Buckling Behavior of Sandwich Cylindrical Shells Covered by Functionally Graded Coatings with Clamped Boundary Conditions under Hydrostatic Pressure

The buckling behavior of sandwich shells with functionally graded (FG) coatings operating under different external pressures was generally investigated under simply supported boundary conditions. Since it is very difficult to determine the approximation functions satisfying clamped boundary conditions and to solve the basic equations analytically within the framework of first order shear deformation theory (FOST), the number of publications on this subject is very limited. An analytical solution to the buckling problem of FG-coated cylindrical shells under clamped boundary conditions subjected to uniform hydrostatic pressure within the FOST framework is presented for the first time. By mathematical modeling of the FG coatings, the constitutive relations and basic equations of sandwich cylindrical shells within the FOST framework are obtained. Analytical solutions of the basic equations in the framework of the Donnell shell theory, obtained using the Galerkin method, is carried out using new approximation functions that satisfy clamped boundary conditions. Finally, the influences of FG models and volume fractions on the hydrostatic buckling pressure within the FOST and classical shell theory (CT) frameworks are investigated in detail.


Introduction
The most important applications of sandwich composites are found in advanced technology industries such as the aviation, aerospace, automotive, railroad and marine industries, due to their high stiffness/weight and strength/weight ratios which increase the load carrying capacity of structures and improve their performance while consuming less energy. The main disadvantage of sandwich structures made of traditional composites is that delamination cannot be prevented due to the different material properties on the contact surfaces of the core and the coating [1,2]. The ability to prevent such disadvantages in the applications of sandwich structural elements has led materials scientists to seek the creation of a new generation composite materials. The development of new technologies such as structural optimization and additive manufacturing has made it possible to realize their applications as functionally classified materials and microelectromechanical systems. These developments allow one to take into account the material properties of structural elements and extend representation beyond geometry. such material compositions and microstructures make object heterogeneous. Heterogeneous objects are primarily classified sandwich plates subjected to various nonuniform compressions using a finite element model based on the high-order shear deformation theory. Alsebai et al. [31] presented the semi-analytical solution to the problem of the thermo-piezoelectric bending of FG porous plates reinforced with graphene platelets. Sofiyev and Fantuzzi [32] solved the stability and vibration problem of clamped cylindrical shells containing FG layers within ST under axial loads. Hu et al. [33] presented a new analytical solution to the problem of the free vibration of non-Lévy-type functionally graded doubly curved shallow shells.
In the studies reviewed above, solutions to the eigen value problem of FG-coated sandwich shells were usually obtained for simply supported boundary conditions. It is very difficult to determine the approximation functions that satisfy the clamped boundary conditions in the framework of shear deformation theory (ST). In addition to this main difficulty, deriving the basic equations in the framework of ST for FG-coated sandwich cylindrical shells under the effect of a hydrostatic pressure load presents an additional difficulty. For this reason, analytical investigations of the mechanical behavior of FG sandwich shells under clamped boundary conditions are very limited. To address this shortcoming, in this study, the modeling and solution of the buckling problem of cylindrical shells with an FGM coating and isotropic core under external pressures under clamped boundary conditions are presented.
The study is constructed as follows: after the introduction, the material and geometric model of the problem is presented in Section 2, the basic relations and basic equations are derived in Section 3, the approximation function and the solution are obtained in Section 4, and Section 5 includes comparisons and original analyses. Figure 1 presents two sandwich cylindrical shells of length L and radius r covered with coatings of functionally graded material whose core consists of two different isotropic materials: (a) a ceramic-rich core and (b) a metal-rich core. We assumed that the FG sandwich cylindrical shell with clamped edges was subjected to hydrostatic pressure. The thickness of the FG coatings, h coat , is equal with the thickness of the core, h core , and the total thickness of the sandwich cylindrical shell is h, i.e., h = 2h F + h core . The origin of the coordinate system (Ox 1 x 2 x 3 ) is located on the reference surface of the core at the left end of the sandwich cylinder, with the x 1 -axis pointing along the length of the cylinder, the x 2 -axis in the circular direction, and the x 3 -axis in the perpendicular direction to the x 1 x 2 surface towards the center of curvature. One of the advantages of FG coatings in the preparation of sandwich structural elements is the formation of one surface from the metal-rich and the ceramic-rich surface, and the continuous and smooth change in properties from one surface to the other. Since the material properties are almost the same on the contact surfaces of the coatings as in the core in the formation of the sandwich structural elements, this advantage ensures that the layers do not break from each other at different loadings. In FG 1 /C/FG 1 sandwich cylinders, the core is ceramic rich and the material properties of the FG coatings continuously change from metal-rich surface to ceramic-rich surface in the thickness direction (Figure 1a). In FG 2 /M/FG 2 sandwich cylinders, on the other hand, the core is metal rich, and the material properties of the FG coatings constantly change from ceramic-rich surface to metal-rich surface (Figure 1b).

Basic Relations and Equations
The volume fractions ( ( ) ( 1,2,3) ) of the coatings and core are obtained from a simple mixing rule of materials and are expressed as follows [10][11][12]: 3  3  2  3  2  3  2   (3)  3  3  3  3   0.5  ,  /2, ; 1 , , ; 0.5 where d is the power law index and dictates the property dispersion profile and The Young's modulus and Poisson's ratio of the FG coatings are mathematically modeled as follows [10,11]: The material properties of the sandwich shells covered by the coatings with ceramicrich or metal-rich cores are expressed as [11]:

Basic Relations and Equations
The volume fractions (V (k) c (k = 1, 2, 3)) of the coatings and core are obtained from a simple mixing rule of materials and are expressed as follows [10][11][12]: where d is the power law index and dictates the property dispersion profile and V The Young's modulus and Poisson's ratio of the FG coatings are mathematically modeled as follows [10,11]: The material properties of the sandwich shells covered by the coatings with ceramicrich or metal-rich cores are expressed as [11]: where core i are the Young moduli and Poisson ratios of the FG 1 and FG 2 coatings and the ceramic-rich and metal-rich cores, respectively. The variations in the dimensionless Young moduli of the sandwich cylinders covered by the FG 1 and FG 2 coatings with ceramic-rich and metal-rich cores are illustrated in Figures 2 and 3 are the Young moduli and Poisson ratios of the FG1 and FG2 coatings and the ceramic-rich and metal-rich cores, respectively. The variations in the dimensionless Young moduli of the sandwich cylinders covered by the FG1 and FG2 coatings with ceramic-rich and metal-rich cores are illustrated in Figures 2 and 3, respectively. Similar graphs can be drawn for other mechanical properties of the FG-coated sandwich shells.  The constitutive relationships of the elastic and isotropic layers of the FG-coated sandwich cylinders based on the FOST can be written as [11]: [ ] are the Young moduli and Poisson ratios of the FG1 and FG2 coatings and the ceramic-rich and metal-rich cores, respectively. The variations in the dimensionless Young moduli of the sandwich cylinders covered by the FG1 and FG2 coatings with ceramic-rich and metal-rich cores are illustrated in Figures 2 and 3, respectively. Similar graphs can be drawn for other mechanical properties of the FG-coated sandwich shells.  The constitutive relationships of the elastic and isotropic layers of the FG-coated sandwich cylinders based on the FOST can be written as [11]: The constitutive relationships of the elastic and isotropic layers of the FG-coated sandwich cylinders based on the FOST can be written as [11]: ij (i = 1, 2, j = 1, 2, 3) and e ii (i = 1, 2), γ ij (i = 1, 2, j = 2, 3) are the stress and strain components, respectively, and q (k) ij (i, j = 1, 2, 6) are the coefficients depending on the normalized thickness coordinate and are defined as: It is assumed that the transverse shear stresses proposed by Ambartsumian [34,35] for homogeneous structural members and generalized to FG structural members in this study vary as follows depending on the thickness coordinate [11,34,35]: Since the expression (6) is taken into account in the fourth and fifth of the system of Equation (4), the following expressions are obtained for shear strains γ 13 and γ 23 : Considering the assumptions of the FOST, the following relations are used [34,35]: When Equation (8) is integrated with respect to x 3 in the interval (0,x 3 ), and when , the expressions of displacements of any point of the shell are obtained as follows: where u and v are the displacements of the axial and circumferential directions on the mid-surface, respectively, w is the deflection, φ 1 (x 1 , x 2 ) and φ 2 (x 1 , x 2 ) are the transverse normal rotations about the x 2 and x 1 axes, respectively, and the following definitions apply: The strain components (e 11 , e 22 , γ 12 ) with u x 1 , u x 2 , w of any point of the cylindrical shell can be defined by the following relations [35]: Substituting the expression (9) for the displacements of u x 1 and u x 2 into Equation (11), the following relations are obtained: where e 0 11 , e 0 22 , γ 0 12 are the strain components on the mid-surface and are defined as: The force and moment components T ij , Q i and M ij of the FG-coated cylindrical shells are derived from the following integrals [32,[34][35][36]: The stress function Φ is related to the forces as [34][35][36]: Taking the pre-buckling state of the sandwich cylinder for the membrane, the resultants T 0 11 , T 0 22 , T 0 12 are determined as [37]: The stability and compatibility equations of the FG-coated cylindrical shells subjected to hydrostatic pressure are expressed as [36,37]: By using the Equations (4), (12), (14)-(16) together, the expressions for the strains at the mid-surface, and forces and moments are obtained, and when the resulting expressions are substituted into the system of Equation (17), the basic equations of the FG-coated sandwich cylindrical shells subjected to hydrostatic pressure in the FOST framework take the following form: where C i , B i , J l (i = 1, 2, ..., 12, l = 3, 4) are given in Appendix A.
The Galerkin method is applied to the system of Equation (18): Substituting (20) into Equation (21), after integration and some mathematical operations, we obtain the following expression for the dimensionless hydrostatic buckling pressure (DHBP) of the FG-coated sandwich cylindrical shells with homogeneous isotropic cores (ceramic-or metal-rich) under clamped boundary conditions based on the FOST: where P ST Hbuc = P ST 1Hbuc Y c is the dimensional hydrostatic buckling pressure (in Pa) within the ST and the following definitions apply: in which Ignoring the transverse shear strains, the following expression is obtained for the DHBP of the FG-coated sandwich cylindrical shells with homogeneous isotropic cores under clamped boundary conditions based on the CT: where P CT Hbuc = P CT 1Hbuc Y c is the DHBP within CT. The minimum values of the DHBP of the FG-coated cylinders with clamped edges based on the FOST and CT are found by minimizing according to the m and n wave numbers.

Numerical Results and Discussion
This section consists of two subsections. The accuracy of the analytical formulas is confirmed under the first subheading. Under the second subheading, the effects of the FG coatings on the DHBP are examined in detail within the framework of the FOST and CT by performing original analyses and providing comments. In all computations, the values in parentheses are the circumferential wave numbers (n cr ) corresponding to the minimum values of the dimensionless hydrostatic buckling pressure (DHBP). Furthermore, it has been determined that the number of longitudinal waves corresponding to the minimum value of the hydrostatic buckling pressure is equal to one (m = 1). Table 1 presents the magnitudes of the DHBP of the cylindrical shells consisting of homogeneous isotropic material under clamped boundary conditions. Our calculations are made according to Equation (25), and the following material properties and geometric characteristics of the single-layer cylindrical shells are: Y m = 2 × 10 11 Pa, ν m = 0.3, L/r = 2, r/h = 100. The P CT 1Hbuc values for the clamped boundary conditions are taken from Singer et al. [39]. As can be seen from Table 1, our results seem to be in agreement with the results obtained in the study of Singer et al. [39].  (11)  1 12.89 (9) 11.03 (8) 11.7789 (9)  2 6.52 (7) 5.026 (7) 5.759 (7) Table 2 presents the magnitudes of the hydrostatic buckling pressure (in kPa) of the homogeneous isotropic cylindrical shells under clamped boundary conditions. Our calculations are made according to Equation (25), and the following material properties and geometric characteristics of the single-layer cylindrical shells are used: Y m = 5.455 × 10 10 Pa, ν m = 0.3, L = 1, 2, 3 m, r = 0.5 m. The P CT Hbuc (kPa) values in the second and third columns are taken from Tables 2 and 3, presented in Ref. [40]. Table 2 shows that our results are in agreement with those obtained in ref. [40].

Novel Applications
In numerical analysis, cylindrical shells with two kinds of functionally graded coatings, cylindrical shells with two kinds of homogenous coatings and two kinds of single-layer cylindrical shells are used (see Figures 2,4 and 5). The FG coatings are composed of a mixture of silicon nitride (Si 3 N 4 ) and stainless steel (SUS304), forming two kinds of sandwich cylindrical shells, designated FG 1 /Si 3 N 4 /FG 1 and FG 2 /SUS304/FG 2 or FG 1 /C/FG 1 and FG 2 /M/FG 2 , respectively ( Figure 1). In addition, metal (SUS304)-and ceramic (Si 3 N 4 )coated sandwich cylindrical shells are designated as M/C/M and C/M/C, respectively ( Figure 4). In addition, single-layer cylindrical shells made of ceramic (Si 3 N 4 ) and metal (SUS304) are designed and used for comparisons ( Figure 5). In all calculations, the ratio of core thickness to coating thickness is indicated by the symbol: η = h core /h coat . The shear stress shape functions are as follows:   The properties of the FGMs are taken from the monograph of Shen [36]. The Young's moduli and Poisson's ratios of the FG coatings as a function of temperature and their values are presented as follows, when T = 300 K: E Si 3 N 4 = 3.4843 × 10 11 (1 − 3.07 × 10 −4 T + 2.16 × 10 −7 T 2 − 8.946 × 10 −11 T 3 ) = 322.271(Gpa) E Sus304 = 2.0104 × 10 11 (1 + 3.079 × 10 −4 T − 6.534 × 10 −7 T 2 ) = 207.788(GPa) ν Sus304 = 0.3262(1 − 2.002 × 10 −7 T + 3.797 × 10 −7 T 2 ) = 0.317756, ν Si 3 N 4 = 0.24 The distribution of the magnitudes of DHBP or P CT 1Hbuc and P ST 1Hbuc for the M/C/M, FG 1 /C/FG 1 , C/M/C and FG 2 /M/FG 2 sandwich, ceramic and metal single-layer cylindrical shells against r/h are tabulated in Table 3 with r/L = 2, η = 0.25 and d = 1. The P CT 1Hbuc and P ST 1Hbuc values for the cylindrical shells covered by the FG 1 and FG 2 coatings decrease, while the number of circumferential waves increases depending on the increase in the r/h. When the P ST 1Hbuc of the FG 1 -and FG 2 -coated sandwich cylinders are compared with the metal-and ceramic-coated homogeneous sandwich cylinders in the framework of the ST, the effects of the FG 1 and FG 2 coatings on the dimensionless hydrostatic buckling pressure reduce from (+18.11%) to (+14.29%) and from (−17.7%) to (−13.5%), respectively, as the r/h increment increases from 20 to 50. As the FG 1 -and FG 2 -coated sandwich shells are compared with pure ceramic and pure metal cylindrical shells in the framework of the ST, respectively, the effects of the FG 1 and FG 2 coatings on the P ST 1Hbuc increase from (−15.72%) to (−16.76%), and from (+17.57%) to (+21.55), respectively, as the r/h ratio increases from 20 to 50. The most significant effect of the transverse shear strains on the DHBP of the FG 1and FG 2 -coated sandwich cylindrical shells occurs with 18.34% of the shell covered by the FG 2 coating at r/h = 20 and decreases by up to 2.84% when r/h = 50. In the shell covered by the FG 1 coating, this effect is lower than in the FG 2 -coated sandwich shell with the metal core, decreasing from 8.46% to 1.39% as the r/h ratio increases from 20 to 50. Although these influences are evident at small values of r/h in pure ceramic and pure metal shells, they are reduced from 11.4% to 1.73% and from 12.68% to 1.72%, respectively, when r/h increases from 20 to 50.  Table 4. The following data and volume fraction index are used: L/r = 0.5, r/h = 25 and d = 1. The magnitudes of P CT 1Hbuc and P ST 1Hbuc for the three-layered cylinders with ceramic cores increase, while they decrease for the three-layered cylinders with the metal cores, as the η increases. The circumferential wave number corresponding to the DHBP increases with the increase in η. When the P ST 1Hbuc of the FG 1 -and FG 2 -coated sandwich cylinders are compared with those of the M/C/M and C/M/C shells, the respective effect on the P ST 1Hbuc decreases from (+13.94%) to (+7.86%) for the FG 1 coating and, although it shows disorder, from (−12.72%) to (−9.42%) for the FG 2 coating as the η ratio increases from 2 to 8. Furthermore, when the FG 1 -and FG 2 -coated sandwich shells are compared with the pure ceramic and metal single-layer shells, the respective effect on the P ST 1Hbuc decreases from (−18.07%) to (−8.98%) for the FG 1 coating and from (+23.66%) to (+11.52%) for the FG 2 coating as the η increases from 2 to 8. The most significant effect of the transverse shear strains on the DHBP of the FG 1 and FG 2 -coated sandwich cylindrical shells occurs at 33.4% in the FG 2 -coated sandwich shell with the metal core at η = 8, and that effect is 18.53% when η = 2. In the FG 1 -coated shell, this effect is lower than in the FG 2 -coated shell, reducing from 8.51% to 8.13% as the η ratio increment from 2 to 8.  (10) 1.569 (10) 1.717 (10) 1.915 (10)  4 1.320 (10) 1.483 (10) 1.532 (10) 1.655 (10)  6 1.391 (10) 1.561 (10) 1.581 (10) 1.708 (10) (9) 1.349 (10) 1.599 (9) 1.146 (10) 1.293 (10)  4 1.540 (10) 1.725 (9) 1.242 (10) 1.521 (9)  6 1.469 (10) 1.648 (9) 1.159 (10) 1.474 (10)  8 1.418 (10) 1.592 (9) 1.081 (10) 1.442 (10) The variations in the magnitudes of P CT 1Hbuc and P ST 1Hbuc for the FG 1 -and FG 2 -coated sandwich cylindrical shells against the d are presented in Table 5. The following data are used: L/r = 0.5, r/h = 25, η = 0.25 and d = 1. The magnitudes of P CT 1Hbuc and P ST 1Hbuc for the FG 1 kind sandwich cylindrical shells decrease, while they increase for the FG 2 sandwich cylindrical shells, as the volume fraction index increases. Within the framework of these data, the circumferential wave numbers are independent of the change in d. When the FG 1and FG 2 -coated cylinders are compared with the pure ceramic and pure metal single-layer cylinders, the respective effect on the P ST 1Hbuc decreases from (−11.79%) to (−4.26%) for the FG 1 coatings, but increases from (+5.16%) to (+11.57%) for the FG 2 coatings as the d increases from 0.5 to 2. It is thus revealed that the effect of material heterogeneity on the DGBP decreases significantly with the increase of the d ratio from 0.5 to 2 in both kinds of FG coating. In addition, the coating with the greatest effect on the DHBP is the FG 2 coating, when compared with the single-layer shells. When the values of the dimensionless hydrostatic buckling pressure of the FG 1 and FG 2 -coated sandwich cylindrical shells are compared, the values of the DHBP are lower in the ST than in the CT. The most significant effect of the transverse shear strains on the DHBP occurs with 18.78% in FG 2 -coated sandwich shell at d = 2. In the FG 1 -coated sandwich shell, it is lower than in the FG 2 -coated sandwich cylindrical shell, decreasing from 5.96% to 4.34% as the d index increases from 0.5 to 2.

Conclusions
In this study, the buckling of FG-coated sandwich cylindrical shells was investigated. The most important aspect of this study is the solution of the buckling problem of clamped FG-coated sandwich cylindrical shells subjected to hydrostatic pressure by determining a new approximation function in the framework of the FOST. The basic equations were derived based on the Donnell shell theory, and new analytical expressions for the hydrostatic buckling pressure under clamped boundary conditions were found within the FOST and CT by applying Galerkin's procedure. Finally, the findings of the present study were verified by comparing with those presented in the literature, and the effects of the FG profiles, shear stresses, volume fractions and shell characteristics on the DHBP were examined in detail.
Numerical analyses and comments revealed the following generalizations: 1.
The P CT 1Hbuc and P ST 1Hbuc values for the cylindrical shells covered by the FG 1 and FG 2 coatings decrease, while the number of circumferential waves increases depending on the increase in the r/h.

2.
As the P ST 1Hbuc of the FG 1 -and FG 2 -coated sandwich cylinders are compared with the metal-and ceramic-coated homogeneous sandwich cylinders in the framework of the FOST, the influence of the FG 1 and FG 2 coatings on the dimensionless hydrostatic buckling pressure decreases as the r/h increases.

3.
As the FG 1 and FG 2 -coated sandwich shells are compared with pure ceramic and pure metal cylindrical shells in the framework of the ST, the effect of the FG 1 and FG 2 coatings on the P ST 1Hbuc increases as the r/h increases. 4.
The most significant effect of the transverse shear strains on the DHBP of the FG 1and FG 2 -coated sandwich cylindrical shells occurs in the shell covered by the FG 2 coating at r/h = 20. 5.
The magnitudes of P CT 1Hbuc and P ST 1Hbuc for the FG 1 sandwich cylindrical shells decrease, while they increase for the FG 2 sandwich cylindrical shells, as the volume fraction index increases. 6.
When FG 1 -and FG 2 -coated shells are compared with the pure ceramic and pure metal single-layer cylinders, respectively, the effect of the FG 1 coating on the P ST 1Hbuc decreases, whereas the influence of the FG 2 coating on the P ST 1Hbuc increases, as the d increases. 7.
The most significant effect of the transverse shear strains on the DHBP occurs in FG 2 -coated sandwich shell at d = 2. 8.
As the FG 1 -and FG 2 -coated sandwich cylinders are compared with the pure ceramic and metal single-layer cylinders, the influence of FG 1 and FG 2 coatings on the P ST 1Hbuc decreases as the η increases.

Conflicts of Interest:
The authors declare no potential conflict of interest with respect to the research, authorship and publication of this article.