastic medium. Zhang et al.³⁴ applied the element-free Kp-Ritz method and non-local elasticity theory to buckling analysis of single-layer graphene sheet, emphasizing the influence of nonlocal parameters and surrounding elastic support on critical buckling capacity. Omar et al.³⁵ investigated the static stability and buckling behavior of functionally graded porous nanoplates embedded in elastic media, incorporating nonlocal strain gradient theory for size effects. Based on that study, the nanoplate’s critical buckling load is influenced by parameters such as porosity distribution, boundary conditions and the stiffness of the elastic environment, offering valuable insights into nanoplate stability under different loading conditions. Arpanahi et al.³⁶ studied the buckling behavior of a nanoplate subjected to fluid flow, using Navier-Stokes equations and surface effects theory. Their findings highlighted how fluid viscosity, flow velocity, and surface energy influence the critical buckling load of nanoplates. Farajpour et al.³⁷ studied the buckling behavior of graphene sheets with various thicknesses, showing that the buckling characteristics of a single-layer graphene sheet are highly sensitive to the nonlocal parameter.

Understanding probable effects of atomic scale issues such as defects is necessary in applied viewpoint as allows prediction of their behavior and proper designing based on the realistic conditions. Regarding theoretical studying of such matters, molecular dynamics methods is a good approach however, it is computationally expensive and takes long time for simulations. On the other hand, continuum-based approaches cannot fully capture atomic-scale phenomena, such as atomic vacancies. Therefore, a methodology that has advantages of atomic-based approaches but less computational cost and time can be a favorable tactic. In this regard, a combination of atomic-based approaches and FEM has been developed that is called atomic finite element method (AFEM) in the open literature¹⁶. In this strategy, the potential functions defined in molecular mechanics are employed to obtain the physical characteristics of the elements required in FEM. Related references are put in another category and as will be reviewed hereafter, has been applied in different aspects such as elastic deformation, buckling, free and forced vibration. Liu et al.³⁹ applied AFEM to analyze the mechanical behavior of single-walled carbon nanotubes⁴⁰. The authors of the current work have previously utilized this approach to study the elastic response behavior of graphene sheets and determination of its elastic coefficients⁴¹. Gu et al. employed AFEM to investigate local buckling in bent single-walled carbon nanotube⁴². Sakhaee-Pour⁴³ conducted elastic buckling analysis and found a nonlinear relationship between buckling force and sheet width while observing no significant effect from the aspect ratio. Rouhi and Ansari, using AFEM, studied the influence of boundary conditions and the aspect ratio on buckling of single-layer graphene sheets. That study revealed that increasing the aspect ratio causes reduction of the buckling load⁴⁴. Giannopoulos⁴⁵ examined the critical buckling load of graphene nanoribbons, revealing that chirality generally affects the critical load, which increases almost linearly. Additionally, as the length of the nanoribbon increases, the critical buckling load decreases significantly in a nonlinear trend. Firouz abadi et al.⁴⁶ explored the critical buckling load of graphene sheets through molecular structural mechanics, molecular dynamics, and finite element models under different boundary conditions. Jafari et al.⁴⁷ studied the mechanical and piezoelectric properties of boron nitride nanotubes via refined AFEM as incorporating piezoelectric effect and concentrated spring elements. They observed that as the nanotube diameter increased, mechanical properties stabilized, with Young’s modulus converging from approximately 0.7 TPa at smaller diameters to about 1.1 TPa at larger diameters.

In summary, research into graphene buckling behavior remains essential, particularly regarding the impact of defects on structural stability. While various studies have examined single-layer and multi-layer graphene sheets under different conditions, limited work has comprehensively addressed impact of atomic defects on the buckling performance. This study aims to develop an enhanced AFEM model to analyze single-layer and double-layer graphene buckling in both defect-free and defective forms. The key innovation of this research is its exploration of how parameters such as the number of layers, size, aspect ratio, defect concentration and boundary conditions influence the buckling behavior—factors that have received limited attention in previous studies.

Governing equations and buckling analysis

Graphene, a two-dimensional nanomaterial, consists of carbon atoms arranged in a hexagonal lattice via covalent bonding. These bonds exhibit specific lengths and angles in three-dimensional space. When an external force is applied to a graphene nanoplate, atomic displacements are governed by these bonds, leading to an overall deformation dictated by interatomic interactions. Within this modeling framework, interatomic forces are represented as load-bearing beam elements, while atoms serve as the nodes of the structural system. (Fig. 1) illustrates the unit cell of graphene nanoplates, where each covalent bond is modeled as a structural component within the space frame, and single vacancy defects are introduced by removing one atom from selected lattice sites.

Fig. 1.

Typical space-frame model of graphene nanoplates with single vacancy defects.

Beam elements can effectively represent covalent bonds between carbon atoms. By making an analogy between beams in structural mechanics and atomic bonds in molecular mechanics, a relationship is formulated between beam stiffness parameters and potential function constants. This correlation enables the derivation of beam element properties, allowing for an accurate simulation of the mechanical behavior of graphene and other nanostructures.

The potential energy of a carbon nanostructure is mainly determined by strong covalent bonds within each individual layer and weak van der Waals forces acting between layers. The effect of long-range electrostatic forces is negligible. In this study, beam-link elements are used to represent the covalent bonds between carbon atoms (Fig. 1), while mechanical springs are employed to model the interlayer van der Waals interactions, as shown in (Fig. 2). It is important to mention that the range of long-range forces follows a finite potential function and is taken as 2.5σ, based on the findings in Ref⁴⁸. As depicted in (Fig. 2), although an atom is located near vacancies, it remains constrained through connections to several atoms of the adjacent layer.

Fig. 2.

Mechanical springs as equivalent of the out-of-plane van der Waals bonds between atoms of the layers of the two‑layer graphene nanoplate, with limitation of cutoff distance.

More detail of determining the elastic constants of the beam elements following the mentioned procedure is presented in Appendix A. The buckling solution in finite element framework is a two-step procedure consisting a step of static elastic solution under external compression load depicted in (Fig. 3), and a step of numerical solution of an eigenvalue problem.

Fig. 3.

Double-layer graphene under compression load causing buckling.

To condense the article, more detail of this procedure is presented in Appendix B however, the main equation of the second step, eigenvalue problem, is solution of the following equation^49–51:

1

In the above equation, [K] represents the global elastic stiffness matrix, is the i th eigenvalue, is the total geometric matrix, and {} is the i th displacement eigenvector. The global stiffness and geometry matrices depend on the element characteristics as explained in Appendix B. To have a non-trivial solution for the above equation, determinant of the coefficient matrix has to be zero as follows.

2

By solving this equation, the roots, , as eigenvalues are obtained; and by substitution of each eigenvalue into Eq. 1, relevant buckling mode shapes, , is obtained. It should be noted that the smallest eigenvalue is labeled and gives the critical buckling load through the following equation.

3

In the above, F represents the total external compressive force applied on the edge of the nanoplate in the static solution as the first step of the buckling analysis procedure. The critical buckling load per unit area of the graphene is determined using the following equation⁴⁹:

4

In this equation, A is equivalent cross-sectional area of the nanoplate as a rectangle that carries the external load depicted in (Fig. 3). This area is obtained via the following equation showing dependance on the nanoplate width, b, the equivalent thickness of each layer,t, (0.34 nm) and number of layers, w.

A=w.t.b 5

Conducting FEM simulations

As aforementioned, the covalent bonds within the graphene structure were modeled using the three-dimensional beam elastic element. Each node in this model possesses three translational degrees of freedom along x, y, and z axes, as well as three rotational degrees of freedom around these axes. Additionally, interlayer springs were incorporated to represent van der Waals interactions between adjacent layers. These interactions were considered only when the interatomic distance was equal to or less than the cutoff radius, denoted by σ (the Lennard-Jones parameter)^44,48. Beyond this threshold, no interaction was assumed between atoms. Once the springs were defined between atomic sites, their stiffness values were assigned based on an equivalence principle linking the elastic energy of the springs to the chemical bond energy.

To conduct the numerical simulations, a combination of ANSYS and MATLAB is employed. Regarding FEM model generation, BEAM4 as a 2-node 3D beam element is used between the atoms of each layer⁵². On the other hand, spring element COMBIN14 was used to create the van der Waals interaction between the layers⁵². To facilitate the modeling process, required MATLAB codes were developed to create the model geometry and the data then imported into ANSYS for applying boundary conditions and performing the solution. Moreover, a MATLAB script was developed to generate a command code, that is subsequently executed within ANSYS. The model represents a two-dimensional structure with length a and width b and a given thickness t, as shown in (Fig. 4) along with the coordinate system used. The modeling process begins by defining points that correspond to carbon atoms, followed by connecting these points with lines representing covalent bonds. To construct the planar structure, a single unit cell is generated and replicated along X- and Y- global axes, forming a complete two-dimensional lattice.

Fig. 4.

Applying fixed constrains under simply supported and clamped boundary conditions in buckling analysis pf the graphene using FEM.

To investigate the buckling behavior, several nanoplates with two values of and for the width and various length in range of to were studied. In fact, for each width, nanoplates with different length were generated to provide possibility to study effect of the aspect ratio, a/b. Moreover, to reveal the impact of atomic defects on the nanoplate buckling capacity, various vacancy concentration were incorporated into its geometry. Boundary condition is another key factor that is investigated in this study. As depicted in (Fig. 4), the plate has four boundaries on X=0 , b and Y=0 ,a. The boundaries on X=0 and X=b are free, while those on Y=0 and Y=a are defined as free, simply supported, or clamped, as illustrated in (Table 1). For briefness, the free, clamped and simply supported conditions are denoted by the abbreviations F, C, and S, respectively. To clarify applying essential BCs in FEM, relevant detail is presented in (Fig 4). It is seen that each boundary node of the beam element has 6 DOFs (3 translation and 3 rotation). For a simply supported boundary condition, only the translational DOFs are zero-constrained. On the other hand, a clamped boundary condition requires all six DOFs (translations and rotations) to be zero. It should be clarified that the compressive load P in all cases is applied on Y=a.

Table 1.

Boundary conditions in buckling analysis

Boundary condition type	Applied constrains
Free (F)	No constrain
Clamped (C)	All DOF=0 (All translations and rotations are fixed)
Simply supported (S)	(All translations are fixed)

Validation

To validate the proposed model for the buckling analysis of graphene nanoplates, its results have been compared with those from reference ⁵³. In addition to this section, more relevant comparisons with other references are made and presented in the subsequent sections^54–56. In this validation, a perfect single-layer nanoplate with a length of 201.68 Å and various aspect ratios was analyzed. The comparison, shown in (Fig. 5), indicates excellent agreement with the reference data, confirming the accuracy and reliability of the developed model.

Fig. 5.

Comparison and validation of the critical buckling load of a single-layer nanoplate with a length of 201.68 Å against Ref.⁵³.

To compare our results with the reference in more detail, buckling load obtained here and the reference beside their difference in percent are given in (Table 2). It is seen that the maximum difference of 13.3% belongs to the nanoplate with aspect ratio of a/b=0.5 under CC boundary conditions. On the other hand, the minimum difference of −0.5% regards to the nanoplate with aspect ratio of a/b=1.0 under CF boundary conditions. The results in (Table 2) show that the differences fall within a limited range, which directly reflects the randomness of defect distributions in the nanoplates.

Table 2.

Buckling load obtained in the current study and in Ref.⁵³; and their difference for the nanoplate with various aspect ratios under different boundary conditions.

	Boundary condition: CF			Boundary condition: CC			Boundary condition: SS
Aspect ratio [a/b]	Current	Ref.53	Difference in %	Current	Ref.53	Difference in %	Current	Ref.53	Difference in %
0.5	0.1188	0.1253	5.2	1.9013	2.1929	13.3	0.4629	0.4729	2.1
1	0.0574	0.0571	−0.5	0.9304	1.0113	8	0.2225	0.2353	5.4
2	0.0285	0.0305	6.6	0.4688	0.4863	3.6	0.1105	0.1151	4
8	0.0069	0.0070	1.4	0.1118	0.1051	−6.4	0.0272	0.0264	−3
12	0.0047	0.0053	11.3	0.0738	0.0790	6.6	0.0181	0.0200	9.5

Results and discussion

This section presents the results of the buckling analysis for both single-layer and double-layer graphene nanoplates, considering the presence and absence of defects. Nanoplates with widths of and , and varying aspect ratios, are subjected to three different boundary conditions (CF, CC, and SS) under a uniform compressive force (P) applied at one end.

Defect-free nanoplates

Figure 6 illustrates the variation of the critical buckling load per unit cross sectional area (​) for single- and double-layer graphene nanoplates with widths of b=30 Å and b=70 Å versus the aspect ratio, a/b. Pertaining to BC effects, the largest buckling load is obtained under CC boundary condition, followed by SS and CF conditions. However, the curves tend to converge as the aspect ratio increases. This suggests that the influence of boundary conditions diminishes for larger nanoplates. Another factor examined is the impact of the aspect ratio on buckling capacity, which exhibits non-linear descending trends across the three introduced boundary conditions.

Fig. 6.

Variation of the critical buckling load () per unit cross sectional area for defect free single- and double-layer graphene as a function of aspect ratio for the widths: (a) 30 Å, and (b) 70 Å.

To understand the role of the graphene width directly, ratio of the buckling load for b=70 Å over the buckling load of b=30 Å is provided in form of graphs in (Fig. 7). It is seen that for single-layer nanoplate, buckling load of the wider graphene (b=70 Å) is smaller than that belong to the narrower (b=30 Å) in almost constant ratio (0.18) for all BCs and all aspect ratios. On the other hand, for double-layer nanoplates, nonlinear trends are observed due to the influence of van der Waals bonds between the layers. The graphs show that for small nanoplates, the bonds are not strong enough to enhance the buckling capacity however, by increasing the size, their contribution enhance the nanoplate stability. As the aspect ratio increases, the influence of van der Waals forces on the overall stiffness decreases.

Fig. 7.

The ratio of versus the aspect ratio under three BCs for the widths: (a) single-layer and (b) double-layer graphene.

Defective nanoplate

To investigate the impact of defects on the buckling behavior of the nanoplate, adequate simulations were conducted with different concentrations of vacancies defects. It should be noted that the studies were done on single- and double-layer graphene with b=30 Å and b=70 Å with various values of the length, a, meaning various aspect ratios, a/b under different BCs. For each case, three different random distributions of vacancies were simulated and their average values were also calculated and reported as the result of that case.

Effects of the defect’s distribution

As a matter of fact, the vacancies can be placed at different positions and probably affecting the buckling load. The numerical results revealed that for a given percentage of atomic vacancy defects, the critical buckling load exhibits a certain degree of sensitivity to the random distribution of these defects. As examples, for two cases, ranges of the buckling load obtained for different random defects distributions are given in (Fig. 8). It is seen that the results belong to different defects distributions are not identical as some variances are exhibited between them. It seems that the variances range for the smaller graphene (b=30Å) is greater than the variance range happened for the wider graphene (b=70Å) however, no meaningful pattern was observed for the range of the variances belonged to different sizes and boundary conditions.

Fig. 8.

The average and variances of the buckling load obtained for different defect random distribution for two specific problem cases.

It should be emphasized that the positions of the atomic vacancies are distributed randomly. The program selects defect positions at random, ensuring no two defects are immediate neighbors. If a randomly picked atom has adjacent neighbors that were already selected, the choice is rejected and replaced by a new random selection. To have a wide view to the effect of the defect’s distribution on the buckling capacity under different conditions, Variance factor is defined as follows.

6

The above factor represents that how the defects distribution affects the buckling capacity. In this regard, relevant graphs are provided in (Fig. 9). It is seen that the variance range depends on the boundary conditions in agreement with what reported in Ref.⁵⁵. Comparing these graphs indicates that by increasing the size and adding the number of graphene layers, the range of all variance ranges are decreased meaning weakening the impact of defects distribution on the buckling load. From a physical perspective, this dependency arises because increasing the size and number of layers reduces the relative share of defects with respect to the total number of atoms, thereby diminishing the impact of defect positions. However, when we talk about the buckling load of the nanoplate in a specific defect concentration, it refers to the average of the three results obtained from different random defects distributions.

Fig. 9.

The range of the variance factor (%) of the buckling load for different sizes, number of layers and boundary conditions.

Effects of the defect’s concentration

The most important factor that is under study in the current research is the impact of the defect’s concentration on the buckling capacity of the nanoplate. To have a clear sense about this matter, the following dimensionless parameter, called buckling degradation factor, BDF, is defined. In this expression, the numerator represents the reduction in the critical buckling load due to the presence of the defects, relative to the ideal (perfect) nanoplate, while the denominator denotes the critical buckling load of the perfect type. For each defect concentration, the average result of the three random distribution is reported here.

7

To observe variations of the degradation factor versus different variables, adequate graphs are provided. For instance, for different values of the aspect ratio, buckling degradation factor versus the defect concentration is plotted in (Fig. 10). The left graphs belong to the narrower width of and the right ones belong to the wider width of . It is seen that the graphs corresponding to the width of b=30Å are slightly higher than those corresponding to the wider width of . The effect of atomic defects is more pronounced in smaller nanoplates than in larger ones, since smaller plates contain fewer atoms, making each defect contribute more significantly to overall instability. Regarding effect of the boundary conditions, comparing the graphs reveals that under SS boundary condition, the degradation is slightly less than what happened under CC condition however, this difference is not noticeable. Another conclusion of these graphs is about their trends that are very close to the molecular dynamics results reported in Ref.⁵⁶. It is seen that for single-layer graphene, they almost obey bilinear pattern. It is seen that in almost all cases, the slope of the graphs decreases at a defect level of 6% however, for the narrower type with , this reduction is more pronounced. Nevertheless, for double-layer nanostructure, the graphs almost vary linearly. These graphs also provide possibility to compare single- and double-layer types. Accordingly, these data reveal that buckling degradation factor in double-layer type is generally less than overall degradation of single-layer type. The long-range van der Waals bonds between atoms of adjacent layers connect the atoms surrounding each vacancy with atoms of the neighboring layer, thereby influencing buckling resistance. Therefore, these bonds can compensate the degradation effect of the vacancies in double-layer nanoplates.

Fig. 10.

Variations of the buckling degradation factor for single-and double-layer nanoplates versus the defect concentration for different aspect ratios and boundary conditions.

It is seen that the graphs relevant to different aspect ratios in single-layer are close together whilst, there are some deviations between them in double-layer types. The observed differences stem from the greater number of potential defect locations in double-layer structures, which alters the buckling response across different aspect ratios.

To have a quantitative sense about the impact of defects on the buckling capacity of the graphene, a few cases are introduced here. For instance, in single-layer type, 6% vacancy concentration leads to an approximate value of around 26~28% decrease in the buckling capacity and for 10% vacancy, around 37~39% degradation is happened. On the other hand, for double-layer types, 6% defect cause around 18~25% degradation and 10% defect brings around 27~37% degradation in the buckling capacity. These values demonstrate how atomic vacancies affect the buckling capacity of nanoplates.

Buckling load of defective nanoplates versus the size (aspect ratio)

To observe direct role of the nanoplate size, or in other term, the aspect ratio, on the buckling load of the defective single- and double-layer nanoplates, adequate graphs are given in Figs. 11 and 12, respectively. It is revealed that all graphs, belong to both narrow and wide nanoplates, follow similar trends so that by increasing the aspect ratio the buckling load is decreased nonlinearly. However, it is seen that for both single- and double-layer types, under both CC and SS boundary conditions and for all aspect ratios and defect’s concentration, the buckling load (per unit cross section area) of the narrow nanoplate, , is larger than those obtained for the wider ones, . Increasing the width raises the likelihood of local buckling, which reduces the overall buckling capacity of the structure. Comparing the graphs relevant to single- and double-layer types indicates that the buckling load per unit cross section area of the double-layer structure is larger than the familiar values obtained for single-layer ones.

Fig. 11.

Variations of the buckling load of defective single-layer nanoplate for sizes: b=30 Å and b=70 Å, containing various defect concentrations under CC and SS boundary conditions.

Fig. 12.

Variations of the buckling load of defective double-layer nanoplate for sizes: b=30 Å and b=70 Å, containing various defect concentrations and under CC and SS boundary conditions.

As a matter of fact, by increasing number of layers, thickness of the nanoplate and its cross-section modulus increases and in classical viewpoint, larger buckling capacity is expected. To reveal the role of this factor clearly, a buckling load ratio as is defined and relevant graphs are provided, for b=30 Å and b=70 Å and presented in (Fig. 13). It is seen that for both widths, the buckling load capacity per the cross-section area of 2-layer nanoplate is larger than 1-layer as the introduced ratio is more than one. It is interesting that for wider nanoplate, the introduced dimensionless ratio is larger than what happened for narrower one. It is observed that the graphs show ascending trend but do not follow a monotonic trend. Actually, the variations of the introduced quantity in terms of the aspect ratio are much more gradual, for the narrower specimen compared to the graphs related to the wider nanoplates. Based on the numerical results, in the narrow graphene (nanoribbon) (width of ), even at an aspect ratio of 10, the graphs are still changing, this suggests that for narrow ribbons, the effects of the length remain significant, requiring larger aspect ratio for convergence of the graph. On the other hand, in the wider plate (width of ), the graphs converge to a stable value once the aspect ratio reaches approximately 5. This indicates that in wider ribbons, the influence of the ribbon’s length becomes negligible more readily. It is noticed that the final convergence values are dependent on the edge boundary conditions. In this regard, under CC boundary conditions, convergence occurred at a value of around 7, while under SS boundary conditions, convergence occurred at a value of around 27.” This behavior is attributed to the size-dependent elastic modulus of nanoplates at small scales⁴¹. In other words, by increasing the size, the Young’s modulus approaches a constant value asymptotically, and therefore the nanoplates approach a classical column. It is of particular interest that under CC boundary conditions, the ratio is significantly greater than the corresponding ratio under SS boundary conditions, as also reported in Ref⁵⁷.

Fig. 13.

Variations of the ratio versus the nanoplate aspect ratio for sizes: b=30 Å and b=70 Å, with various defects concentrations, under CC and SS boundary conditions.

buckling mode shapes

Here, buckling was analyzed as an eigenvalue problem, and the smallest eigenvalue was reported as the critical buckling load. However, observing the deformed geometry provides a better understanding of the buckling state. Hence, images of the first three buckling modes for a few cases are presented and discussed as follows. In this regard, the three first buckling mode shapes of single- and double-layer graphene nanoplate with , aspect ratio of , under CC and SS boundary conditions, in two cases of defect-free and 10% vacancy defects, are presented in Figs. 14,15,16,17. It is notable that in both types of boundary conditions and in both the defect-free and defective types of graphene, their main (first) buckling modes are completely similar. However, the shapes of the second and third buckling modes are interchanged between the two types of boundary conditions, SS and CC. This result stems from the fact that SS boundary condition allows greater movement in the boundary regions; therefore, the shape of the second buckling mode is such that larger displacements occur in the upper and lower regions. In contrast, under CC boundary condition, the shape of the second mode is such that the displacement in the regions near the boundary is more restricted. Simultaneously, the images show that in both the defect-free and 10% defect, global buckling have occurred in all buckling modes. These findings suggest that global buckling remains the primary failure mode, preceding local buckling near vacancies even in defective graphene. This situation is due to the fact that, here, the vacancies are distributed across all regions, preventing accumulation of defects in a region. However, it is possible that if a major concentration of imperfections let to occur in one region, local buckling would occur there faster than global buckling. This matter can be the subject of further complementary research in this field.

Fig. 14.

The first three buckling modes of defect-free single layer graphene nanoplate (SLGNP) under SS and CC boundary conditions, for aspect ratio ( and ).

Fig. 15.

The first three buckling modes of single layer graphene nanoplate (SLGNP) containing 10% defect, under SS and CC boundary conditions, for aspect ratio ( and ).

Fig. 16.

The first three buckling modes of defect-free single layer graphene nanoplate (2LGNP) under SS and CC boundary conditions, for aspect ratio ( and ).

Fig. 17.

The first three buckling modes of double layer graphene nanoplate (2LGNP) containing 10% defects under SS and CC boundary conditions for aspect ratio ( and ).

Conclusion

One of the strengths and novelties of the present study is that the effects of various parameters on the buckling behavior of graphene have been investigated simultaneously. This approach makes it possible to examine the interaction effects among different factors. Such a comprehensive study has not been observed in previously published research. In this regard, buckling behavior of single- and double-layer rectangular nanoplates was analyzed under different types of boundary conditions, different sizes and different aspect ratios. In addition, effect of the vacancy defects was also studied. Nevertheless, in order to provide further details of the results, the most important findings are summarized below together with their numerical values.

1. Size dependence: Based on the obtained results, in addition to the length, width of the nanoplate affects the buckling capacity per unit cross section area, so that for 70 Å-width, the buckling capacity is less than the capacity obtained for 30 Å-width. As the sheet width increases, the probability of local buckling increases, leading to a reduction in the buckling load capacity of graphene.

2. Effect of the aspect ratio: Under the studied boundary conditions, at low values of the aspect ratio, dependency of the buckling load to the aspect ratio is considerable however, the dependency becomes weaker as the aspect ratio increases. Moreover, almost the same functionality was observed for different defect’s concentration and different sizes.

3. Effect of the boundary conditions: Among the boundary conditions, CC yields much more buckling load than SS type that is similar to what happened in classical structures. It was observed that as the size and the aspect ratio increase, the influence of the boundary conditions reduces. From a physical perspective, it is important to emphasize that as the size increases, the proportion of boundary edges relative to the entire domain decreases, leading to reduction in the influence of the boundary conditions.

4. Effect of the defects: To identify this dependency, a dimensionless factor called buckling degradation factor (BDF), was introduced. In all cases, the presence of defects reduces the buckling load so that with 10% vacancy defect, a degradation around 35% is happened. However, reduction of the single-layer nanoplates capacity is more significant than double-layer nanoplates. Moreover, the impact of defects in smaller nanoplates is more noticeable compared to the larger ones. From a physical viewpoint, this dependency arises because larger graphene sheets provide more space for defect distribution, which reduces the likelihood of defect clustering.

5. Effect of the number of layers: To explore effects of the number of layers on the buckling load, this dependency was investigated quantitatively. In this regard, buckling capacity ratio was defined and its dependencies to different factors were studied. The numerical results demonstrate that the buckling capacity of the double-layer nanoplate is consistently greater than that of the single-layer; however, the buckling capacity ratio is also affected by other parameters. Specifically, it was observed that in narrow graphene, the double-layer exhibits a superior capacity compared to the single-layer, and this ratio increases with the aspect ratio (length-to-width). On the other hand, for the wider nanoplate, by increasing the aspect ratio, the buckling capacity ratio grows more significantly. Interestingly, as the dimensions and aspect ratio increase, the capacity ratio of the double-layer to the single-layer graphene asymptotically approaches a constant value. This behavior reflects the size effect, whereby the mechanical response of the nanostructure progressively converges toward classical continuum behavior as the size increases.

6. Future work: In this paper, we presented a part of our research regarding graphene buckling, but it must be acknowledged that there is still room for research development in this field. For example, certain assumptions were considered here which can be changed in future research developments according to the (Table 3).

Table 3.

Assumptions of the current study and suggested directions for future work

Assumptions in the current study	Suggestions for future work
Defect type: vacancy	Other defect types
Defect distribution: random and no accumulation	To let accumulation
Number of layers: 2	More number of layers
layer interactions: van der Waals	Other interaction types.
Boundary conditions: two sides are supported and two sides are free	Boundary conditions: all sides are supported.
Cutoff radius: a fixed value	Different values of the cutoff radius.

List of symbols

a

Nanoplate length []

A

Nanoplate equivalent cross-sectional area []

b

Nanoplate width []

d

Cross-sectional area of the circular beam []

E

Beam elastic modulus []

F

Total external compressive force [N]

G

Beam shear modulus []

I

Second moment of area []

J

Polar second moment of area []

K

Global elastic stiffness matrix []

Total geometric matrix []

Axial spring stiffness []

Van der Waals constant []

Bending spring stiffness []

Torsional spring stiffness []

L

Beam length [nm]

Critical buckling load [N/ ]

Critical buckling load of the perfect nanoplate [N/]

Critical buckling load of the defective nanoplate [N/]

r

Distance between adjacent atoms []

Δ r

Changes of distance between adjacent atoms []

t

Equivalent thickness of nanoplate layer []

u

Nodal displacement vector [nm]

Bond energy related to out-of-plane twisting [J]

Bond energy related to in-plane twisting [J]

Bond energy related to bending [J]

Bond energy related to axial deformation [J]

Strain energy under pure bending [J]

Strain energies related to stretching [J]

Strain energy under torsion [J]

Total elastic potential energy [J]

Van der Waals energy [J]

w

Number of layers [-]

Greek symbols

\varepsilon

Lennard-Jones parameters [N.nm]

σ

Cut of distance [nm]

i Th eigenvalue [-]

i Th displacement eigenvector [-]

Smallest eigenvalue [-]

Changes in bond angle [rad]

Changes in dihedral angle [rad]

Subscripts and superscripts

i, J, K

Unit vector along the x, y, and z axes

x, y, z

X, y, and z axes in a three-dimensional space

CF

Clamped-Free boundary condition

CC

Clamped-Clamped boundary condition

SS

Simply supported-Simply supported boundary condition

BC

Boundary condition

Appendix A: Elements elastic constants determination

The total static potential energy of a carbon nanostructure, excluding electrostatic interactions, is the sum of the energies resulting from both bonding and non-bonding interactions, as given by the following expression²³:

1A

where , , , and ​ represent the bond energies associated with bond stretching, bond angular bending, in-plane angular twisting, and out-of-plane twisting, respectively. ​ denotes the non-bonding van der Waals energy, which is often neglected in covalent systems. A schematic of the interatomic interactions corresponding to Eq. (1A) is shown in (Fig. 18).

Fig. 18.

Interatomic interactions at the molecular level⁵³.

Different functional forms are available for these energy terms, depending on the material and loading conditions. However, including all these terms complicates the model and the solution process. To simplify, Lee and Chou²³ adopted the simplest harmonic form and combined the in-plane angular twisting and out-of-plane twisting into a single equivalent term. Accordingly, the potential energies due to interactions between carbon atoms with covalent bonds are expressed as follows²³:

2A

3A

4A

In these equations, , and ​ represent elastic constants in axial deformation, bending, and torsion, respectively. , and denote the elastic changes in bond length, bond angle, and dihedral angle from the equilibrium state.

According to structural mechanics, the deformation of a space frame is the result of the change in strain energy due to tension, bending, and torsion. The various loadings that produce these energies are shown in (Fig. 18). If the elastic modulus, shear modulus, length, cross-sectional area, moment of inertia, and polar moment of inertia of the beam are , , , , , and , then its strain energy under pure tension is^23,39,58,59:

5A

Additionally, the strain energy under pure bending moment is expressed as follows:

6A

The strain energy resulting from the torsion of the beam under a torque is given by:

7A

As shown, and ​ represent tensile energy, and ​ represent bending energy, and and ​ represent torsional energy. By comparing the energy terms in molecular mechanics with the corresponding terms in structural mechanics, it is reasonable to assume that corresponds to , the 2α rotation is equivalent to the change in bond angle , and corresponds to . Therefore, by comparing these two sets of equations, a direct relationship is established between the structural mechanics parameters , , and and the molecular mechanics parameters , , and ​​. These relationships form the foundation for applying structural mechanics theory to model graphene nanoplates and similar structures²³:

8A

To determine the properties and constants of the beam elements representing covalent bonds between carbon atoms, it is assumed that the cross-sectional area of these bonds is uniform and circular. In this case, , and ​, with only the parameters , and needing to be determined. By substituting these relationships into Eqs. (8A), the properties of the beam elements are derived²³.

9A

By substituting the values of , , and from (Table 4), the elastic properties of the beam elements are determined as follows²³:

10A

In multilayer structures, in addition to the bonding terms, there is also potential energy resulting from weak van der Waals interactions (the last term in Eq. (1A)). This attraction is a relatively weak force that exists between two adjacent layers, enabling the layers to slide over one another. Van der Waals interactions are pair potentials between particles, and the Lennard-Jones potential is commonly used to represent these pair potentials, which is given by the following expression³⁸

11A

In Eq. (11A), represents the distance between atoms in the bonding pair, while σ and \upvarepsilon are the Lennard-Jones parameters, with values provided in (Table 4). To model these interactions, linear spring elements are used. The stiffness of these elements is determined by calculating the second derivative of the potential function (Eq. 11A) with respect to , as follows⁴⁸:

12A

Table 4.

Force constants, bond lengths, and Lennard-Jones parameters⁴⁴.

Value	Parameter
	\upvarepsilon
	σ

Appendix B: Elements stiffness matrices determination

The geometry and nodal forces of a beam element is depicted in (Fig. 19). Each node of the beam has three translation and three rotation degree of freedom and relevant force and moments depicted in (Fig. 19).

Fig. 19.

Two-node beam element indicating nodal force and moments.

Finite element solution of buckling problem consists of two parts: static solution and eigenvalue problem. The static solution of the structure under static external loading is obtained via the following equation. Solution of this equation provides nodal displacements and internal nodal forces, required for the second step as eigen value problem.

1B

In the above, is the global stiffness matrix, is the nodal displacement vector, and is the global vector of external force. The global matrices are obtained via assembly of the element’s matrices introduced as follows.

2B

Moreover, the vector of nodal forces and moments are as follows.

3B

The global stiffness matrix, is assembly of the beam’s elements stiffness and interlayer Van der Waals springs. The beams element’s stiffness matrices is a combination of four sub-matrices as follows Refs^44,60.

4B

The components of the above sub-matrices are related to the beam element geometry and its elastic properties in the following form⁴⁴.

5B

6B

7B

In addition to the beams, interlayer Van der Waals bonding also contribute to the overall stiffness, represented in form of spring element with the following stiffness matrix⁶⁰.

8B

The sub-matrices in the above equation are determined via the following equations.

9B

10B

The sub- matrix is a 3× 3 null matrix and components are related to the equivalent Van der Waals stiffness as follows.

11B

As mentioned previously, the last part of the buckling solution is an eigenvalue problem with the following governing equation.

12B

In the above equation, is the global stiffness matrix introduced before in the step of static solution, is global geometry matrix, is ith component of eigenvalues vector and is ith eigenvector. Global geometry matrix is obtained via assembly of the geometry matrices of the beams and springs elements. The geometry matrix of the elements is related to the data obtained in the step of static solution. For the beam elements, the geometry matrix is obtained as follows which is a combination of four sub-matrices.

13B

the sub-matrices in the above are determined as follows.

14B

15B

16B

In addition to the beam elements, springs are existed between the nodes of the layers with the following geometry matrix.

17B

in the above equation is static force in the spring obtained in the step of static solution; and is distance between end nodes of the spring. Furthermore, the above sub-matrices are determined as follows.

18B

19B

In the above, is a 3× 3 null matrix and is as follows.

20B

Author contributions

Author contributions (1) Karim Aliakbari: Investigation, Writing - Original Draft, Writing - Review & Editing, Visualization, Conceptualization. (2) Akbar Jafari: Investigation, Methodology, Resources, Supervision, Project administration. (3) Mina Amiri: Software, Validation, Formal analysis, Data Curation, Methodology.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability

All data generated or analysed during this study are included in this published article.

Declarations

Competing interest

The authors declare no competing interests.