Nonlinear Post-Buckling Stability of Graphene Origami-Enabled Auxetic Metamaterials Plates

1  Introduction

Auxetic metamaterials increase in the lateral direction when elongated axially, contract laterally, and compressed axially, thereby resulting in a negative Poisson’s ratio. Auxetic word is originally from the Greek word “auxetos” that means “tends to expand”. It was first defined by Evans and colleagues in 1991 [1]. Auxetic metamaterials have lower densities, reduction in material usage, high shear resistance, high energy absorption capacity, good acoustic behavior, and crashworthiness [2]. These merits make them very favorable in many applications such as aerospace, biomedical scaffold, and defense areas [3,4].

Wen et al. [5] proved the microstructural hierarchy of pristine chemically modified graphene films renders remarkable negative Poisson’s ratios ranging from −0.25 to −0.55. Dutta et al. [6] developed single cell auxetic beams with different geometries using a finite element (FE) approach. Meng et al. [7] investigated the self-folding of 2D graphene into 3D graphene Miura origami helped by hydrogenation and identified the origin of their deviation. Zhao et al. [8] conducted bending tests and numerical simulations to study the static bending behavior of the sandwich auxetic honeycomb beams. Wu et al. [9] presented a review article for metallic metamaterials focusing on design process, fabrication and property testing and classification. Fang et al. [10] presented a comprehensive review of bi/multi-stable phenomenon of composite origami metamaterial structures. Liu et al. [11] presented yield area, yield criterion, and loading history as design tools of plasticity in modifying buckling load and sequence in metamaterials. Jiang et al. [12] estimated crushing response of 3D reentrant circular auxetic metamaterials under quasi-static loads. Lu et al. [13] and Wang et al. [14] studied the quasi-static bending response and energy absorption of sandwich beam with auxetic core. Francisco et al. [15] examined the vibration and bending of sandwich auxetic tubes. They noticed that the vertical length of cell has effects on natural frequencies and buckling loads. Wu et al. [16] exploited the gradient-free method to get the topology optimization of metamaterial microstructures for negative Poisson’s ratio under large deformation. Lv et al. [17] explored analytically mechanical and thermal post-buckling of FG-GOEMA auxetic metamaterials plates. Du et al. [18] developed a new approach for the layout design of cuts in kirigami metamaterials attain desired deformation patterns under buckling and post-buckling behaviors. Li et al. [19] studied the nonlinear vibration of the sandwich beam with auxetic honeycomb core under thermal shock. Vineyard et al. [20] employed the asymptotic homogenization approach to obtain the effective thermoelastic properties of the multiphase metamaterials.

Response of a structure regarding its load bearing capacity may fall into two general categories: stable or unstable. Buckling and post-buckling are unstable responses of thin structures those must be understood comprehensively by researchers. The long history of buckling theory for structures begins with the studies by Euler in 1744 of the stability of flexible compressed beams [21]. In 1993, Wang et al. [22] studied the buckling response of rectangular Mindlin plates with internal line supports. Shariyat et al. [23] investigated uniaxial and biaxial postbuckling behaviors of the longitudinally graded plates by using the exact 3D elasticity theory. Wehmeyer et al. [24] presented reduced-order numerical solutions to describe nonlinear buckling of elastic struts inclined relative to the loading direction. Eltaher et al. [25] illustrated the influence of periodic/nonperiodic imperfections modes on buckling and postbuckling of beam with nonlinear elastic foundations. Mao et al. [26] examined buckling and postbuckling of FG piezoelectric plate under the effect of electrical and mechanical loads. Mehrabi et al. [27] examined post buckling of the composite eccentric annular sector auxetic metamaterial plate under a thermal load. Derveni et al. [28] explored buckling response of plate-lattice architectures under uniaxial compression. Moradi-Dastjerdi et al. [29] examined the biaxial buckling of an active sandwich plate. Liu et al. [30] studied nonlinear vibration response of auxetic honeycomb plates via Galerkin and multiple scale methods. Abbaspour et al. [31] predicted the thermal buckling of sandwich microshell with piezoelectric layers. Long et al. [32] studied the nonlinear stability of porous FG spherical caps and circular plates. Yan et al. [33] presented a comprehensive overview of recent articles related to snapping metamaterials and their design strategies.

Based on refined plate theory, Ezzati et al. [34] investigated free vibration GOAM plate rested on Winkler–Pasternak elastic substrate analytically. Mahinzare et al. [35] proposed DQM to examine the nonlinear vibration of GOAM sandwich pates numerically. Mohamed et al. [36] developed a novel mathematical model for nonlinear buckling, postbuckling, and snap-through of bidirectional FG porous plates rested on elastic foundation. He et al. [37] investigated buckling and postbuckling of anisogrid lattice-core sandwich plates with nanocomposite skins. Milazzo et al. [38] exploited Ritz method to study buckling and postbuckling responses of variable stiffness plates with cutouts. Vaghefi [39] used meshless approach to analyze the elastoplastic postbuckling of porous FG plates with elastic foundations. Dai et al. [40] exploited the Floquet exponent method to study dynamic stability characteristics of rotating truncated conical shells under periodic axial loads. Nasri et al. [41] studied nonlinear bending and buckling stability of curved metamaterial beam and optimized the results by using the response surface methodology.

Post-buckling behavior of metamaterial structures has received limited attention from researchers. Shen et al. [42] illustrated the effect of in-plane negative Poisson’s ratio on the post-buckling response of graphene-reinforced metal matrix composite plates. An et al. [43] studied bending and buckling response of FG-GOAMs irregular plates using generalized finite difference method. Murari et al. [44,45] investigated nonlinear vibration and post-buckling responses of GOAMs tapered beams in fluid by using physics-embedded machine learning model. Ghasemi et al. [46] presented the small/large amplitude vibration of the pre/post-buckled geometrically imperfect re-entrant auxetic tube modeled by von-Kármán nonlinear strain and Timoshenko beam theory. Based on the first order shear deformation and the two-step perturbation method, Lv et al. [17] examined the mechanical and thermal post-buckling of FG-GOAMs. Sengar et al. [47] presented stability of FG skew sandwich plates having temperature-dependent properties and a re-entrant auxetic core. Ghasemi et al. [48] investigated analytically nonlinear postbuckling and static bending behaviors of 3D auxetic tubes.

To our knowledge, this study presents the first application of the numerical differential quadrature method (DQM) to investigate the nonlinear post-buckling behavior of FG auxetic metamaterial higher-order plates. The primary objective is to determine the optimal combination of graphene content, folding degree, and distribution patterns to enhance the postbuckling performance of FG-GOAM plates. In the following sections, the problem formulation will be presented by including the high shear deformation theory, von Karman nonlinear strain, and micromechanical models used to evaluate material properties of the FG-GOAM. DQM, linearization and novel iterative approaches used in the solution of developed models are discussed through Section 3. Model validation with previous respectable works is proved through Section 4. Numerical parametric studies to present the effects of graphene content, folding degree and GO distribution patterns on the postbuckling responses of GOAM plates are discussed in Section 5. Observations and concluding remarks are summarized in Section 6.

2  Problem Formulation

2.1 Mechanical Properties of Metamaterials

The considered GOAM plate consists of an even number of layers arranged symmetrically about its mid surface. Each layer is composed of cupper matrix enforced by graphene origami (GO) material. The GO content is constant within each layer but it changes in different layers according to a specified distribution of GO weight fraction ().

Three GO distribution patterns are considered [44,45]:

(1)

U-type where the same GO content in all layers, X-type where higher GO content at the top and bottom layers and O-type where higher GO content near the mid surface. Where is the average GO volume fraction in the layer that is related to the weight fraction by [44]:

(2)

meanwhile and are the densities of graphene and cupper, respectively.

These types are schematically represented in Fig. 1 where the darker color stands for a higher GO content. The plate has length , width and thickness . The plate is described in a rectangular coordinate system in which the x- and y-axes lie in the mid-plane (z = 0) while the z-axis is perpendicular to the mid-plane.

Figure 1: Schematic of considered three GO distribution types of GOAM plates

The composite Young’s modulus and Poisson’s ration of GOAM at temperature are evaluated by genetic micromechanical model [49] as:

(3)

where , , and are the length and thickness of graphene, respectively, and is the atom coverage quantify the folding degree of GO.

2.2 Kinematic and Constitutive Equations

The displacement field of four variables shear deformation theory without shear correction is portrayed by [50–53]:

(4)

where the von Karman strains are:

(5a)

(5b)

According to Reddy theorem, and . Based on equivalent single layer theory, that lowers 3D to 2D shear deformation theory , the stress–strain relationships for the layer can be characterized by:

(6a)

where plane stress stiffnesses at layer are:

(6b)

and and are computed by Eq. (3).

2.3 Nonlinear Governing Equation

Nonlinear governing equilibrium equations of composite-GOAM plates in terms of stress resultants are drew by the virtual work principle as follows:

(7a)

(7b)

(7c)

(7d)

where the nonlinear term and the in-plane force terms are defined as:

(8a)

(8b)

The present work considers composite GOAM plates consisting of symmetrically arranged layers in the thickness direction (Eq. (1)). Accordingly, the stress resultants can be expressed by:

(9a)

(9b)

(9c)

(9d)

where are defined in Eq. (5). The rigidity terms are obtained as [54]:

(10a)

(10b)

The present study investigates the responses of rectangular GOAM plate with different boundary conditions (BCs). Each edge of the plate is symbolized by: C, S, or F, to denote, respectively: clamped, simply supported, or free edge with the BCs defined in Table 1.

3  Solution Methodology

3.1 DQM Discretization of Partial Differential Equations

Several research [25,28,29,32] demonstrated that the DQM can effectively solve systems of differential equations with a relatively small number of grid points, which leads to less computational cost. Furthermore, DQM provides matrix-based operators to approximate the solution’s derivatives. This transforms the original system of differential equations into a system of algebraic equations, which can be readily solved.

Based on DQM [55], a function with domain is discretized such that the domain and function are represented as vectors: and , respectively. Additionally, the approximation of its first derivative is computed as , where is the weighting matrix of the first order derivative. Higher order derivatives are derived using matrix multiplication: .

The DQM is developed here to discretize the governing set of partial differential equations (Eq. (7)). The plate domain is discretized in the x- and y-directions using - and -nodes, respectively. The discrete values of an unknown function is generally arranged as a two-dimensional array but can be rearranged as a -vector constructed from the array vectors one after another such that . According to this vector representation of the unknowns, partial derivative and are approximated by vectors and , respectively, such that

(11)

where is the identity matrix of dimension , are the first order derivative matrices of single variable functions in and , respectively, and indicates Kronecker product. The resulting partial derivative matrices and have dimensions . Approximation of higher and mixed partial derivatives such as , are computed as and , respectively, where .

The governing equations for the GOAM plate, see Eqs. (5), (7) and (9), consist of four nonlinear partial differential equations in the unknowns and . They are discretized by DQM as the unknown vectors each of dimension and arranged as global unknown vector .

3.2 Linearization and Iterative Approach

Since the governing nonlinear set of partial differential equations for the GOAM plate is homogeneous (without any external transversal force), it is a nonlinear eigenvalue problem. Investigation of the plate response is based on critical buckling load and postbuckling paths that describe the relation between the applied axial load and buckling deflection. In the present study, the axial load intensity is plotted vs. the maximum buckling deflection of the plate.

A simple linearization approach is adopted to solve Eq. (7). The nonlinear terms in the strain , Eq. (5a), and the nonlinear function , Eq. (8a), are discretized by DQM and linearized such that

(12)

where is an approximate -vector. Through an iterative procedure, is computed from the previous iteration. According to the above data structure and using the notation: and applying the DQM, the linearized stress resultants can be written as:

(13)

where an overbar indicate a linearized stress, is the zero square matrix of dimension and operator ‘’ is defined for a vector and matrix , each with the same number of rows, such that implies that .

Now, the linearized discrete governing algebraic system for Eq. (7) is written in an iteration step ‘’ as a linear generalized eigenvalue problem

(14)

where and are presented in Appendix A. Upon application of the boundary conditions to Eq. (14), the resulting linear eigenvalue problem has to be solved for the unknown deflection vector as well as the compressive load . The iteration procedure is described in the following Algorithm 1 in which the maximum of a buckling deflection is computed as

4  Model Validation

For GOAM plates, the material properties of matrix Cu and graphene are presented in Table 2 while the geometric parameters of graphene are

First, an isotropic plate is considered. Table 3 compares the values of dimensionless critical buckling loads of a square plate under uniaxial and biaxial compression assuming simply supported boundary conditions and Poisson’s ratio v = 0.3. Different solution methods are compared: DQM [26], analytical solution based on Mindlin plate theory [22] and the two-step perturbation method [17]. The results demonstrate good consistency for different thickness to length ratios.

Comparisons with Prakash et al. [56] and Mohamed et al. [36] of the dimensionless postbuckling equilibrium paths of a square isotropic plate are depicted in Fig. 2. The dimensionless deflection is plotted vs. dimensionless axial pressure . The material properties are: GPa. Good agreement is observed especially with those of [36,56].

Figure 2: Comparison with [36,56] for postbuckling paths of simply supported isotropicsquare plate under uniaxial compression.

Next, to validate the present formulation and solution methodology of GOAM plates, critical buckling loads of square GOAM plates under biaxial compression are compared with [17]. For different GO folding degrees and weight fractions under three distribution patterns, the critical buckling loads are compared in Table 4. It is clear that the current research results are in good agreement with those of [17].

The postbuckling response of an X-type GOAM plate under biaxial loading is compared with Lv et al. [17] in Fig. 3 showing good agreement.

Figure 3: Comparison with Lv et al. [17] The postbuckling response of the X-type GOAM plates under biaxial loading. .

5  Numerical Parametric Studies

5.1 Influence of Graphene Origami Parameters on the Material Mechanical Properties

Fig. 4 illustrates the mutual effect of graphene origami contents ( and folding degrees parameter on material mechanical characteristics of the studied GOAM plate: As observed from Fig. 4a, Young’s modulus of GO/Cu composite material is generally increased with the increase of and/or reduction of the folding degree . However the rate of this increase is not uniform but depends on the values of both and For example, for small graphene origami content , has no significant change while it increases significantly for higher values of especially for smaller folding degree values. Fig. 4b illustrates that increasing either of or reduces the Poisson’s ratio, which can reach negative values leading the GO/Cu composite to be transformed into an auxetic metamaterial. The influence of and on the plane stress stiffnesses , defined in Eq. (6b), are shown in Fig. 4c–e, respectively.

Figure 4: Influence of Graphene Origami parameters on the mechanical properties of a single layer

5.2 Influence of GO Volume Fraction Distribution Types

Herein, the effect of GO distribution patterns defined in Eq. (1) on the postbuckling response of GOAM plates are examined. Fig. 5 presents the postbuckling curves of simply supported GOAM plates under biaxial compression. The plate is composed of 10 layers and arranged according to different distribution types. Compared with other distribution types of GO volume fraction, the GOAM plate with X-distribution type exhibits the greatest critical buckling load and best postbuckling performance, followed by the U-type, then the O-type GOAM plate. This performance demonstrates that dispersing more GO near the surfaces of the plate is an effective way to achieve good structural performance of the metamaterial plate.

Figure 5: Influence of GO volume fraction distribution types on postbuckling behavior of a SSSS GOAM plate under biaxial compression

5.3 Influence of Graphene Folding Degrees

Fig. 6 highlights the influences of graphene folding degree on the dimensionless buckling and postbuckling behavior of FG-GOAM composite plates, assuming constant GO weight fraction . A simply supported square GOAM plate under uniaxial and biaxial compression is considered. Fig. 6 demonstrates that the buckling loads in case of uniaxial compression are double those computed in case of biaxial loading. Furthermore, the figure reveals that increasing results in decreasing the postbuckling loads. This behavior is understood since the stiffness of GOAM composite decreases with increasing due to the reduction of Young’s modulus, Poisson’s ration, and stiffness coefficient as can be observed from Fig. 4. In general, increasing folding degree gives rise to the transformation of GO/Cu composite into GOAM with NPR and reduced Young’s modulus, making the metamaterial structure more flexible.

Figure 6: Influence of GO folding degrees on postbuckling behavior of GOAM plate under (a) uniaxial and (b) biaxial compression for X-distribution type

5.4 Influence of Graphene Weight Fraction

Fig. 7 plots the postbuckling paths of X-type GOAM simply supported square plate under uniaxial and biaxial compression for different values of graphene-origami weight fraction . In the current parametric study, the folding degree is assumed constant . As can be seen from Fig. 7, for large graphene origami content , a rise in GO content results in increased postbuckling loads. However, this characteristic is violated for GOAM plates when GO weight fraction increases in the range where the buckling loads change slowly and may even reduce. This can be interpreted since, as can be noticed from Fig. 4a,b in the range the increased graphene content leads to a larger Young’s modulus and smaller negative Poisson’s ratio (or greater ). This results in larger values of . Accordingly, increasing graphene-origami weight fraction in the range leads to increased plate stiffness and higher postbuckling loads. On the other side, in the smaller range , the GO/Cu composite cannot show NPR features. The increased graphene content leads to a larger Young’s modulus but smaller (positive) Poisson’s ratio, giving rise to the nearly constant or even decreased value of .

Figure 7: Influence of Graphene weight fraction W_Gr on postbuckling behavior of GOAM plate under (a) uniaxial and (b) biaxial compression for X-distribution type

5.5 Mutual Influence of GO Properties

In order to investigate the mutual influence of Cu\GO parameters and on postbuckling behavior of GOAM plate, a simply supported U-type square plate under biaxial compression is considered. The postbuckling paths for different and are presented in Fig. 8. As can be observed, at , the postbuckling paths are identical and as expected are independent on the value . Another observation is driven form Fig. 8a–c for that the postbuckling loads considerably increase with increasing the content. The rate of this increase reduces as the folding degrees increase. Alternative behavior is observed in Fig. 8d for . The postbuckling loads decrease with increasing from up to but they increase as increases from up to . To understand these responses and theoretically investigate mutual influences of and on postbuckling behavior, the mechanical properties of the Cu/GO composite are computed based on Eq. (3) and plotted in Fig. 9a,b. Fig. 9a indicates that Young’s modulus increases with the increase of but it decreases with the increase of . With respect to Poisson’s ratio, Fig. 9b shows that increases with increasing , but it decreases as increases. The most important is to notice that becomes negative at some specific values of and . For example, in the considered range , is always positive in cases of , however it reaches negative values in cases of . Since is a reasonable measure of plate stiffness, we plot and in Fig. 9c,d, respectively. It is clear that, is a critical value that can reverse the performance of GOAM plates. Fig. 9d indicates that the plate stiffness increases as increases in the cases of . However, in case of , the plate stiffness decreases in the range of , where but it increases as becomes negative. These theoretical conclusions are consistent with the numerical results shown in Fig. 7 and interpret mutual influence of Cu\GO parameters and on postbuckling behavior of GOAM plates.

Figure 8: Mutual influence of GO properties on postbuckling behavior of GOAM plate under biaxial compression for U-distribution type

Figure 9: Mutual influence of GO parameters on mechanical properties of GOAM plate (U-distribution type)

As can be observed from Figs. 5–8, the postbuckling paths are of the pitchfork bifurcation type, which is characterized by zero lateral deflection whenever the axial load is less than a critical value . The dimensionless critical buckling load corresponds to in these figures. Table 5 reports values of of simply supported GOAM square plates under uni/bi-axial compression for different values of and different FG-distribution types.

Results in Table 5 demonstrate that, for both of uniaxial and biaxial loading, for all material distribution types, and for , increasing the GO folding degree decreases the critical buckling load. Influence of GO content on critical buckling load depends on the value of . At the critical buckling increases as increases. However, at higher folding degrees the critical buckling reduces then rises as increases.

5.6 Influence of GOAM Plate Thickness

To study influence of GOAM plate thickness on postbuckling behavior, a different normalization of the axial load is defined as to exclude from the previous normalization. A simply supported GOAM plate under biaxial compression is considered. Fig. 10 presents the postbuckling paths of plates with different thicknesses assuming two different GO distribution types. As the figure reveals, the postbuckling loads considerably increase with the increase of plate thickness. Moreover, X-type GOAM plate exhibits better postbuckling performance compared with the U-type. Table 6 reports the normalized buckling loads of simply supported (SSSS) and fully clamped (CCCC) X-type GOAM plates with different thickness and different GO parameters .

Figure 10: Influence of plate thickness on postbuckling behavior of GOAM plate under biaxial compression for GO distribution patterns: (a) U-type and (b) X-type

5.7 Influence of Boundary Conditions

In almost all previous results, a simply supported GOAM plates was considered. Only Table 6 presents comparison of first critical buckling loads of SSSS and CCCC corresponding to different values of: ratio, and . In the current subsection, the first four critical buckling loads with the associated buckling modes of GOAM plates are presented in Figs. 11–13 for different boundary conditions. In these figures, U-type GOAM plates are considered and the critical buckling loads are normalized by . The first four modes entitled by the values of corresponding normalized critical buckling load are shown for: (1) simply supported GOAM plate under biaxial loading in Fig. 11, (2) CCCC plate under uniaxial compression in Fig. 12, and (3) a CFCF plate with two opposite sides clamped and other two opposite sides free under uniaxial loading in Fig. 13.

Figure 11: First four linear mode shapes of U-type GOAM SSSS plate under biaxial loading

Figure 12: First four linear mode shapes of U-type GOAM CCCC plate under uniaxial loading

Figure 13: First four linear mode shapes of U-type GOAM CFCF plate under uniaxial loading

6  Conclusions

This paper studies the characteristics of critical-buckling, and nonlinear post-buckling behavior of multi-layer GOAM plate, subjected to uniaxial or biaxial compressive loads. The mechanical properties, within each layer, are modeled as a function of graphene origami content, and the folding degree. Three symmetric distributions of GOAM (i.e., U-pattern, X-pattern, and O-pattern) are considered in the analysis. Based on four variables of nonlinear high shear deformation theory, the virtual work principle is implemented to derive the governing equations assets of nonlinear eigenvalue problems. DQM is applied to discretize the governing equations, and the resulting algebraic nonlinear eigenvalue problem is solved iteratively. Theoretical and numerical analyses present the effects of graphene origami content, degree of folding, and distribution pattern on the post-buckling response of GOMA plates. The main remarks from the following analysis can be summarized as follows:

•   Increasing the folding degree increases the transformation of GO/Cu composite into GOAM with NPR, and reduces Young’s modulus, thus making the metamaterial structure more flexible.

•   For different folding degrees , a rise in graphene origami content results in increasing Young’s modulus and decreasing Poisson’s ratio.

•   For a smaller range of the folding degrees, where the composite Poisson’s ratio is still positive (), the critical and post-buckling loads of the plate with any GO distributions are increased by increasing . In contrast, as the folding degree increases, the negative Poisson’s ratio is reached and the GOAM plate exhibits a counter-intuitive buckling behavior.

•   Buckling loads under uniaxial compression are twice those calculated for biaxial loading.

•   The GOAM plate with an X-distribution type demonstrates the greatest resistance to buckling and the most favorable post-buckling behavior.

•   Thicker plates demonstrate the higher-load-bearing capacity

•   The following model is limited to flat plates without curvature and imperfections.

Acknowledgement: Not applicable.

Funding Statement: The authors received no specific funding for this study.

Author Contributions: Salwa A. Mohamed: Validation, Supervision, Resources, Methodology, Investigation, Formal analysis. Mohamed A. Eltaher: Conceptualization, Validation, Investigation, Formal analysis. Nazira Mohamed: Resources, Conceptualization, Investigation, Formal analysis. Rasha Abo-bakr: Visualization, Validation, Investigation, Formal analysis. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: Data can be provided according to reasonable requirements.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest to report regarding the present study.

Appendix A

(A1)

(A2)

(A3)

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