Document Type : Research Article
Authors
^{1} Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak, Iran
^{2} Associate Professor of Mechanical Engineering, Faculty of Engineering, Arak University, Arak, 3815688349, Iran
Abstract
Keywords
Buckling Analysis of Sandwich Structures with Metamaterials Core Integrated by Graphene Nanoplatelets Reinforced Polymer Composite
^{a} Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak, 3815688349, Iran
^{b} Associate Professor of Mechanical Engineering, Faculty of Engineering, Arak University, Arak, 3815688349, Iran
KEYWORDS 

ABSTRACT 
Sandwich structures; Nanocomposites; Graphene nanoplatelets; Buckling analysis; Metamaterials; 
A study on buckling analysis of Marine sandwich panels for interior partition walls with multilayer graphene nanoplatelet (GPL)/polymer composite facesheets is presented in this paper. Three different shapes of square, honeycomb, and reentrant cellular shape with negative poison ratio are considered for the core layer. It is assumed that facesheets be composed of a polymer matrix reinforced by graphene nanoplatelet (GPL). HalpinTsai's micromechanical approach is used to determine the effective Young’s modulus of the top and bottom layers and the rule of mixture for effective Poisson’s ratio and mass density. The wall sandwich plate is modeled based on a new fifthorder shear deformation theory. Hamilton principle is employed to obtain the governing differential equations of motions of plates. The accuracy of the proposed formula and results are verified and proven accurate by the high agreement with the available results in the literature. Based on our results, we indicated the effect of cell configurations of the cellular core on the critical buckling load of marine interior wall sandwich plates. Moreover, the effect of thickness, aspect ratios, graphene nanoplatelet weight fraction, and geometrical parameters on the critical buckling load by the use of Galerkin’s method is illustrated. The findings of this research may be beneficial in creating more efficient engineering applications, especially in the marine and ship industries. 
There is a continuous demand in using sandwich structures due to the need for lowweight, highperformance structures in the industry[1, 2]. There has been a rapid increase in the use of sandwich structures in an array of applications, including ships, automobiles, aircraft, satellites, and wind energy systems[37]. The rising demand in the shipbuilding industry is because the use of sandwich structures may lead to substantially lower production costs, relatively shorter product delivery times, improved ship performance owing to a lower structural weight, and improved fatigue and corrosion characteristics [8]. In theory and experiment, it has been shown that adding even a very small amount of graphene to a pure polymer matrix improves its mechanical properties dramatically[913]. Nanofillers, such as carbon nanotubes (CNTs)[1416] and graphene platelets (GPLs)[17], have been introduced to reinforced structures for engineering purposes (i.e., keeping a relatively high stiffness while losing weight). in a new methodology for structure comparison and design, Palomba et al.[18] suggested the use of aluminum honeycomb sandwich structures (AHS) in place of common marine structures, owing to their lightweight and environmental friendliness. Honeycombs have been used successfully in sandwich construction since the middle of the 1940s[19]. A finite element method can be used for modeling and analyzing honeycomb sandwich panels, but the method requires many computations because it is difficult to mesh a whole structure[20]. In the present study, we used a method that simulates the sandwich cores with a solid orthogonal plate to simplify the modeling of honeycomb, reentrant, and rectangular core layers.
The buckling behavior of engineering structures made of the advanced materials mentioned above has been investigated by many researchers. The ChebyshevRitz method was used by Yang et al. [21] to determine the uniaxial, biaxial, and shear buckling loads of FGGPLRC plates. Kiani [22] used the TSDT to study the thermal buckling and postbuckling behavior of FGGPLRC plates. An isogeometric analysis was used to study the static buckling of porous plates FGGPLRC by Li et al. [23]. Ji and Waas [24] introduced a finite element formulation for buckling analysis of thick sandwich panels. The study by Song et al. [25, 26] examines bending and buckling analyses for FG graphenereinforced polymer plates by using the FSDT and Navier solutions. Modeling and analysis of the postbuckling behavior of graphenereinforced composite (GRC) laminated plates are presented by Shen et al. [27]. Shahverdi et al.[28] investigated the postbuckling analysis of geometrically perfect/imperfect honeycomb core sandwich panels having graphene platelet (GPL) reinforced face sheets based on a general higherorder plate model. Thermal buckling analysis of annular/circular microplates, which are made from functionally graded Graphene nanoplatelets (GNPs) reinforced porous nanocomposite is presented by Arshid et al.[29].
In this study, the buckling analysis of sandwich walls in the interior portion of ships and switchboard rooms in marine applications with a threeshape Models core, including positive, zero, and negative Poisson's ratio, surrounded by reinforced composite faces are investigated. It is assumed that facesheets be composed of an epoxy matrix reinforced by GPLs. The effect of geometric parameters of the core layer, besides the aspect ratio, lengthtothickness ratio, and corethicknesstototalthickness ratio on critical buckling load for all models in research is conducted. In this regard, the HalpinTsai micromechanical approach and rule of a mixture are used to determine the effective mechanical properties of composite layers, and a new fifthorder shear deformation theory was employed to derive the governing equation of the marine sandwich walls. The Galerkin method is used to solve the equations and the accuracy of the formulation and introduced theory is assessed by comparing available results in the literature with the calculated critical buckling loads. The findings of this research may be beneficial in creating more efficient engineering applications, especially the wall structure of ship engine control rooms (ECR) and switchboard rooms (SBR).
The ship structures panel with length a, width b, and thickness h are considered. The core layer is covered by multilayer GPL/polymer nanocomposite facesheets in the paper. The composite material has uniform distribution (UD) pattern of graphene sheets through the thickness in the facesheets of the wall panels. The facesheets with the UD pattern consist of layers of the same GPL weight fraction, as shown in Figure 1. Considering the HalpinTsai model, the effective Young's moduli of the facesheets will be defined as follows[30]:









where and are Young’s modulus of the polymer matrix, Young’s modulus of the GPLs, GPL volume fraction, and the parameters characterizing both the geometry and size of GPL nanofillers, respectively. Besides, and In formulations are the average width, thickness, and length of the GPLs. The volume fraction of GPLs is defined as:

where and are the mass densities of GPLs and polymer matrix, and is GPL weight fraction in the nanocomposite facesheets. The effective Poisson's ratio ( ) and Mass density ( ) of GPL/ polymer nanocomposite can be calculated by applying the rule of mixture:



where are the volume fraction and Poisson's ratio, respectively. The mass fraction of GNPs for UD pattern can be expressed as follows:

Three different types of wall sandwich plates are considered: rectangular core cell plates (Type A), honeycomb core plates (Type B), and reentrant core cell plates (Type C). The sandwich plate of Type A has an aluminum rectangular core cell, as shown in Figure 2. In all Types, hc is the thickness of the core, and h is the total thickness of the sandwich wall. In sandwich plate Type A, the Poisson's ratio of the core layer is equal to zero[31].


Fig. 1. Sandwich plate Type A with rectangular core cell 
The Young’s modulus (E), the shear elastic modulus (G), Poisson’s ratios(ν), and density(ρ) of the rectangular core cell can be calculated from[31]:













where , and and superscript “c” represents core material. In the figures, L1 is the length of the inclined cell rib, L2 is the length of the vertical cell rib, and t is the rib thickness. Type B sandwich plates have an aluminum honeycomb core (Figure 2) that possesses a positive Poisson's ratio. The mechanical properties of the honeycomb core are defined in Eqs. 1724.

















Fig. 2. Sandwich plate Type B with honeycomb core cell 
Figure 3 indicates wall sandwich plate Type C, where the middle layer of this type is made of metamaterial reentrant cells with negative Poisson's ratio. The mechanical properties of the core of Type C are described in Eqs. (2532):


Fig. 3. Sandwich plate Type C with reentrant core cell 















In all types, the Cartesian coordinate system (x,y,z) is assumed at the corner of the midplane of the panel.
The constitutive equations of the marine sandwich panels can be written based on Hook’s law as [32]:
,


, , , , , 
In equations 33 and 34, stress(σ) is related to the strain(ε, ) with stiffness coefficients(Q). Superscript (k) refers to the layer of the sandwich plate. k=t,c, and b denote the top layer, core layer, and bottom layer, respectively. The displacement field of wall sandwich plates based on a new fifthorder shear deformation theory is represented as:





where , and are midplane displacement in the x, y, and z directions, and are denoted for rotation of the midsurface around the x and y axes, respectively[33]. In addition, and are , .Linear straindisplacement relations are given by:









The distribution of transverse shear stress along the thickness of the structure for Trigonometric theory(TSDPT)[34], Hyperbolic theory(HSDPT)[35, 36], Parabolic theory(PSDPT)[37], and the present introduced theory is illustrated in Fig 4. The modified shear deformation theory satisfies free stress conditions at z = h/2 and the z = h/2 surfaces of the plate. The governing equations are extracted based on Hamilton's principle [38, 39]:

Fig. 4. A comparison of transverse shear stress distributions of the plate based on various theories 

where Us, T, and W refer to strain, kinetic, and work done by external nonconservative forces. Also, δ and t are the variation operator and time. The variation of strain energy of the wall sandwich plates with GPLRC facesheets is calculated as follows[40]:



where









the total variation in kinetic energy is calculated as follows [41, 42]:



where I_{1}, I_{2}, I_{3}, I_{4}, I_{5}, and I_{6} are the inertia coefficients calculated from:



where ρ is the mass density of each layer. Employing Hamilton’s principle, the equations of motion for the threelayer wall sandwich plate can be written as:









In this section for achieving the critical buckling load of Sandwich plates used on marine interior walls, the Galerkin method is implemented. The following admissible functions which satisfy the simply supported boundary conditions are used to approximate the displacement field [43].









In Eqs.(6064) , um,n, vm,n, wm,n, ζm,n, and ψm,n are unknown coefficients, m and n are the numbers of halfwaves of mode shapes in the x and y directions. By multiplying governing equations by trial functions and integrating over the domains and using equation 65, the critical buckling load can be reached.

where [K] and [M] refer to the stiffness and buckling matrix, respectively, {Γ} is the vector of unknown coefficients and is the buckling parameter for the sandwich plate in the marine structure.
In this section, numerical results are presented to examine the effect of various parameters on the critical buckling load of the GPLRC. Material properties of facesheets are considered as , , and for epoxy and , , and for GPLs. To verify the accuracy of the presented solution consider a simply supported GPLRC plate of b=0.4 (m), h/b=0.1, and a/b=1 reinforced with GPLs of , , and . For different GPL weight fraction dimensionless critical buckling (Ncr) load plates under uniaxial (β1 = 1, β2 = 0) and equal biaxial (β1 = 1, β2 = 1) compressions is presented in Table 1.
The dimensionless definition is considered as in table 1 and the percentage below each value represents the difference with the critical buckling load of pure epoxy. As you can see in Table 1, a good agreement for different is observed between the present work and the obtained results by Song et al [26].
Parametric studies have been conducted on the effects of the wall sandwich plate aspect ratio (a/b), plate thicknesstowidth ratio (H/b), and GPL weight fraction ( ) on the dimensionless critical buckling load. The effects of these parameters were investigated for three Types of ship partition wall sandwich plates: rectangular core cell plates (Type A), Honeycomb core plates (Type B), and reentrant core cell plates (Type C). The geometric parameters are chosen as follows, the lengths a=b, t1=t2, t1=0.001 m, h=0.02 m. The core of the sandwich plate is made of aluminum, and its modulus is Es = 69GPa, the shear modulus is Gs = 26GPa, Poisson's ratio is νs=0.33, and density ρs = 2700 kg/m3 and dimensionless definition of critical buckling load is assumed as . Figure 5. shows the effect of cellular parameters on dimensionless critical buckling load of wall sandwich plates with hc/H=0.6 in three cases: rectangular core cell plates (Type A), honeycomb core plates (Type B), and reentrant core cell plates (Type C).

Fig. 5. The effect of vertical to inclined cell rib length 
The results show that the lower ratio of L2/L1 decreases the dimensionless critical buckling load of the plates with honeycomb (Type B) and Auxetic (Type C) cores. On the contrary, in Type A with a rectangular core by changing the length of the vertical to inclined cell rib ratio (l2/L1) dimensionless critical buckling load increases. As shown in Figure 5, the dimensionless critical buckling load of plates Types A and C around L2/L1=3 are the same. It means it is possible to reach a similar critical buckling load by changing the core cellular shape.
In Figure 6, the effects of the aspect ratio (a/b) on the dimensionless critical buckling load of all three types are shown (γ1=2, hc/H=0.6). It is shown that with an increase in the aspect ratio (a/b), the dimensionless critical buckling load decreases. It is illustrated that in general, there aren't many differences between them but in low aspect ratios the dimensionless critical buckling load of plates with reentrant cell shape core is higher than those of the other two types, and in low ratios, it is the opposite.

Fig. 6. The effects of the aspect ratio(a/b) 
Figure 7 compares the dimensionless critical buckling load, , versus lengthtothickness ratio, a/H; which shows that with an increase in the lengthtothickness ratio (constant length and thickness decrease), the dimensionless critical buckling load increases. Figure 7 also shows that at a low lengthtothickness ratio (a/H), the critical buckling load of Type C is greater than that of Type A As the lengthtothickness ratio (a/H) increases, the critical buckling load of plates types B and C rises more than for plates types A.

Fig. 7. The effects of the lengthtothickness ratios(a/H) on the dimensionless critical buckling 
In Figure 8, the lengthtothickness ratio is assumed as h/b=0.02, , , and hc indicates the thickness of the core layer. The dimensionless critical buckling loads of all Types at the beginning of the graph are close to each other because the facesheets are more effective than the core at low hc/h ratios. In both Figures 8 and 9 according to the assumption, the dimensionless critical buckling load of panel type C is less than type A, and Type A less than Type B. the dimensionless critical buckling load of panel type B is higher than others due to the effect of the honeycomb core. It can also be seen in Figure 8 that values of the critical buckling load in uniaxial loading are higher than corresponding ones of biaxial loading which is an evident conclusion. The effect of the width and thickness of the GNPs on the critical buckling load of sandwich ship walls is illustrated in Figure 9. In figure 9, similar to figure 8, panel type B has a higher dimensionless critical buckling load than the two other types due to its core properties. The average length of GPL nanofillers lGPL is kept constant and other assumptions for the plates are similar to figure 8. Figure 9 displays that an increase in the width of GNPs leads to a rise in the values of the critical buckling load. Furthermore, the figure demonstrates that increasing the thickness of GNPs lowers the values of critical buckling load. Therefore, in order to have better reinforcing effects, it will be necessary to use graphene nanoparticles with a higher surface area and fewer monolayer graphene sheets.

Fig. 8. The effects of the core thickness to total thickness ratios(hc/H) on the dimensionless critical biaxial and uniaxial buckling load of sandwich ship walls 

Fig.9. The effects of GPL lengthtothickness and lengthtowidth ratios on the dimensionless critical buckling 
Table 1. Comparisons of dimensionless critical buckling loads of GPL/epoxy plates under uniaxial and equal biaxial compressions


Pure Epoxy 






Uniaxial compression 
Present 
35.1 
58.4 
81.8 
105.1 
128.4 
151.7 
175 


166.3% 
233% 
299.4% 
365.8% 
432.1% 
498.5% 

Song et al.[26] 
35 
58 
82 
105 
128 
152 
175 


165.7% 
234.3% 
300% 
365.7% 
434.3% 
500% 

Biaxial compression 
Present 
17.5 
29.2 
40.9 
52.5 
64.2 
75.8 
87.5 


166.8% 
233.7% 
300% 
366.8 
433.1 
500% 

Song et al.[26] 
18 
29 
41 
53 
64 
76 
88 


161.1% 
227.8% 
294.4% 
355.6% 
422.2% 
488.9% 
In this paper, the Buckling behavior of three interior marine sandwich wall structure Types with rectangular, honeycomb, and reentrant metamaterial cores was investigated based on a new fifthorder shear deformation theory. The effect of all three core shapes is established critical buckling loads as covered with multilayer graphene nanoplatelet (GPL)/polymer composite facesheets. The HalpinTsai model and the rule of the mixture were employed to achieve the effective material properties of facesheets. The Galerkin method was implemented to solve the governing equations of the sandwich walls that were derived using Hamilton’s principle. The accurate results show that the presented fifthorder shear deformation theory is efficient in predicting the Buckling of sandwich wall structures. Numerical results have shown a significant increase in the critical buckling load with an increase in the mass fraction of GNPs. With constant , decreasing the will increase the dimensionless critical buckling load, while increasing the will decrease it. Generally, the value of the dimensionless critical buckling load of sandwich wall Type B is higher than that of the other Types with different core shapes. It is possible to increase the dimensionless critical buckling load of plate Type C by increasing the length of the vertical to inclined cell rib ratio (γ1) in the core layer, while it is the opposite in the other two types. When a very small amount of GPLs is added to the polymer matrix, the critical buckling load can be increased significantly. The critical buckling load in all types is increased when the lengthtothickness ratio (a/H) is increased and it is increased at low values of the aspect ratio (a/b).
Nomenclature
a 
Length of plate 
b 
Width of plate 
H 
Total thickness 
h_{c} 
Thickness of the core layer 

GPL volume fraction 

size parameter of GPL nanofillers 

geometry parameter of GPL nanofillers 

Young’s modulus of the GPLs 

Young’s modulus of the polymer matrix 

average width of the GPLs 

average Thickness of the GPLs 

average length of the GPLs 

effective Poisson's ratio of facesheets 

Mass density 

Length ratio of cell ribs 

Thickness ratio of cell ribs 

Length to Thickness of cell ribs 

Plate inertias 

Normal strains 

Normal stresses 
U,V,W 
Displacement components 
t1, t2 
Thickness of ribs of auxetic’s cell 
L1, L2 
Length of ribs of auxetic’s cell 

Dimensionless critical buckling load 
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