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Abstract

The main aim of this research is to establish the algorithm for nonlinear thermo-mechanical buckling of sandwich functionally graded graphene platelet reinforced composite (FG-GPLRC) shallow spherical caps and circular plates with porous core under external pressure and/or uniformly distributed thermal load according to the higher-order shear deformation theory considering the von Karman nonlinearities. Sandwich spherical caps and circular plates are made by the porous core and two FG-GPLRC coatings and are assumed to be rested on an elastic foundation modeled by the Pasternak model. The equilibrium equation system in the form of nonlinear algebra can be approximately obtained using the Ritz energy method. The critical buckling loads and postbuckling curves can be explicitly determined. The effects of material parameters, geometrical parameters, porous core, and elastic foundation on thermo-mechanical buckling of sandwich spherical caps and circular plates with porous core and FG-GPLRC coatings are investigated and discussed in detail in the numerical investigation section.

Keywords

Shallow spherical cap and circular plate nonlinear buckling and postbuckling sandwich functionally graded graphene platelet reinforced composite porous core higher-order shear deformation theory

Introduction

Sandwich functionally graded materials (FGMs) are typically three-layer materials, in which the two outer coatings are made of high-strength FGM that function to withstand almost all of the loads of the structure. The core layer is usually made of porous and lightweight materials, which increases the eccentricity of the two coatings, thereby increasing the overall stiffness of the whole structures. In addition, due to two FGM coatings, FGM sandwich materials can overcome the phenomenon of stress concentration between layers as ordinary sandwich materials. Therefore, FGM sandwich structures are increasingly being applied in many engineering fields. The analysis of the mechanical behavior of these structures has received many good and meaningful results.

For the FGM and sandwich FGM rectangular plates, studies on the buckling, postbuckling, and bending behavior of the plates^1–8 were performed using the classical shell theories (CST), first-order shear deformation theory (FSDT), and higher-order shear deformation theory (HSDT) and basing the analytical, semi-analytical or numerical approaches. Many types of loads were considered such as: axial compression load, external pressure load, thermal load. By applying different theories and methods, the natural frequency, vibration, and nonlinear dynamics of sandwich FGM panels have also been studied by Wang and Shen,^8,9 Hadji et al.,¹⁰ Singh and Harsha,¹¹ and Vyacheslav et al.¹² Based on the CST, FSDT, HSDT, and improved shear deformation theory, Alibegloo and Liew,¹³ Pandey and Pradyumma,^14,15 Dong et al.,¹⁶ Long and Tung,¹⁷ Dung and Dong,¹⁸ Dong and Dung,¹⁹ Hao et al.,²⁰ Alibegloo²¹ and Karakoti et al.²² examined the buckling, free vibration, forced vibration and nonlinear transient responses of sandwich FGM plates, doubly-curved panels and shells subjected to different load types with different sandwich FGM models.

For the sandwich FGM cylindrical shells,^23–33 the structure received a lot of attention in the world, the problems of buckling and nonlinear static postbuckling,^{23,24,26–30} nonlinear vibration problems^{23–25,27,31} and nonlinear dynamic responses^32,33 were considered with different load types, theories and methods.

For sandwich FGM spherical caps and circular plates, the bending, buckling, vibration and nonlinear dynamic problems were analyzed by Reddy et al.,³⁴ Phuong et al.,³⁵ Minh et al.,³⁶ Alipour and Shariyat,³⁷ Anh and Duc,³⁸ and Fu et al.³⁹

Based on the idea of FGM, in recent years, the types of nanocomposite including functionally graded graphene platelet reinforced composite (FG-GPLRC) and functionally graded carbon nanotube reinforced composite (FG-CNTRC) have also started to be studied.^40–42 These types of material also have the advantages of FGM such as lightness and good load capacity. In addition, for the designs in sandwich forms of these materials, the top and bottom coatings are reinforced by graphene sheets (GPLs) or carbon nanotubes (CNTs) which have many outstanding mechanical properties such as high strength, and good electrical and thermal conductivity while being light in density. Therefore, sandwich FG-GPLRC and sandwich FG-CNTRC can be considered potential materials for engineering designs in the future. In addition, shear deformation theories are increasingly showing advantages in the thermo-mechanical analysis of composite structures.⁴³

Some remarkable studies on the thermo-mechanical behavior of FG-GPLRC structures were performed. By applying HSDT, Mohammadi and Nematollahi⁴⁴ studied forced vibration of sandwich FG-GPLRC beam with metal core taking into account the effects of external excitation via nonlinear dispersion pattern using the Galerkin procedure. By using the four-variable shear and normal deformable quasi-3D beam theory, the free and forced vibration behavior of FG-GPLRC beams subjected to moving loads was mentioned by Jafari and Kiani.⁴⁵ For the FG-GPLRC plates and doubly curved shells, by using the FSDT, the dynamic response and free vibration of rectangular and skew plates were also mentioned.^46–48 The investigations of the free and forced vibrations of the sandwich cylindrical panels made from two FG-GPLRC coatings and the honeycomb auxetic core was carried out by Karimiasl and Alibeigloo.⁴⁹ In which, the basic formulations are established based on the HSDT and the gorvening equations are obtained through the generalized differential quadrature method (GDQM) and pertubation technique. The quasi-3D plates theory was also used to analyse the mechanical behavior of FG-GPLRC rectangular plates.^50,51 For the spherical caps, based on the FSDT and Galerkin method, the nonlinear dynamic responses of the porous core sandwich FG-GPLRC shell were reported by Anh et al.⁵² For FG-CNTRC plates, the postbuckling behavior and dynamic responses of rectangular and skew plates were studied using the FSDT,^53,54 and using the HSDT⁵⁵ By using the self-consistent model, the degraded stiffness of cracked layers was predicted, and the multilayer composite plates and shells containing CNTRC layers and matrix cracked fiber reinforced composite layers were considered.^56–58

It can be seen, in comparison with the rectangular plate, cylindrical panel, or cylindrical shell, only less number of studies of the spherical caps and circular plates can be observed. On the other hand, analyzing the mechanical behavior of these structures requires solving the partial differential equation system with coefficients as the function, therefore, studies on this structure, especially studies using shear deformation theories are often mathematically difficult. However, this is also a type of structure commonly used in civil engineering, space equipments, or fusion reactors. In addition, only a less number of studies on the thermo-mechanical behavior of sandwich FG-GPLRC structures, especially sandwich FG-GPLRC structures with porous core, were published.

According to the authors’ knowledge, there is no research on spherical caps and circular plates made of sandwich FG-GPLRC with porous core using the HSDT. Compared with the FSDT, the HSDT satisfies the boundary conditions for shear stress and strain at the top and bottom surfaces of the structure, leading to it not requiring the shear correction factor, and it describes the kinematics better. Therefore, in this study, an analytical approach for nonlinear buckling and postbuckling behavior of the sandwich FG-GPLRC shallow spherical caps and circular plates with porous core under mechanical and thermal loads based on HSDT is presented for the first time. Basic relations such as strain-displacement, stress-strain; internal force, and moment expressions are formulated. The potential energy expression of buckling and postbuckling of the spherical caps and circular plates is established. By applying the Ritz energy method, the load-deflection equations that are used to determine the buckling and postbuckling of the sandwich FG-GPLRC spherical caps and circular plates were obtained. In the numerical results section, the significant effects of the geometrical parameters, material parameters, elastic foundation parameters, and porous core on the nonlinear buckling behavior of the sandwich FG-GPLRC spherical caps and circular plates were found and validated.

Sandwich FG-GPLRC shallow spherical caps and circular plates with porous core

In this study, spherical caps and circular plates with thickness $h$ were made of sandwich FG-GPLRC with porous core. The upper and lower coatings are made of FG-GPLRC with the thicknesses $h_{f s}$ , and the core layer is made of porous material with a thickness $h_{p c}$ . To ensure the continuity, the core material is made from the matrix material of the FG-GPLRC. The five laws of mass fraction distribution of GPLs of upper and lower coatings are considered as follows:

+) The upper coating

‐ U D t y p e : W_{G P L} = W_{G P L}^{*};

(1)

‐ X t y p e : W_{G P L} = (| \frac{8 z + 4 h}{h - h_{p c}} - 2 |) W_{G P L}^{*} f o r \frac{- h}{2} \leq z \leq \frac{- h_{p c}}{2}

(2)

‐ O t y p e : W_{G P L} = (2 - | \frac{8 z + 4 h}{h - h_{p c}} - 2 |) W_{G P L}^{*} f o r \frac{- h}{2} \leq z \leq \frac{- h_{p c}}{2}

(3)

‐ V t y p e : W_{G P L} = (\frac{4 z + 2 h_{c}}{h_{p c} - h}) W_{G P L}^{*} f o r \frac{- h}{2} \leq z \leq \frac{- h_{p c}}{2}

(4)

‐ A t y p e : W_{G P L} = (\frac{4 z + 2 h}{h - h_{p c}}) W_{G P L}^{*} f o r \frac{- h}{2} \leq z \leq \frac{- h_{p c}}{2}

(5)

+) The lower coating

‐ U D t y p e : W_{G P L} = W_{G P L}^{*};

(6)

‐ X t y p e : W_{G P L} = (| \frac{8 z - 4 h}{h - h_{p c}} + 2 |) W_{G P L}^{*} f o r \frac{h_{p c}}{2} \leq z \leq \frac{h}{2}

(7)

‐ O t y p e : W_{G P L} = (2 - | \frac{8 z - 4 h}{h - h_{p c}} + 2 |) W_{G P L}^{*} f o r \frac{h_{p c}}{2} \leq z \leq \frac{h}{2}

(8)

‐ V t y p e : W_{G P L} = (\frac{2 h_{p c} - 4 z}{h_{p c} - h}) W_{G P L}^{*} f o r \frac{h_{p c}}{2} \leq z \leq \frac{h}{2}

(9)

‐ A t y p e : W_{G P L} = (\frac{2 h - 4 z}{h - h_{p c}}) W_{G P L}^{*} f o r \frac{h_{p c}}{2} \leq z \leq \frac{h}{2}

(10)

The elastic modulus of the FG-GPLRC coating layer is determined by the extended Halpin-Tsai model as below

E = (3 \frac{1 + χ_{1} ϑ_{1} V_{G P L}}{1 - ϑ_{1} V_{G P L}} + 5 \frac{1 + χ_{2} ϑ_{2} V_{G P L}}{1 - ϑ_{2} V_{G P L}}) \frac{E_{m,}}{8},

(11)

where

V_{G P L}

is the volume fraction of GPLs, defined as follows

V_{G P L} = \frac{W_{G P L}}{W_{G P L} + (\frac{ρ_{G P L}}{ρ_{m}}) (1 - W_{G P L})},

(12)

with

W_{G P L}

is mass fraction of GPL, defined by equations 1–10,

ρ_{m}

and

ρ_{G P L}

are the densities of the matrix and of the GPLs, respectively, and

χ_{1} = 2 \frac{a_{G P L}}{h_{G P L}}, χ_{2} = 2 \frac{b_{G P L}}{h_{G P L}}, ϑ_{1} = \frac{E_{G P L} - E_{m}}{E_{G P L} + χ_{1} E_{m}}, ϑ_{2} = \frac{E_{G P L} - E_{m}}{E_{G P L} + χ_{2} E_{m}},

(13)

with

a_{G P L}, b_{G P L}, h_{G P L}

are the length, width and thickness of the GPLs, respectively.

E_{G P L}

and

E_{m}

are the Young’s modulus of GPLs and the matrix, respectively.

The core layers of the sandwich spherical caps and circular plates are made of the same material as the matrix of the FG-GPLRC. Effective properties of the core such as Young’s modulus $E$ , coefficient of thermal expansion $α$ are determined as follows

[E, α] = [E_{m}, α_{m}] [1 - e_{0} \cos (π z / h)], - \frac{h_{p c}}{2} < z < \frac{h_{p c}}{2},

(14)

where

e_{0}

is the porosity coefficient (

0 \leq e_{0} \leq 1

With the law of material distribution of two coatings and the porous core (PC), five material types of the FG-GPLRC sandwich spherical caps and circular plates are obtained as: X-PC-X; O-PC-O, UD-PC-UD, V-PC-A, A-PC-V (Figure 1).

Figure 1.

Configuration, geometrical parameters, and material design of sandwich FG-GPLRC shallow spherical caps and circular plates with porous core.

Fundamental formulations

Consider the sandwich FG-GPLRC spherical caps with porous core with radius of cap’s curvature $R$ and radius of base $a$ , resting on the Pasternak elastic foundation under uniformly distributed external pressure on the upper surface of caps and uniformly distributed thermal load throughout the shell thickness. Assume that the axisymmetrically deformed spherical caps are placed in the coordinate system $(φ, θ, z)$ , where $z$ is in the centripetal direction, and $φ$ and $θ$ are respectively in the meridian and parallel directions of the caps. (Figure 2). For the shallow spherical caps, the parallel radius variable $r$ is introduced, defined by $r = R \sin φ$ . On the other hand, due to the shallowness of the shell, the approximations $\cos φ \approx 1$ and $R d φ = d r$ can be applied.

Figure 2.

Comparison of postbuckling curve of S-FGM spherical cap without foundation under external pressure.

By applying the HSDT and the von Kármán nonlinearities, the basic formulations and equations of the nonlinear buckling problem of the spherical caps can be established. The basic formulas and equations of the sandwich FG-GPLRC circular plates with porous core are obtained by applying $R \to \infty$ .

The displacements of any point on the spherical caps and circular plates at a distance z from the middle surface according to the HSDT are respectively as^34,39

\begin{array}{l} \bar{u} (r, z) = u (r) + z ψ (r) - z^{3} \frac{λ}{3} (ψ (r) + w (r),_{r}), \\ \bar{v} (r, z) = 0, \\ \bar{w} (r, z) = w (r) + w_{0} (r), \end{array}

(15)

where

λ = 4 / h^{2}

ψ (r)

is the rotation of a normal to the middle surface, and the initial imperfection of the spherical caps and circular plates

w_{0} (r)

is taken into account.

The strains of the spherical caps at a distance z from the middle surface are presented as

{\begin{cases} ε_{r} \\ ε_{θ} \\ ε_{r z} \end{cases}} = {\begin{cases} ε_{r}^{0} + z ψ_{, r} - \frac{λ}{3} z^{3} (ψ_{, r} + w_{, r r}) \\ ε_{θ}^{0} + z \frac{ψ}{r} - \frac{λ}{3} z^{3} (\frac{ψ}{r} + \frac{w_{, r}}{r}) \\ ε_{r z}^{0} - λ z^{2} (ψ + w_{, r}) \end{cases}},

(16)

where the relations between the strains and displacements at the middle surface of the spherical caps, taking into account the von Kármán nonlinearities, are presented as

{\begin{array}{c} ε_{r}^{0} \\ \begin{array}{l} ε_{θ}^{0} \\ ε_{r z}^{0} \end{array} \end{array}} = {\begin{array}{c} u_{, r} - \frac{w}{R} + \frac{1}{2} w_{, r}^{2} + w_{, r} w_{0, r} \\ \begin{array}{l} \frac{u}{r} - \frac{w}{R} \\ ψ + w_{, r} \end{array} \end{array}},

(17)

where

ε_{r}^{0}

and

ε_{θ}^{0}

are the normal strains at the middle surface in the radius and circumference directions, respectively, and

ε_{r z}^{0}

is the shear strain at the middle surface of spherical caps.

The relations between stresses and strains are presented by Hooke’s law for the sandwich FG-GPLRC spherical caps with porous core taking into account the temperature effect is as

{\begin{array}{c} σ_{r} \\ σ_{θ} \end{array}} = [\begin{array}{c} Q_{11} & Q_{12} \\ Q_{12} & Q_{22} \end{array}] {{\begin{array}{c} ε_{r} \\ ε_{θ} \end{array}} - {\begin{array}{c} 1 \\ 1 \end{array}} α Δ T}, σ_{r z} = Q_{44} ε_{r z .}

(18)

where

Δ T

is the uniformly temperature change from the thermally undeformed state to the thermally deformed state, and

Q_{i j}

are the reduced stiffnesses of spherical caps can be determined as

Q_{11} = Q_{22} = \frac{E}{1 - ν^{2}}, Q_{12} = \frac{E ν}{1 - ν^{2}}, Q_{44} = \frac{E}{2 (1 + ν)} .

The expressions of the internal forces including: extension forces, moments, higher-order moments, shear forces and higher-order shear forces are determined by integrating the expression of stresses in equation (18) along the thickness of the shell/plate as follows

\begin{array}{l} (N_{r}, M_{r}, L_{r}) = \int_{- \frac{h}{2}}^{\frac{h}{2}} (1, z, z^{3}) σ_{r} d z, \\ (N_{θ}, M_{θ}, L_{θ}) = \int_{- \frac{h}{2}}^{\frac{h}{2}} (1, z, z^{3}) σ_{θ} d z, \\ (Q_{r}, S_{r}) = \int_{- \frac{h}{2}}^{\frac{h}{2}} (1, z^{2}) σ_{r z} d z . \end{array}

(19)

Substituting equation (16) into (18), then substituting the resultant equations into equation (19) the force and moment expressions are obtained as

\begin{array}{l} {\begin{array}{c} N_{r} \\ N_{θ} \\ M_{r} \\ M_{θ} \\ L_{r} \\ L_{θ} \end{array}} = {\begin{array}{c} A_{11} & A_{12} & 0 & 0 & 0 & 0 \\ A_{12} & A_{22} & 0 & 0 & 0 & 0 \\ 0 & 0 & D_{11} & D_{12} & {\bar{D}}_{11} & {\bar{D}}_{12} \\ 0 & 0 & D_{12} & D_{22} & {\bar{D}}_{12} & {\bar{D}}_{22} \\ 0 & 0 & {\bar{D}}_{11} & {\bar{D}}_{12} & E_{11} & E_{12} \\ 0 & 0 & {\bar{D}}_{12} & {\bar{D}}_{22} & E_{12} & E_{22} \end{array}} {\begin{array}{c} ε_{r}^{0} \\ ε_{θ}^{0} \\ ψ_{, r} \\ ψ / r \\ - λ (ψ_{, r} + w_{, r r}) \\ - λ (ψ / r + w_{, r} / r) \end{array}} - {\begin{array}{c} Φ_{a} \\ Φ_{a} \\ Φ_{b} \\ Φ_{b} \\ Φ_{c} \\ Φ_{c} \end{array}} Δ T, \\ {\begin{array}{c} Q_{r} \\ S_{r} \end{array}} = {\begin{array}{c} H_{11} & H_{11} \\ H_{22} & H_{22} \end{array}} {\begin{array}{c} ε_{r z}^{0} \\ - λ z^{2} (ψ + w_{, r}) \end{array}}, \end{array}

(20)

With

\begin{array}{l} (A_{i j}, D_{i j}, {\bar{D}}_{i j}, E_{i j}) = \int_{- \frac{h}{2}}^{\frac{h}{2}} Q_{i j} (1, z^{2}, z^{4}, z^{6}) d z; i j = 11,12,22, \\ H_{11} = \int_{- \frac{h}{2}}^{\frac{h}{2}} Q_{44} d z - λ \int_{- \frac{h}{2}}^{\frac{h}{2}} Q_{44} z^{2} d z, H_{22} = \int_{- \frac{h}{2}}^{\frac{h}{2}} Q_{44} z^{2} d z - λ \int_{- \frac{h}{2}}^{\frac{h}{2}} Q_{44} z^{4} d z, \end{array}

(21)

and

Φ_{a} = \int_{- \frac{h}{2}}^{\frac{h}{2}} (Q_{11} + Q_{12}) α d z, Φ_{b} = \int_{- \frac{h}{2}}^{\frac{h}{2}} (Q_{11} + Q_{12}) α z d z, Φ_{c} = \int_{- \frac{h}{2}}^{\frac{h}{2}} (Q_{11} + Q_{12}) α z^{3} d z,

(22)

Boundary conditions, solution forms and the Ritz energy method

In this paper, the spherical caps and circular plates are considered with the clamped edges at the base circumference. The boundary conditions in these cases are

\begin{array}{l} ψ = 0, u = 0, w_{, r} = 0, w = f i n i t e, a t r = 0, \\ w = 0, ψ = 0, u = 0, w_{, r} = 0, a t r = a . \end{array}

(23)

To satisfy the boundary condition (23), the approximate solutions of the problem are chosen as^35,36,38

u = U \frac{r (a - r)}{a^{2}}, ψ = Ψ \frac{r (a^{2} - r^{2})}{a^{3}}, [w, w_{0}] = [W, ξ h] \frac{{(a^{2} - r^{2})}^{2}}{a^{4}},

(24)

where

U, Ψ,

and

W

are the amplitudes of the displacement, rotation and deflection components, respectively, and

ξ

is the size of the imperfection and is assumed to be small.

The Ritz energy method is applied by minimizing the total potential energy expression $Π$ , i.e.

\frac{\partial Π}{\partial U} = \frac{\partial Π}{\partial Ψ} = \frac{\partial Π}{\partial W} = 0,

(25)

where the total potential energy

Π

is determined to be the difference of strain energy

Π_{i n}

and work done of external forces

Π_{e x}

. Their expressions are presented as

Π_{i n} = π \int_{- \frac{h}{2}}^{\frac{h}{2}} \int_{0}^{a} [σ_{r} (ε_{r} - α Δ T) + σ_{θ} (ε_{θ} - α Δ T) + σ_{r z} ε_{r z}] r d r d z,

(26)

and

Π_{e x} = 2 π \int_{0}^{a} q w r d r - π \int_{0}^{a} {[K_{1} w - K_{2} (w_{, r r} + \frac{1}{r} w_{, r})] w} r d r,

(27)

where

K_{1}

(N/m³) and

K_{2}

(N/m) are the stiffnesses of Pasternak elastic foundation.

The expression of total potential energy $Π$ can be obtained as

Π = Π_{i n} - Π_{e x}

(28)

By using the equations 17–20, the total potential energy (28) can be rewritten by three unknown functions $u, ψ$ and $w$ . Subtituting the solution forms (24) into the obtained equation and then applying the equation (25), obtained as

l_{11} U + l_{12} W + l_{13} W^{2} + l_{14} W ξ h = 0,

(29)

l_{21} Ψ + l_{22} W = 0,

(30)

\begin{array}{l} l_{12} U + l_{31} Ψ + l_{32} W + l_{33} K_{1} W + l_{34} K_{2} W + l_{35} W^{2} + l_{36} W^{3} + l_{37} W U + l_{38} W Ψ + l_{39} Ψ ξ h \\ + l_{310} W ξ h + l_{311} W ξ^{2} h^{2} + l_{312} W^{2} ξ h + l_{313} Δ T + l_{314} Δ T ξ h + l_{314} W Δ T - \frac{π}{3} q a^{2} = 0, \end{array}

(31)

Where

l_{11} = \frac{A_{11}}{3} + \frac{A_{22}}{6}, l_{12} = - \frac{153 A_{11} + 1122 A_{12} + 969 A_{22}}{5355 R} a, l_{13} = \frac{82 A_{12} - 46 A_{11}}{315 a}, l_{14} = \frac{164 A_{12} - 92 A_{11}}{315 a}, l_{21} = (\frac{a^{2}}{12} H_{22} + \frac{12 E_{11} + 4 E_{22}}{108}) λ^{2} - [(H_{11} + H_{22}) \frac{a^{2}}{12} + \frac{24 {\bar{D}}_{11} + 8 {\bar{D}}_{22}}{36}] λ + \frac{a^{2} H_{11} + 12 D_{11} + 4 D_{22}}{12},

l_{22} = - H_{22} \frac{λ^{2}}{3} a + (H_{11} + H_{22}) \frac{λ}{3} a - \frac{H_{11}}{3} a - (3 E_{11} + E_{22}) \frac{4 λ^{2}}{27 a} + (3 {\bar{D}}_{11} + {\bar{D}}_{22}) \frac{4 λ}{9 a}, l_{31} = \frac{36 A_{11} + 72 A_{12} + 36 A_{22}}{180 R^{2}} a^{2} + \frac{4}{3 a} [H_{22} λ^{2} - (H_{11} + H_{22}) λ + H_{11}] + \frac{λ^{2}}{9 a^{2}} (12 E_{11} + 4 E_{22}), l_{32} = \frac{a^{2}}{5}, l_{33} = \frac{4}{3}, l_{34} = - \frac{4}{5 R} (A_{11} + A_{12}), l_{35} = \frac{128 A_{11}}{105 a^{2}}, l_{36} = \frac{2 a^{2}}{9}, l_{37} = \frac{164 A_{12} - 92 A_{11}}{315 a},

l_{38} = - \frac{16}{15 R} (A_{11} + A_{12}), l_{39} = \frac{256 A_{11}}{105 a^{2}}, l_{310} = \frac{128 A_{11}}{35 a^{2}}, l_{311} = \frac{2 a^{2} Φ_{a}}{3 R}, l_{312} = - \frac{4 Φ_{a}}{3} .

The expressions of $U$ and $Ψ$ are obtained from equations (29) and (30), substituting them into the equation (31), the nonlinear algebraic equilibrium equation of the shell can be rewritten as

\begin{array}{l} n_{1} h^{3} {\bar{W}}^{3} + n_{2} h^{2} {\bar{W}}^{2} + (n_{3} + K_{1} l_{32} + K_{2} l_{33} +) h \bar{W} + n_{4} \bar{W} ξ h^{2} \\ + n_{5} \bar{W} ξ h^{3} + n_{6} \bar{W} ξ^{2} h^{3} + l_{312} \bar{W} h Δ T + l_{311} ξ h Δ T + l_{310} Δ T - q \frac{a^{2}}{3} = 0, \end{array}

(32)

where

\bar{W} = W / h,

and

n_{1} = l_{37} m_{12} + l_{35}, n_{2} = l_{13} m_{12} + l_{37} m_{11} + l_{34}, n_{3} = l_{13} m_{11} - l_{22} + l_{31},

n_{4} = l_{13} m_{13} + l_{37} m_{11} + l_{38}, n_{5} = l_{37} m_{12} + l_{37} m_{13}, n_{6} = l_{37} m_{13} + l_{39},

m_{11} = - l_{13} l_{21} / l_{11} l_{22}, m_{12} = - l_{14} l_{21} / l_{11} l_{22}, m_{13} = - l_{15} l_{21} / l_{11} l_{22},

The relation between mechanical load-deflection is obtained from equation (32) is

\begin{array}{l} q = \frac{3}{a^{2}} [n_{1} h^{3} {\bar{W}}^{3} + n_{2} h^{2} {\bar{W}}^{2} + (n_{3} + K_{1} l_{32} + K_{2} l_{33} +) h \bar{W} + n_{4} \bar{W} ξ h^{2} \\ + n_{5} \bar{W} ξ h^{3} + n_{6} \bar{W} ξ^{2} h^{3} + l_{312} \bar{W} h Δ T + l_{311} ξ h Δ T + l_{310} Δ T], \end{array}

(33)

Similarly, from equation (32), the thermal load–deflection relation of the FG-GPLRC spherical caps with porous is obtained as

\begin{array}{l} Δ T = - \frac{1}{l_{312} \bar{W} h + l_{311} ξ h + l_{310}} [n_{1} h^{3} {\bar{W}}^{3} + n_{2} h^{2} {\bar{W}}^{2} + (n_{3} + K_{1} l_{32} + K_{2} l_{33}) h \bar{W} \\ + n_{4} \bar{W} ξ h^{2} + n_{5} \bar{W} ξ h^{3} + n_{6} \bar{W} ξ^{2} h^{3} - q \frac{a^{2}}{3}] . \end{array}

(34)

In the case of a perfect circular plate subjected only to thermal loads, the critical thermal buckling load is obtained by applying $\bar{W} \to 0$ and $ξ = 0$ , as

Δ T_{c r} = - (K_{1} l_{32} + K_{2} l_{33} + n_{3}) / l_{312} .

(35)

In order to analyze the nonlinear buckling of the sandwich FG-GPLRC spherical caps and circular plates with porous core under mechanical and thermal loads placed on the Pasternak elastic foundation, equations 33–35 are used. Equations (33) and (34) are used to investigate the postbuckling curves of the sandwich FG-GPLRC spherical caps and circular plates with porous core, and equation (35) are used to determine the critical thermal buckling loads of the perfect sandwich FG-GPLRC circular plates with porous core.

Numerical results and discussions

Validation results

To validate the present approach, the comparisons of deflection of isotropic circular plates with the results of Reddy⁵⁹ are presented in Table 1. The results of Reddy⁵⁹ are exactly obtained using the classical plate theory for linear bending problem. The nonlinear results of the present research can be obtained from equation (33), and the linear results are determined by eliminating the nonlinear terms of equation (33). The postbuckling curve of the Sigmoid FGM (S-FGM) spherical cap without foundation under external pressure of the present approach is compared with that of Anh and Duc³⁸ in Figure 2. The work of Anh and Duc³⁸ applied the FSDT and solved the equilibrium equations by using the Galerkin method.

Table 1.

Comparison of maximal deflection $W$ (mm) of isotropic circular plates under external pressure ( $E =$ 207.78 GPa, $h =$ 0.01 m, q = 10 kN/m2).

$h / a$ .	0.01	0.02	0.04	0.06	0.08	0.1
Reddy⁵⁹	8.212	0.513	3.208 × 10⁻²	6.336 × 10⁻³	2.005 × 10⁻³	8.212 × 10⁻⁴
Present (linear)	8.212	0.513	3.208 × 10⁻²	6.336 × 10⁻³	2.005 × 10⁻³	8.212 × 10⁻⁴
Present (nonlinear)	6.423	0.512	3.208 × 10⁻²	6.336 × 10⁻³	2.005 × 10⁻³	8.212 × 10⁻⁴

Clearly, the results of these comparisons affirm the validity of the present approach.

Results and discussions

In this paper, the matrix is made of copper, and the material properties and efficiency parameters of GPLs and matrix are chosen as reported by Wang et al.⁶⁰

In this section, the effects of mass fraction of GPLs, porosity coefficient, core thickness, and foundation stiffnesses on the critical thermal buckling loads of sandwich FG-GPLRC circular plates with porous core for five distributed laws of GPLs are investigated in Tables 2–5. It can be seen, in all examined cases, that the maximum critical thermal buckling loads are obtained with the V-PC-A circular plates, next are X-PC-X, UD-PC-UD, and O-PC-O circular plates, respectively, and the smallest corresponds to the A-PC-V circular plates. All five distributed laws of the GPLs are designed to be symmetrical about the middle surface, that is, in these cases, the middle surface coincides with the neutral surface. Therefore, the further away the GPLs are placed from the middle surface, the greater the stiffnesses of the structure, which can lead to larger critical thermal buckling loads of the structure.

Table 2.

Effects of mass fraction of GPLs on the critical thermal buckling loads $Δ T_{c r}$ (K) of sandwich FG-GPLRC circular plates with porous core ( $a / h = 20$ , $h = 0.03$ (m), $h_{p c} = h_{f s} = h / 3 ξ$ = 0, $e_{0} = 0.3$ , $K_{1} = 10$ MN/m³, $K_{2} = 0.5$ MN/m).

$W_{G P L}^{*} (%)$ .	V-PC-A	X-PC-X	UD-PC-UD	O-PC-O	A-PC-V
1	214.5271	198.8875	196.0500	194.3994	178.6217
2	240.6123	217.1617	212.2734	210.4066	186.5703
3	261.3063	233.0453	226.7070	224.8774	195.9621
5	294.5025	260.5149	252.6877	250.6367	215.4757
7	321.4739	284.1766	276.2405	273.2863	234.3501

Table 3.

Effects of porosity coefficient $e_{0}$ on the critical thermal buckling loads $Δ T_{c r}$ (K) of sandwich FG-GPLRC circular plates with porous core ( $a / h = 20$ , $h = 0.03$ (m), $h_{p c} = h_{f s} = h / 3 ξ$ = 0, $W_{G P L}^{*}$ = 3%, $K_{1} = 10$ MN/m³, $K_{2} = 0.5$ MN/m).

$e_{0}$	V-PC-A	X-PC-X	UD-PC-UD	O-PC-O	A-PC-V
0.1	249.3736	222.4308	216.7796	214.6439	187.0774
0.2	255.4817	227.8648	221.8666	219.8831	191.6269
0.3	261.3063	233.0453	226.7070	224.8774	195.9621
0.4	266.7865	237.9179	231.2514	229.5744	200.0376
0.5	271.8605	242.4278	235.4502	233.9213	203.8072

Table 4.

Effects of core thickness on the critical thermal buckling loads $Δ T_{c r}$ (K) of sandwich FG-GPLRC circular plates with porous core ( $a / h = 20$ , $h = 0.03$ (m), $h_{f s} = (h - h_{p c}) / 2$ , $ξ$ = 0, $W_{G P L}^{*}$ = 3%, $K_{1} = 10$ MN/m³, $K_{2} = 0.5$ MN/m).

$h_{p c}$ (m)	V-PC-A	X-PC-X	UD-PC-UD	O-PC-O	A-PC-V
0.005	244.0107	212.2919	202.6234	198.2568	165.4138
0.01	261.3063	233.0453	226.7070	224.8774	195.9621
0.015	275.0139	253.2214	249.6890	249.2302	227.1180
0.02	280.0018	266.7105	265.5230	265.3055	251.9017

Table 5.

Effects of foundation stiffnesses on the critical thermal buckling loads $Δ T_{c r}$ (K) of sandwich FG-GPLRC circular plates with porous core ( $h = 0.03$ (m), $h_{p c} = h_{f s} = h / 3$ , $ξ$ = 0, $e_{0} = 0.3$ , $W_{G P L}^{*}$ = 3%).

$a / h$	Distributed law of GPLs	Without foundation	$K_{1} = 10$ MN/m³, $K_{2} = 0$ MN/m	$K_{1} = 10$ MN/m³, $K_{2} = 0.5$ MN/m
20	UD-PC-UD	219.91	223.43 (1.60%)^a	226.70 (3.09%)
	X-PC-X	225.96	229.64 (1.63%)	233.04 (3.13%)
	V-PC-A	254.22	257.90 (1.45%)	261.30 (2.78%)
	A-PC-V	188.88	192.55 (1.94%)	195.96 (3.75%)
	O-PC-O	217.79	221.47 (1.69%)	224.87 (3.25%)
30	UD-PC-UD	97.73	105.68 (8.13%)	108.94 (11.47%)
	X-PC-X	100.43	108.69 (8.22%)	112.10 (11.62%)
	V-PC-A	112.98	121.26 (7.33%)	124.66 (10.34%)
	A-PC-V	83.94	92.21 (9.85%)	95.62 (13.91%)
	O-PC-O	96.79	105.07 (8.55%)	108.47 (12.07%)
40	UD-PC-UD	54.97	69.09 (25.69%)	72.36 (31.64%)
	X-PC-X	56.49	71.19 (26.02%)	74.59 (32.04%)
	V-PC-A	63.55	78.25 (23.13%)	81.66 (28.50%)
	A-PC-V	47.22	61.92 (31.13%)	65.32 (38.33%)
	O-PC-O	54.44	69.15 (27.02%)	72.55 (33.27%)

^aEffects of foundation in comparison with the without foundation cases.

From Tables 2–4, it can be observed that, with all five material distribution laws, the critical thermal buckling loads of the sandwich FG-GPLRC circular plates with porous core are directly proportional to the mass fraction of GPLs, porosity coefficient, and the thickness of the porous core. Clearly, all the examined results in this paper show a large influences of mass fraction of GPLs, porosity coefficient, and core layer thickness on the critical thermal buckling loads of the circular plates can be observed.

Table 5 investigates the critical thermal buckling loads of the FG-GPLRC plates with porous core when the $a / h$ geometrical parameter changes with three cases of foundation stiffnesses ( $K_{1} = 0$ MN/m³ and $K_{2} = 0$ MN/m, $K_{1} = 10$ MN/m³ and $K_{2} = 0$ MN/m, $K_{1} = 10$ MN/m³ and $K_{2} = 0.5$ MN/m). The critical thermal buckling loads of the circular plates FG-GPLRC plates with porous core strongly increases when the foundation stiffnesses increase. The effects of the elastic foundation increases largely as the $a / h$ ratio increases and the effect of the foundation is greatest in the case of A-PC-V circular plates. In addition, as the $a / h$ ratio increases, i.e., as the plate becomes thinner, the critical thermal buckling loads of the sandwich circular plate with porous core decreases.

Figure 3(a)–(d) examine the $q - W$ postbuckling curves (Figure 3(a)) and $Δ T - W$ postbuckling curves (Figure 3(b)) of sandwich FG-GPLRC spherical caps with porous core for different distributed laws of GPLs. The obtained results show that the mechanical postbuckling strength of UD-PC-UD spherical cap is the largest in the small domain of deflection, oppositely, in the large domain of deflection, the postbuckling strength of V-PC-A spherical cap is dominant (Figure 3(a)). For thermal postbuckling strength, the largest postbuckling strengths are obtained in the cases of V-PC-A spherical caps, and the weakest postbuckling strengths correspond to the cases of A-PC-V spherical caps (Figure 3(b)). The extreme buckling type is observed in the cases of spherical caps subjected to external pressure in thermal environment (Figure 3(a)), and the bifurcation buckling type is obtained for circular perfect plates subjected to thermal loads.

Figure 3.

Effects of the GPLs distributed types and mass fraction of GPLs on the postbuckling curves of sandwich caps and plates with porous core (a) (b) (c) (d) (e) (f).

Additionally, in the cases of the spherical caps subjected to both external pressure and thermal load, the pre-deflection is created (Figure 3(c)), the postbuckling curves are derived from a point on the horizontal axis. Figure 3(c)–3(f) show the significant effects of GPL mass fraction on the $q - W$ and $Δ T - W$ postbuckling curves. While the GPL mass fraction increases the external pressure load carrying capacity of the FG-GPLRC with porous core.

The mechanical and thermal postbuckling strength of sandwich FG-GPLRC spherical caps increases when the mass fraction of GPLs increases. Especially, the $Δ T - W$ postbuckling curves are in the negative region of deflection only for spherical caps. In the remaining cases, the postbuckling curve is in the positive region of the deflection. It can be explained that due to the curvature of caps, the increase of temperature causes the spherical caps to expand and the deflection to develop in the negative region.

Figure 4 shows the effects of porous core thickness $h_{p c}$ (Figure 4(a)–(d)) and porosity coefficient $e_{0}$ (Figure 4(e) and (f)) on the $q - W$ and $Δ T - W$ postbuckling curves of the sandwich spherical caps with porous core. Clearly that, as the porous core thickness or porosity coefficient increases, the $q - W$ postbuckling curves become significantly lower, in other words, the porous core thickness or porosity increases significantly reduces the pressure carrying capacity of shells/plates (Figure 4(a),(c),(e)).

Figure 4.

Effects of core thickness on the postbuckling curves of sandwich caps and plates with porous core (a) (b) (c) (d) (e) (f).

However, the opposite trend occurs for the $Δ T - W$ postbuckling curves (Figure 4(b),(d),(f)). That is, when the porous core thickness or the porosity coefficient increase, the thermal postbuckling strengths of the sandwich FG-GPLRC spherical caps and circular plates with porous core increase. A similar tendencies can be observed in Tables 3 and 4 for the critical thermal buckling loads of plates.

Figure 5(a)–(d) investigate the influence of $a / h$ geometrical parameter on the $q - W$ and $Δ T - W$ postbuckling curves and of the sandwich FG-GPLRC spherical caps with different cases of the $a / R$ ratio. When $R \to \infty$ , the cases of circular plates are obtained (Figure 5(e) and (f)). Obviously, for all investigated values of the $a / R$ ratio, the pressure load carrying capacity of the perfect or imperfect spherical caps/circular plates is directly proportional to the $a / h$ geometrical parameter. In addition, it can be seen that the $q - W$ and $Δ T - W$ postbuckling curves and of the imperfect circular plates and spherical caps have the same form as the postbuckling curves of the perfect circular plates and spherical caps, except for the case of Figure 5(d). In this figure, the bifurcation type of buckling behavior of the perfect circular plates under thermal loads can be observed. The $Δ T - W$ postbuckling curves of the imperfect circular plate start from the origin, and the postbuckling curves of the perfect circular plate start from the buckling value at the vertical axis.

Figure 5.

Effects of $a / h$ ratio on the postbuckling curves of sandwich FG-GPLRC spherical caps and circular plates with porous core (a) (b) (c) (d) (e) (f).

The influences of foundation stiffnesses on the $q - W$ and $Δ T - W$ postbuckling curves and of the sandwich FG-GPLRC circular plates and spherical caps with porous core are shown in Figure 6(a)–(d). In all the investigation cases, including: perfect and imperfect spherical caps, perfect and imperfect circular plates, the postbuckling curves of the plate/shell increase as the elastic foundation stiffnesses increase. The lowest $q - W$ và $Δ T - W$ postbuckling curves correspond to the case of circular plates/spherical caps. When the elastic foundation stiffnesses increase, the general stiffnesses of the structures increase, thereby the load carrying capacity of the circular plates and spherical caps increases.

Figure 6.

Effects of foundation stiffnesses on the postbuckling curves of sandwich FG-GPLRC spherical caps and circular plates with porous core.

Conclusions

A new analytical approach to analyze the buckling and postbuckling behavior of circular plates and shallow spherical caps made of two FG-GPLRC coatings with porous core has been presented. In which, the basic equations and relations are established based on the HSDT. By applying the Ritz method, the expressions for determining the critical thermal loads of the circular plates and the mechanical and thermal postbuckling curves and of the circular plates and the spherical caps are obtained. Some obtained notable conclusions are as follows:

(i) Perfect and imperfect sandwich FG-GPLRC circular plates and spherical caps with porous core with V-PC-A distribution law have the best load-carrying capacity in almost investigated cases. The only case with an adverse outcome for V-PC-A spherical cap is the mechanical postbuckling strength in the small domain of deflection.

(ii) Porous core characteristics such as: core thickness and porosity coefficient are inversely proportional to the pressure load capacity of sandwich FG-GPLRC circular plates and spherical caps with porous core, oppositely, those are directly proportional to the thermal load capacity of the circular plates and spherical caps.

(iii) The influences of the geometrical, material and elastic foundation on the buckling and postbuckling responses of the sandwich FG-GPLRC circular plates and spherical caps with porous core are significant.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is funded by University of Transport Technology (UTT) under grant number DTTD2021-05.

ORCID iDs

Minh Duc Vu

Thi Phuong Nguyen

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Nonlinear thermo-mechanical buckling and postbuckling of sandwich FG-GPLRC spherical caps and circular plates with porous core by using higher-order shear deformation theory

Abstract

Keywords

Introduction

Sandwich FG-GPLRC shallow spherical caps and circular plates with porous core

Fundamental formulations

Boundary conditions, solution forms and the Ritz energy method

Numerical results and discussions

Validation results

Results and discussions

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References