Improved Meshfree Moving-Kriging Formulation for Free Vibration Analysis of FGM-FGCNTRC Sandwich Shells

1  Introduction

Many new materials have been found that have contributed to developments in numerous areas of science and engineering. Functionally graded materials (FGMs) are among those proposed, and many studies have been carried out on them. An FGM is a type of composite material created by mixing two distinct material phases, such as ceramic and metal. As a result, an FGM inherits the characteristics of both material components. For example, an FGM can possess important properties, like good strength, good thermal or electrical insulating properties, or high resistance to fatigue or corrosion. Consequently, such materials are now widely used in semiconductor technologies, aerospace engineering, medical applications, nuclear reactors, etc. Numerous studies have therefore been performed to investigate the behaviors of FGM structures such as beams [1], plates [2–4], and shells [5]. Another modern material- carbon nanotubes (CNTs), which are known as “materials for the 21st century” [6], have attracted considerable attention from researchers in many fields because of their notable mechanical, thermal, and electrical characteristics [7,8]. The discovery of CNTs [9] opened a new area of materials science. Practically, CNTs are very suitable for replacing conventional reinforcements like glass fibers, steel, etc., in composite structures. This is the case because CNTs possess outstanding properties such as low density but with high stiffness, strength, and aspect ratio. A typical study of CNT structures can be found in [10]. In this study, free vibration and buckling behaviors of carbon nanotube-reinforced cross-ply laminated composite plates were obtained by using the method of discrete singular convolution and a first-order shear deformation theory. In structural engineering, sandwich structures are constructed using layers of different materials, and the resulting structures fully inherit the characteristics of each constituent material. This is very useful for improving structural performance, and many types of sandwich beams, plates, and shells have therefore been proposed and investigated. In this article, FGMs and functionally graded carbon nanotube-reinforced composite (FGCNTRC) materials were used to create sandwich shells. The proposed sandwich structure consists of skins of FGM layers and an FGCNTRC core. As mentioned above, this structure possesses all the advantages of both the FGM and the FGCNTRC material. In real-world applications, shells are widely used in mechanical, civil, and aerospace engineering. Some notable applications include the roofs of buildings and stadiums, and shells of ships, cars, submarines, airplanes, and spacecrafts. Thus, the analysis and design of FGM-FGCNTRC sandwich shells are necessary and practical. Another type of FG structure-reinforced with functionally graded graphene platelet has attracted great attention due to its outstanding advantages of being lightweight but with a high load-carrying capacity [11]. In these structures, the graphene platelets are used as reinforcements to enhance the structural stiffness. Results for the vibrations of shells reinforced with graphene platelets considering non-linearity and external load can be found in [12,13]. Results for the thermomechanical free-vibration buckling of FG graphene-reinforced doubly-curved sandwich shells can be found in [14]. Additional studies of functionally graded graphene platelet-reinforced structures should be carried out in the future.

Analytical approaches are only appropriate for problems with simple geometries, boundaries, and loads. Due to these limitations, many numerical methods have been developed for structural analyses, such as the finite element method (FEM) [15], meshfree methods [16–19], isogeometric analysis [20–22], the smoothed FEM [23], etc. In addition, many FEM-based software programs have been successfully developed, such as Abaqus, Sap2000, Ansys, etc. Many meshfree methods have also been proposed, such as the element-free Galerkin method [16], the reproducing kernel particle method [24], the meshless local Petrov-Galerkin method [25], etc. Most of the meshfree methods have a similar limitation that their interpolation functions do not satisfy the Kronecker-delta property. Essential boundary conditions, thus, can not be straightforwardly imposed as in the FEM. Accordingly, several correction techniques for boundaries have been proposed and developed, including Lagrange multipliers [26], penalty methods [27], or a combination with the FEM [28]. Interestingly, the moving-Kriging (MK) interpolation function naturally satisfies the Kronecker-delta property. The meshfree MK method was first proposed to solve the one-dimensional steady-state heat conduction problem [29]. Further developments of this method can be found in [30] for one- and two-dimensional elasticity problems, thin plates [31], shells [32], piezoelectric structures [33], and laminated composite structures [34]. As shown in [29], the accuracy of the MK solutions strongly depends on the quality of the MK interpolation, which is affected by the choice of the correlation parameter θ. However, it is very difficult to find an optimal value for θ. Consequently, an improved meshfree MK method was proposed in [35] to ensure that the MK solutions remain stable and accurate. As numerically demonstrated, the MK shape functions and the solutions of the improved meshfree MK method are independent of changes in θ [35]. To date, the development of the improved meshfree MK method has been limited to the analyses of plate structures [35–37]. In this article, the improved meshfree MK method is first developed for the analysis of shell structures.

As mentioned earlier, sandwich structures possess some outstanding advantages, and numerous studies on the behaviors of sandwich shells made from FGM or FGCNTRC materials have been performed. Typical studies include analyses of shells with an FGM core and two isotropic skins [38,39], conical FGM sandwich shells [40], truncated conical FGM sandwich shells reinforced with FGM stiffeners [41], a structure with a soft viscoelastic core and FGM layers [42], cylindrical FGM sandwich shells reinforced by FGM stiffeners [43], cylindrical FGM sandwich shells reinforced using periodic eccentric ring stiffeners [44], shells with FGCNT face sheets and an isotropic core [45], the nonlinear stability of sandwich shells consisting of a porous FGM and CNT-reinforced composite layers [46], thermal vibrations of shells made of a sandwich of CNTRC sheets on both sides of a porous FG core [47], and conical sandwich shells with FG face sheets and a FG porous core [48]. In this article, we propose sandwich shells consisting of a skin of FGM layers and an FGCNTRC core. The improved MK meshfree method was first developed for modeling curved structures like shells. In addition, an improved meshfree MK formulation for free vibration analysis of FGM-FGCNTRC sandwich shells is first proposed. A nonlinear CNT distribution that we term FG-nX is proposed, and many new results of FGM-FGCNTRC sandwich shells are provided. In this paper, we investigate two types of boundaries, which are simply supported and clamped boundaries. In the case that some structures are connected by bolts or complex boundaries, the necessary information can be found in [49]. FGM-FGCNTRC sandwich shells can be used in semiconductor technologies, aerospace engineering, nuclear reactors, airplanes, spacecrafts, etc. Frequencies and vibrational mode shapes of these applications can be accurately predicted via the proposed technique in this study.

2  First-Order Shear Deformation Theory for FGM-FGCNTRC Sandwich Shells

2.1 Functionally Graded Carbon Nanotube-Reinforced Composite Materials

Fig. 1 shows five types of CNT distributions. UD is the uniform distribution. FG-V, FG-O, and FG-X are three types of linear CNT distributions. For the FG-O distribution, CNTs are enriched at the mid-surface. The top surface is CNT-rich in the case of FG-V distribution, and CNTs are enriched near both the top and bottom surfaces in the case of FG-X distribution. In this article, a nonlinear CNT distribution that we term FG-nX is proposed to enhance the stiffness of FGCNTRC structures. Similar to FG-X, FG-nX is CNT-rich near the top and bottom surfaces, but it is nonlinear through the structural thickness. Fig. 2 shows the FG-nX distribution along the thickness. The volume fraction of CNTs VCNT are determined as follows [50]:

VCNT=VCNT∗(UD)VCNT(z)=(1+2zh)VCNT∗(FG-V)VCNT(z)=2(1−2|z|h)VCNT∗(FG-O)VCNT(z)=2(2|z|h)VCNT∗(FG-X)VCNT(z)=(π/2)sin(π|z|h)VCNT∗(FG-nX)(1)

where

VCNT∗=wCNTwCNT+(ρCNT/ρm)−(ρCNT/ρm)wCNT(2)

where ρCNT is the density of CNTs, ρm is the density of the matrix, and the mass fraction of CNTs is wCNT. Notably, these five types of FGCNTRC shells have the same value of wCNT. The effective characteristics of materials reinforced using CNTs are affected by the structure of CNTs [51–54]. These characteristics can be determined by using micro-mechanical models like the Eshelby-Mori-Tanaka method [55,56] or the rule of mixture [57,58]. In this article, the rule of mixture is utilized because of its simplification as

E11=η1VCNTE11CNT+VmEm(3)

η2E22=VCNTE22CNT+VmEm(4)

η3G12=VCNTG12CNT+VmGm(5)

where the volume fraction of the matrix is Vm. The notations G12CNT, E11CNT and E22CNT are, respectively, the shear and Young’s modulii of the CNTs while Gm and Em are the corresponding characteristics of the matrix. The efficiency parameters ηi (i=1,2,3) are used in the above formulas to take into account the scale-dependent material characteristics. These parameters are determined by matching the effective characteristics of the CNTs calculated using the rule of mixture with those obtained from molecular-dynamics simulations. In particular, the sum of the volume fractions of the CNTs and the matrix is unity. We determined Poisson’s ratio, the density, and the thermal-expansion coefficients α11 and α22 by utilizing the rule of mixture as

v12=VCNT∗v12CNT+Vmvm(6)

ρ=VCNT*ρCNT+Vmρm(7)

α11=VCNTα11CNT+Vmαm(8)

α22=(1+v12CNT)VCNTα22CNT+(1+vm)Vmαm−v12α11(9)

where Poisson’s ratios are vm, and v12CNT and the thermal-expansion coefficients are αm, α11CNT, and α22CNT. Notably, v12 is constant along the structural thickness.

Figure 1: Constitution of an FGM-FGCNTRC sandwich structure

Figure 2: FG-nX distribution (proposed) along the thickness with VCNT∗=0.11

2.2 Functionally Graded Materials

An FGM is a composite material that consists of two distinct material phases which are ceramic and metal. Thus, an FGM fully inherits the mechanical characteristics of these two materials. The volume fractions of the material phases are determined as in [2]

Vc(z)=(12+zh)n;z∈[−h/2, h/2];Vm=1−Vc(10)

where Vc and Vm are the volume fractions of ceramic and metal, respectively. As we have seen, not only do Vc and Vm continuously change along the thickness, but also they depend on the power-law index n. In this article, the rule of mixture is utilized to determine the effective characteristics of the FGM [2], as follows:

Pe=PcVc(z)+PmVm(z)(11)

here, Pe is an effective characteristic which can be Poisson’s ratio ν or Young modulus E or the mass density ρ. The quantities Pc and Pm, respectively, represent the mechanical characteristics of the ceramic and metal.

2.3 FGM-FGCNTRC Sandwich Shells

Fig. 1 shows an FGM-FGCNTRC sandwich structure with five types of CNT distributions. It consists of two FGM-skin layers and one FGCNTRC core. For the two FGM-skin layers, the metallic component is enriched at the surfaces z=z1 and z=z4 while the ceramic component is enriched at the surfaces z=z2 and z=z3. For the FGM skins, the volume fractions of ceramic and metal are determined as follows [3]

Vc(z)=(z{4}−zz{4}−z{3})n;z∈[z{3},h2],for top skinVc(z)=(z−z{1}z{2}−z{1})n;z∈[−h2,z{2}],for bottom skinVm=1−Vc(12)

where h is the total thickness of the sandwich structure, and hb−hc−ht are the layer-thickness ratios, where hb, hc and, ht, respectively, are the thicknesses of the bottom, core, and top layers. In the case that this ratio is 1-1-1, the thicknesses of the three layers are equal.

2.4 Free Vibration Analysis of FGM-FGCNTRC Sandwich Shells Using First-Order Shear Deformation Shell Theory

We consider a doubly-curved shell with the geometric characteristics shown in Figs. 3 and 4. In these figures, r denotes the position vector of a point (x,y,0) on the mid-surface, while R=r+zn^ denotes that of an arbitrary point (x,y,z) within the shell. In addition, ds represents the distance between (x,y,0) and (x+dx,y+dy,0), computed as follows

(ds)2=dr.dr=ax2(dx)2+ay2(dy)2(13)

and

dr=gxdx+gydy;gx=∂r∂x;gy=∂r∂y(14)

Figure 3: A doubly-curved FGM-FGCNTRC sandwich shell

Figure 4: Geometric properties of a doubly-curved FGM-FGCNTRC sandwich shell

gx and gy are, respectively, the tangents to the x- and y-coordinate lines. The surface metrics ax and ay are given by

ax2=gx.gx;ay2=gy.gy(15)

The linear strains written in the curvilinear coordinate system are as follows [59,60] for the in-plane strains

εxx=1Ax(∂u∂x+1ay∂ax∂yv+axR1w)εyy=1Ay(∂v∂y+1ax∂ay∂xu+ayR2w)γxy=AyAx∂∂x(vAy)+AxAy∂∂y(uAx)(16)

and for the shear strains

γxz=1Ax∂w∂x+Ax∂∂z(uAx)γyz=1Ay∂w∂y+Ay∂∂z(vAy)(17)

where Ax=ax(1+zR1) and Ay=ay(1+zR2) are Lame coefficients. According to the FSDT, the displacements of the structures are given by

u(x,y,z)=u0(x,y)+zβx(x,y)v(x,y,z)=v0(x,y)+zβy(x,y)oru¯=u{0}+zu{1}w(x,y,z)=w0(x,y)(18)

and

u¯={uvw};u{0}={u0v0w0};u{1}={βxβy0}(19)

where u0 and v0 are the tangential displacements, and w0 is the radial displacement. In addition, βx and βy, respectively, are the rotations in y and x axes. For shallow shells, we can assume the following: ax,y=ay,x=0 (constant radii of curvatures), (1+z/R1)≈1 and (1+z/R2)≈1. By using Eq. (18) in Eqs. (16) and (17), the Sanders’ strains are re-expressed in the vector form as [59]

ε={εxxεyyγxy}T=ε0+zκbγ={γxzγyz}T=εs(20)

where

ε0={u0,x+w0R1v0,y+w0R2u0,y+v0,x};κb={βx,xβy,yβx,y+βy,x};εs={−u0R1+w0,x+βx−v0R2+w0,y+βy}(21)

The constitutive equation is written based on Hooke’s law as follows [59]:

{σxxσyyτxyτxzτyz}=[Q11Q12000Q21Q2200000Q6600000Q5500000Q44]{εxxεyyγxyγxzγyz}(22)

and

Q11=E111−υ12υ21;Q12=υ12E221−υ12υ21;Q22=E221−υ12υ21Q66=G12;Q55=G13;Q44=G23(23)

A Galerkin weak form derived from the motion equations for free vibration analysis is expressed as follows

∫Ωδ{ε0κb}T[ABBDb]{ε0κb}dΩ+∫ΩδεsTDsεsdΩ++∫Ωδ{u0u1}T[I1I2I2I3]{u¨0u¨1}dΩ=0(24)

where

(I1,I2,I3)=∫−h/2h/2ρ(z)(1,z,z2)dz(25)

(Aij,Bij,Dijb)=∫−h/2h/2(1,z,z2)Qijdz;i,j=1,2,6Dijs=κ∫−h/2h/2Qijdz;i,j=4,5(26)

and the shear correction factor κ=5/6.

3  The FGM-FGCNTRC Sandwich Shell Formulation Using the Meshfree MK Method and FSDT

3.1 Improved Meshfree MK Method

We consider a domain Ω with the boundary Γ and an arbitrary point x inside Ω. The support domain of point x, Ωx, which contains n nodes, is shown in Fig. 5 (Ωx∈Ω). The MK approximation is expressed as follows [35]

uh(x)=[pT(x)A+rT(x)B]u(27)

where

p(x)={p1(x)p2(x)p3(x)…pm(x)}TA=(PTR−1P)−1PTR−1B=R−1(I−PA)(28)

in which, p(x) is a polynomial. In this article, the second-order polynomial is chosen to establish the MK shape functions. The quantity u is the displacement vector, and {u}h(x) is the approximate value of this vector at point x. The matrix P is achieved from the polynomial as follows

p=[p1(x1)…pm(x1)………p1(xn)…pm(xn)](29)

where rT(x) includes the n correlation functions R(xi,x) between point x and node xi inside the support domain Ωx, computed as

r(x)={R(x1,x)R(x{2},x)⋯R(xn,x)}T(30)

and

R=[R(x1,x1)…R(x1,xn)………R(xn,x1)…R(xn,xn)](31)

Figure 5: Discretization and support domain of a 2D problem utilizing the meshless moving-Kriging method

The MK approximation also can be re-expressed in another form as follows [35]

uh(x)=∑1nNI(x)uI(32)

where NI(x) is the MK interpolation function (or the MK shape function) given by

NI(x)=∑j=1mpj(x)AjI+∑k=1nrk(x)BkI(33)

where the first- and second-order derivatives of the MK shape functions are given by

NI,α(x)=∑j=1mpj,α(x)AjI+∑k=1nrk,α(x)BkINI,αα(x)=∑j=1mpj,αα(x)AjI+∑k=1nrk,αα(x)BkI(34)

An improved meshfree MK method was proposed in [35] to ensure that the MK solutions remain stable and accurate. This improved method uses a quartic-spline correlation function to establish the MK shape functions as

R(xI,xJ)=1−6(θrIJa0)2+8(θrIJa0)3−3(θrIJa0)4;(0≤θrIJa0≤1)(35)

where a0 is the maximum distance between the computational point x and the farthest node in its support domain, and θ is the correlation parameter. As numerically demonstrated, the MK shape functions and the solutions obtained using the improved meshfree MK method are independent of changes in θ [35]. Thus, θ can be fixed at 1 in the numerical implementations in this article. In addition, rIJ is the Euclidean distance between points xI and xJ, which is given by

rIJ=‖xI−xJ‖(36)

The MK shape functions satisfy the Kronecker-delta property. Thus, the essential boundary conditions are simply and straightforwardly imposed as in the FEM. Notably, the main computational procedures of the improved meshfree MK method and the FEM are the same. In meshfree methods, a support domain is utilized to determine a set of nodes that are used for constructing the shape functions. In this article, the support domain with a circular shape as shown in Fig. 5 is utilized. The radius of this circle is determined as follows:

dm=αdc(37)

here, dc is a characteristic length, which is the node spacing in cases with regularly distributed nodes, and α is a scale factor, with α=[2÷4] for elastic problems [35]. The higher the scale factor, the more nodes collected in the support domain. As a result, the computational cost can be high. Accordingly, α is fixed at 2.4 for all problems in this article for the balance between the accuracy and efficiency of the method.

3.2 The FGM-FGCNTRC Sandwich Shell Formulation Using the Meshfree MK Method

The displacement field u of shells can be approximated using the MK interpolation functions as follows

uh(x,y)=∑A=1nNA(x,y)qA(38)

where NA(x,y) is the MK interpolation function, qA={u0Av0Aw0AβxAβyA}T is a vector containing degrees of freedom of node A, and uh is the vector including the approximate displacements. Inserting Eq. (38) into Eq. (21), the membrane, bending, and shear strains are, respectively, expressed in terms of the displacements as

ε0=∑A=1nBA0qA;κb=∑A=1nBAbqA;εs=∑A=1nBAsqA(39)

where

BA0=[NA,x01R1NA000NA,y1R2NA00NA,yNA,x000];BAb=[000NA,x00000NA,y000NA,yNA,x]BAs=[−1R1NA0NA,xNA00−1R2NANA,y0NA](40)

Eq. (38) is inserted into Eq. (19), we obtain the approximate displacement vectors as

u0=∑A=1nNA0qA;u1=∑A=1nNA1qA(41)

where

NA0=[NA00000NA00000NA00];NA1=[000NA00000NA00000](42)

Finally, by using Eqs. (39) and (41) in Eq. (24), the discretized system of equations for free vibration analysis of FGM-FGCNTRC sandwich shells based on MK meshfree method and FSDT is achieved as

(K−ω{2}M)q=0(43)

where K and M are, respectively, the global stiffness and mass matrices as follows

K=∫Ω({B0Bb}T[ABBDb]{B0Bb} + (Bs)TDsBs)dΩM=∫Ω{N0N1}T[I1I2I2I3]{N0N1}dΩ(44)

4  Results and Discussion

For the FGCNTRC core, the matrix material is Poly methyl methacrylate (PMMA) [61] and the reinforcements are the armchair (10,10) single-walled carbon nanotubes [50]. Unless otherwise stated, the parameters of the CNTs are as provided in [50]

•   VCNT∗=0.11: η1=0.149 and η2=0.934

•   VCNT∗=0.14: η1=0.150 and η2=0.941

•   VCNT∗=0.17: η1=0.149 and η2=1.381

It is assumed that η3=η2 and G12=G13=G23. The mechanical properties of materials are provided as follows

•   The matrix material

Em=(3.52−0.0034T) GPa, vm=0.34, ρm=1150kg/m3, T=T0+ΔT, T0=300K

•   The reinforcement material

E11CNT=5.6466 TPa, E22CNT=7.0800 TPa, G12CNT=1.9445 TPa, v12CNT=0.175, ρCNT=1400kg/m3

For the FGM skins, Al and Al2O3 were respectively chosen as the metal and ceramic materials. Their mechanical properties are given as the following: E=70 GPa, ν=0.3, ρ=2707 kg/m3 (for Al); E=380 GPa, ν=0.3, and ρ=3800 kg/m3 (for Al2O3). As mentioned earlier, the rule of mixture was used to compute the effective properties of both the FGCNTRC core and the FGM skins. For the typical shell shown in Fig. 6, we investigate two types of boundary conditions:

Figure 6: Geometry of a cylindrical shell

•   Simply supported (S)

u0=w0=βx=0aty=0,Lv0=w0=βy=0atx=0,αR(45)

•   Clamped (C)

u0=v0=w0=βx=βy=0(46)

In this article, boundary conditions were enforced as in the standard FEM. A mesh of 27 × 27 nodes was used for all problems, and numerical integration was performed utilizing 4 × 4 Gaussian points per each background structure cell. The computational meshes for cylindrical, spherical, and hyperbolic parabolic shells are shown in Fig. 7. Notably, the two key parameters investigated in section Parameter study are defined as the thickness-to-radius ratio (h/R) and the length-to-radius ratio (L/R).

Figure 7: Structural discretization utilizing the meshless moving-Kriging method

4.1 Verification Study

Notably, this is the first study on free-vibration analysis of FGM-FGCNTRC sandwich shells. Thus, solutions for these proposed structures are not available in the literature. However, because square FGM sandwich plates, cylindrical FGCNTRC shells, isotropic spherical and hyperbolic-parabolic shells are particular cases of doubly curved FGM-FGCNTRC sandwich shells. Therefore, the accuracy of the present formulation is, respectively, confirmed by performing free vibration analyses of these structures. Notably, the mechanical properties of FGM and FGCNTRC have been provided at the beginning of this section. In particular, by setting the thicknesses of the two skin layers to zero, the present formulation can be applied to structures with only one layer. This numerical implementation is very simple and can be done quickly using our in-house MATLAB codes.

4.1.1 Square Sandwich Plates with FGM Skins and an Isotropic Core

A square plate is a particular case of a doubly curved shell, with R1=R2=∞. A fully simply supported (SSSS) sandwich square plate (a/b=1 and a/h=10) is first investigated. This sandwich plate consists of FGM skins and a pure ceramic core. The FGM skins (Al/Al2O3) are metal-rich at the top and bottom surfaces. The normalized frequency is defined as ω¯=ωa2/h. Table 1 shows the first normalized frequencies of the plates with various power-law indices n and layer-thickness ratios. As seen, the present results well agree with those obtained using 3D elasticity [3]. The high accuracy of the present formulation for analyses of sandwich plates with FGM skins and isotropic cores is confirmed.

4.1.2 Cylindrical FGCNTRC Shells

Next, we verify the present formulation for the analysis of cylindrical FGCNTRC shells. A cylindrical FGCNTRC panel is shown in Fig. 6, and its geometric properties are α=0.1 rad, h/R=0.002, R=1 m, and L/R=0.1. The results we obtained include the first six normalized frequencies of the panels, which are listed in Table 2 for various types of CNT distributions and boundaries. Here again, the present results agree well with those obtained using the meshfree KP-Ritz method (MKR) [61]. The high accuracy of the present formulation for the analysis of cylindrical FGCNTRC shells is confirmed.

4.1.3 Isotropic Spherical and Hyperbolic-PARabolic Shells

We next carried out a verification of the present formulation for the analysis of isotropic spherical and hyperbolic-parabolic shells. The doubly-curved shell shown in Fig. 8 is considered. Here, Poisson’s ratio is fixed at ν=0.3 and the length-to-thickness ratio is fixed at Ly/h=100. The shell is fully clamped (CCCC) at its boundaries. The non-dimensional frequency parameter is defined as λ=ωLxLyρh/D, with D=Eh3/12(1−ν2). Table 3 shows the first six normalized frequencies of the spherical and hyperbolic-parabolic shells. Notably, the spherical shells correspond to the radii Rx=Ry=1 m while the hyperbolic-parabolic ones correspond to Rx=1 m and Ry=−1 m. As seen, the present results agree well with those obtained using the pb-2 Ritz method [62]. The high accuracy of the present formulation for the analysis of isotropic spherical and hyperbolic parabolic shells is confirmed.

Figure 8: Geometry of a doubly-curved shell

4.2 Parameter Study

4.2.1 Cylindrical FGM-FGCNTRC Sandwich Shells

We next consider the cylindrical FGM-FGCNTRC sandwich shell shown in Fig. 6. Its geometric properties are α=0.1 rad, R=1 m, L/R=0.1, and the shell is fully clamped (CCCC) at its boundaries. The normalized frequency is computed as ω^=10ωL2ρm/Em, where ρm and Em are the mechanical properties of the matrix material of the FGCNTRC core. Table 4 presents the first normalized frequencies of shells with the power-law index n=1, the layer-thickness ratio 1-2-1, various types of CNT distributions, values of VCNT∗, and thickness-to-radius ratios (h/R). As seen, the higher the thickness-to-radius ratio, the higher the stiffness and frequency of the shell. The results in Table 4 are visualized in Fig. 9. It is seen that the greater the quantity of CNTs, the higher the stiffness and frequency of the shell. FG-X is the best CNT distribution which produces the highest stiffness and frequency of the shell compared to the rest CNT distributions. FG-nX (proposed) produces a higher stiffness and frequency of the shell than do the distributions UD, FG-V, and FG-O. Table 5 shows the first six normalized frequencies of shells with various boundaries, power-law indices, and length-to-radius ratios (L/R). It is found that when more constraints are applied at the boundaries, the higher the frequency of the shell. The higher the length-to-radius ratio, the higher the stiffness and frequency of the shell. Note that R was fixed at 1 m for these calculations. The higher the power-law index n, the lower is the amount of ceramic in the shell, which produces the frequency of the shell. Fig. 10 shows the first six mode shapes of a cylindrical CCCC FGM-FGCNTRC sandwich shell with L/R=0.1 and n=1. Table 6 lists the first normalized frequencies of cylindrical CCCC FGM-FGCNTRC sandwich shells with various power-law indices, CNT distributions, and layer-thickness ratios. It is seen that the thicker the FGCNTRC core, the higher the stiffness and frequency of the shell. The results in Table 6 are visualized in Fig. 11. It is found that the effect of the CNT distribution on the frequency of the shell is very small for the layer-thickness ratio 2-1-2, while it is significant for the case layer-thickness ratio 1-2-1. It is recommended to use cylindrical FGM-FGCNTRC sandwich shells with the layer-thickness ratio 1-2-1 (or shells with thicker FGCNTRC cores) to efficiently exploit the CNT distribution.

Figure 9: Effect of the CNT distribution on the first normalized frequency of the cylindrical FGM-FGCNTRC sandwich shell with h/R=0.008, R=1 m, L/R=0.1,n=1 and the layer-thickness ratio 1-2-1

Figure 10: First six mode shapes of a cylindrical FGM-FGCNTRC sandwich shell

Figure 11: Effect of the layer-thickness ratio on the first normalized frequency of the cylindrical FGM-FGCNTRC sandwich shell with h/R=0.008, R=1 m, L/R=0.1,n=10, VCNT∗=0.11

4.2.2 Spherical FGM-FGCNTRC Sandwich Shells

The spherical FGM-FGCNTRC sandwich shell shown in Fig. 8 is next considered. Its geometric properties are Rx=Ry=1 m, Ly/Ry=0.1, and Lx=Ly. The normalized frequency is ω^=10ωLy2ρm/Em, where ρm and Em are again the mechanical properties of the matrix material of the FGCNTRC core. Table 7 lists the first six normalized frequencies of spherical FGM-FGCNTRC sandwich shells with various boundaries, power-law indices, and thickness-to-radius ratios. Similar to our earlier observations in Section 4.2.1, when more constraints are applied at the boundaries, the higher is the frequency of the shell. The higher the thickness-to-radius ratio, the higher are the stiffness and frequency of the shell. Fig. 12 shows the first six mode shapes of a spherical SSSS FGM-FGCNTRC sandwich shell with h/Rx=0.008 and n=1. As seen, the mode shapes of the spherical shell are quite different from those of the cylindrical shell shown in Fig. 10. Table 8 lists the first normalized frequencies of CCCC shells with various CNT distributions, VCNT∗, power-law indices, and length-to-radius ratios. The results in Table 8 are visualized in Fig. 13 for the case Ly/Ry=0.1 and n=1. It is seen that the higher the length-to-radius ratio or the larger the amount of CNTs, the higher the stiffness and frequency of the shell. Here again, FG-X is the best CNT distribution which produces the highest stiffness and frequency of the shell compared to the rest CNT distributions. FG-nX distribution (proposed) produces a higher stiffness and frequency of the shell than do the distributions UD, FG-V, and FG-O. Table 9 presents the first normalized frequencies of CCCC shells with various CNT distributions, power-law indices, and layer-thickness ratios. The results in Table 9 are visualized in Fig. 14 for the case n=10. It is seen that the thicker the FGCNTRC core, the higher are the stiffness and frequency of the shell. It is found that the effect of CNT distribution on the frequency of the shell is very small for the case layer-thickness ratio 2-1-2, while it is significant for the case layer-thickness ratio 1-2-1. It is recommended to use spherical FGM-FGCNTRC sandwich shells with the layer-thickness ratio 1-2-1 (or shells with thicker FGCNTRC cores) to efficiently exploit the CNT distribution.

Figure 12: First six mode shapes of a spherical FGM-FGCNTRC sandwich shell

Figure 13: Effect of the CNT distribution on the first normalized frequency of the spherical FGM-FGCNTRC sandwich shell with h/Rx=0.008, Rx=1 m, Ry=1 m, Ly/Ry=0.1,n=1, Lx=Ly and the layer-thickness ratio 1-2-1

Figure 14: Effect of the layer-thickness ratio on the first normalized frequency of the spherical FGM-FGCNTRC sandwich shell with h/Rx=0.008, Rx=1 m, Ry=1 m, Ly/Ry=0.1,n=10, Lx=Ly, VCNT∗=0.11

4.2.3 Hyperbolic-Parabolic FGM-FGCNTRC Sandwich Shells

The geometries of the shells and the normalized frequency and parameter investigations discussed in this section are the same as those in Section 4.2.2 except that we use Ry=−1 m in this section. The obtained results for hyperbolic-parabolic FGM-FGCNTRC sandwich shells are presented in Tables 10–12 and in Figs. 15 and 16. Similar to our earlier observations in Section 4.2.2, when more constraints are applied at boundaries, the higher is the frequency of the shell. Also, the higher the thickness-to-radius ratio, the higher the length-to-radius ratio, or the greater the amount of CNTs, the higher the stiffness and frequency of the shell. Here also, FG-X is the best CNT distribution which produces the highest stiffness and frequency of the shell compared to the rest CNT distributions. FG-nX distribution produces a higher stiffness and frequency of the shell than do the UD, FG-V, and FG-O distributions.

Figure 15: Effect of CNT distribution on the first normalized frequency of the hyperbolic-parabolic FGM-FGCNTRC sandwich shell with h/Rx=0.008, Rx=1 m, Ry=−1 m, Ly/Ry=0.1,n=1, Lx=Ly and the layer-thickness ratio 1-2-1

Figure 16: Effect of the layer-thickness ratio on the first normalized frequency of the hyperbolic-parabolic FGM-FGCNTRC sandwich shell with h/Rx=0.008, Rx=1 m, Ry=−1 m, Ly/Ry=0.1,n=10, Lx=Ly, VCNT∗=0.11

5  Conclusions

In this work, we have presented free vibration analyses of FGM-FGCNTRC sandwich shells utilizing an improved meshfree moving-Kriging method and first-order shear deformation shell theory. The effective material characteristics of both the FGM skin layers and the FGCNTRC core were determined using the rule of mixture. The accuracy of the present formulation was confirmed by solving some problems. The obtained results agreed well with the reference ones. Some main parameters and factors such as the thickness-to-radius ratio (h/R), the length-to-radius ratio (L/R), the layer-thickness ratio (hb−hc−ht), the CNT distribution, the CNT volume fraction VCNT∗, the power-law index n, and the boundary condition were rigorously studied in section Parameter study. The present formulation can be applied to cylindrical, spherical, hyperbolic-parabolic FGM-FGCNTRC sandwich shells and plates. Especially, a nonlinear CNT distribution named FG-nX was first proposed in this work, and an improved meshfree moving-Kriging method was first developed for shell analyses. In addition, many results for FGM-FGCNTRC sandwich shells were first proposed. The following are some of our main conclusions:

•   The greater the amount of CNTs, the higher the stiffness and frequency of the shell. FG-X is the best CNT distribution that produces the highest stiffness and frequency of the shell compared to the rest of the CNT distributions. FG-nX (proposed) produces a higher stiffness and frequency of the shell than do the UD, FG-V, and FG-O distributions.

•   The thicker the FGCNTRC core, the higher the stiffness and frequency of the shell. It is found that the effect of the CNT distribution on the frequency of the shell is very small for the case where the layer-thickness ratio is 2-1-2, while it is significant for the case where the layer-thickness ratio is 1-2-1. It is recommended to use FGM-FGCNTRC sandwich shells with the layer-thickness ratio 1-2-1 (or shells with thicker FGCNTRC cores) to efficiently exploit the CNT distribution.

•   The higher the thickness-to-radius ratio or the length-to-radius ratio, the higher the stiffness and frequency of the shell.

Although the present approach possesses some advantages and some findings for FGM-FGCNTRC sandwich shells have been recommended, some limitations remain to be overcome. These include the following: 1) The present formulation applies to doubly curved shells, but it needs to be improved for analyzing free-form shells; 2) The meshfree MK method should be improved for reducing the computational cost. Overcoming these limitations can be considered as future work. In addition, some possible future research directions include analyses of the transient vibrations, nonlinear vibrations, and nonlinear forced vibrations of FGM-FGCNTRC sandwich shells.

Acknowledgement: None.

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

Author Contributions: Study conception and design: Tan N. Nguyen, Nuttawit Wattanasakulpong; Literature review and data collection: Tan N. Nguyen, Suppakit Eiadtrong; Analysis and interpretation of literature: Tan N. Nguyen, Suppakit Eiadtrong, Mohamed-Ouejdi Belarbi; Visualization and graphical representation: Tan N. Nguyen, Suppakit Eiadtrong; Draft manuscript preparation: Tan N. Nguyen, Suppakit Eiadtrong; Critical revision of the manuscript: Tan N. Nguyen, Nuttawit Wattanasakulpong, Mohamed-Ouejdi Belarbi. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: Data will be made available on request.

Ethics Approval: No applicable.

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

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