Thermomechanical postbuckling of functionally graded graphene-reinforced composite laminated toroidal shell segments surrounded by Pasternak’s elastic foundation

Abstract

Nonlinear buckling and postbuckling analysis of functionally graded graphene-reinforced composite (FG-GRC) laminated toroidal shell segments subjected to external pressure surrounded by elastic foundations and exposed to thermal environment are presented in this article. Governing equations for toroidal shell segments are based on the Donnell shell theory taking into account geometrical nonlinearity term in von Kármán sense with shell–foundation interaction modeled by Pasternak’s elastic foundation. Three-term solution form of deflection and stress function are chosen, and Galerkin method is applied to obtain the nonlinear load–deflection relation. Numerical investigations show the effects of graphene volume fraction, graphene distribution types, geometrical properties, elastic foundation, and thermal environments on the linear and nonlinear buckling and postbuckling behaviors of FG-GRC laminated toroidal shell segments.

Keywords

Nonlinear buckling and postbuckling functionally graded graphene-reinforced composite toroidal shell segment thermal environment elastic foundation external pressure

Introduction

In recent years, many authors are mostly interested in the linear and nonlinear buckling and postbuckling behavior of many types of revolution shells as cylindrical shells and toroidal shell segments made by isotropic and traditional composite materials. Dutta et al.¹ studied the nonlinear axisymmetric stability of toroidal pipe-reducer system. Numerical solutions are obtained using the multi-segment method of integration, and critical pressure buckling loads of the toroidal reducers are determined according to the Thompson’s theorems I and II. Shen^2
–4 investigated the nonlinear large deflection buckling analysis of stiffened laminated composite cylindrical shells of finite length subjected to thermal loads, combined external pressure and thermal loading, and combined external liquid pressure and axial compression using the singular perturbation technique and the boundary layer theory. A numerical study of linear elastic and elastoplastic buckling behavior of circular and elliptical cross-sectional steel toroidal shells subjected to uniform external pressure with closed cross-sections was proposed by Blachut and Jaiswal.⁵ Du et al.⁶ manufactured a whole welding steel toroidal model with the ring stiffeners, tested it until collapse in pressure chamber, and analyzed it using nonlinear finite element method, and the cause and collapse mode of tested stiffened toroidal shell model were discussed. Large deformation-free vibration of toroidal shells was investigated by Senjanović et al.⁷ using the Rayleigh–Ritz energy method and Fourier series. Using the first-order shear deformation theory, Masumi et al.⁸ studied the thermoelastic stress analysis in composite cylindrical vessel with metallic liner. Using three-dimensional elasticity theory and generalized differential quadrature method, the free vibration response of sandwich cylindrical shells with functionally graded material (FGM) face sheets resting on Pasternak foundation was studied by Kamarian et al.⁹ Bouazza et al.^10,11 studied the thermal buckling of FGM plates under uniform temperature rise and linear temperature rise through the thickness using classical plate theory.

FGMs and functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are the advanced composite materials in which thermomechanical properties are continuously varied through the thickness of structure. Research on thermomechanical behavior of FGM structure has been an exciting field in the last decade. Huang and Han^12,13 studied the nonlinear buckling and postbuckling of FGM cylindrical shells under axial compression and radial pressure using Donnell shell theory and three-term solution form. The nonlinear static and dynamic buckling behavior of imperfect orthogonal eccentrically stiffened FGM cylindrical shells subjected to axial compression was considered in the work of Bich et al.¹⁴ The dynamic critical buckling loads of shells under step loading of infinite duration were found corresponding to the load value of sudden jump in the average deflection and those of shells under linear-time compression were investigated according to Budiansky–Roth criterion. Dung and Nam¹⁵ investigated an analytical approach of orthogonal eccentrically stiffened FGM cylindrical shells surrounded by Pasternak’s elastic foundation and subjected to external pressure. The approximate three-term solution of deflection shape was chosen and the frequency–amplitude relation of nonlinear vibration was obtained in explicit form. Recently, Nam et al.¹⁶ and Phuong et al.¹⁷ studied the nonlinear buckling and postbuckling of FGM and multilayer FGM cylindrical shells reinforced by orthogonal and spiral stiffeners in thermal environment subjected to torsional load with and without elastic foundation. Ninh and Bich^18
–20 investigated the nonlinear buckling and vibration of FGM ring and stringer stiffened and unstiffened FGM cylindrical shells and toroidal shell segments under thermal and mechanical loads surrounded by Pasternak’s elastic foundation using Donnell shell nonlinear theory with Stein and McElman assumption and Galerkin procedure. Thang and Trung²¹ studied the nonlinear dynamic response of sigmoid FGM and FGM toroidal shell segments reinforced by ring and stringer stiffeners, respectively. Donnell shell theory, Galerkin procedure, and Runge–Kutta method were used. Hieu and Tung²² investigated the nonlinear buckling of FG-CNTRC cylindrical shell surrounded by an elastic medium and subjected to mechanical loads in thermal environments. The nonlinear stability of FG-CNTRC cylindrical panels with elastically restrained straight edges subjected to combined thermomechanical loading conditions is investigated in the work of Trang and Tung.²³ Bouazza et al.^24,25 investigated the thermal buckling and free vibration of traditional laminated composite plates using refined hyperbolic shear deformation theory and two-variable simplified nth-higher-order theory.

In recent years, with the extraordinary properties, graphene has attracted more attention of many authors in the world. Based on the idea of FGM, graphene is used to reinforce in the polymeric matrix, and a new material, namely, functionally graded graphene platelet (FG-GPL), is formed. Barati and Zenkour²⁶ analyzed the free vibration behavior of FG-GPL-reinforced cylindrical shells with uniform, symmetric, and asymmetric porosity distributions. Wang et al.^27,28 investigated the eigenvalue buckling and torsional buckling of FG-GPL-reinforced cylindrical shell with cutout through finite element method. Bouazza et al.^29,30 investigated the thermal buckling of nanoplates with and without the elastic foundation effects.

A more complex composite reinforced by graphene is piecewise functionally graded graphene-reinforced composite (FG-GRC) laminated structure, where the GRC layer is reinforced by zigzag or armchair graphene sheet with strong orthotropic behavior in two transversal directions. Shen et al.^31
–33 studied the nonlinear thermal buckling, vibration, and bending of FG-GRC laminated plates resting on elastic foundation by two-step perturbation technique. Shen and Xiang^34
–36 and Shen et al.³⁷ investigated the postbuckling behavior of FG-GRC laminated circular cylindrical shells subjected to axial compression, external pressure, and thermal load, respectively. Kiani and Mirzaei^38,39 and Kiani^40,41 studied the buckling behavior and free vibration of FG-GRC laminated beams, plates, and conical shells under mechanical and thermal loads.

To the best of the author’s knowledge, research studies on the analytical solution for FG-GRC shells have so far been limited. Previous studies only mentioned about FG-GRC cylindrical shells. A more complex structure is considered in this article: the nonlinear buckling analysis of FG-GRC laminated toroidal shell segments surrounded by Pasternak’s elastic foundations, exposed to thermal environments, and subjected to uniform external pressure is investigated by an analytical approach. The equilibrium and compatibility equations are established according to the Donnell’s shell theory with von Kármán nonlinearity sense and Stein and McElman assumption.⁴² The prebuckling and linear and nonlinear postbuckling terms are considered in deflection solution form, and the Galerkin procedure is applied to obtain the critical buckling load expression and the nonlinear external pressure–deflection relation. The effects of graphene volume fraction, distribution types, geometrical parameters, temperature conditions, and elastic foundations on the linear and nonlinear buckling behavior of FG-GRC laminated toroidal shell segments are numerically investigated and discussed.

FG-GRC laminated toroidal shell segments surrounded by elastic foundation

This article investigates the nonlinear buckling behavior of FG-GRC laminated toroidal shell segments subjected to external pressure load of uniform distribution intensity $q_{0}$ as shown in Figure 1. The shells are surrounded by Pasternak elastic foundation as presented in Figure 2. Governing equations of shallow toroidal shell segments are established according to the Stein and McElman assumption,⁴² represented as follows:

Figure 1.

Configuration and coordinate system of FG-GRC laminated toroidal shell segments.

Figure 2.

Two-parameter Pasternak foundation model of toroidal shell segments.

The radius of parallel circle R ₀ can be expressed as

R_{0} = a - R (1 - sin φ)

with the shallow toroidal shell along the meridian, the angle $φ$ approximate $π / 2$ . Therefore, $sin φ \approx 1, cos φ \approx 0$ and $R_{0} \approx R$ , a simpler coordinate is applied

d x = R d φ, d y = a d θ

The origin of coordinate O locates on the midplane and at the end of the shell, and $x, y$ , and z axes are in the longitudinal, circumferential, and inward radial directions, respectively. R and a are the circumferential and longitudinal radii, respectively, $h$ is the total thickness, and $L$ is the longitudinal length of the shell. Toroidal shell segment is surrounded by two-parameter Pasternak elastic foundation and the shell–foundation interaction is expressed by

q_{e} = K_{1} w - K_{2} (w_{, x x} + w_{, y y})

where K ₁ (N m⁻³) is Winkler foundation modulus and K ₂ (N m⁻¹) is the shear layer foundation stiffness of Pasternak’s elastic foundation model.

The graphene reinforcement in the $x$ direction is denoted by (0-layer), while the $y$ direction is denoted by (90-layer). Each layer in the FG-GRC toroidal shell segment structure has different direction arrangement, divided into three samples of toroidal shell segments as (0)₁₀, (0/90/0/90/0)_S, and (0/90)_5T. In addition, five different graphene distribution types, namely, UD, FG-X, FG-V, FG-Λ, and FG-O, are considered (see Figure 3). The graphene volume fractions are gradually increased from inner layer to outer layer for FG-Λ distribution type, and vice versa, for FG-V distribution type, and the 10 layers of the two types are denoted by [(0.11)₂/(0.09)₂/(0.07)₂/(0.05)₂/(0.03)₂] referred to as FG-V and [(0.03)₂/(0.05)₂/(0.07)₂/(0.09)₂/(0.11)₂] referred to as FG-Λ. For FG-O and FG-X distribution types, the inner five layers have graphene volume fractions distributed symmetrically with the outer five layers, [0.11/0.09/0.07/0.05/0.03]_S referred to as FG-O distribution type and [0.03/0.05/0.07/0.09/0.11]_S referred to as FG-X distribution type. For UD distribution type, the graphene volume fraction is distributed uniformly across all layers ( $V_{G} = 0.07$ ). Note that 0.03, 0.05, 0.07, 0.09, and 0.11 are the graphene volume fraction of the constituent layers.

Figure 3.

Distribution types of graphene-reinforced composite laminates: (a) UD, (b) FG-X, (c) FG-O, (d) FG-V, and (e) FG-Λ.

Young’s moduli and shear modulus of the FG-GRC layer are estimated according to the extended Halpin–Tsai model, expressed as

\begin{array}{l} E_{11} = η_{1} \frac{1 + 2 (a_{G} / h_{G}) γ_{11}^{G} V_{G}}{1 - γ_{11}^{G} V_{G}} E^{m} \\ E_{22} = η_{2} \frac{1 + 2 (b_{G} / h_{G}) γ_{22}^{G} V_{G}}{1 - γ_{22}^{G} V_{G}} E^{m} \\ G_{12} = η_{3} \frac{1}{1 - γ_{12}^{G} V_{G}} G^{m} \end{array}

with

\begin{array}{l} γ_{11}^{G} = \frac{E_{11}^{G} / E^{m} - 1}{E_{11}^{G} / E^{m} + 2 a_{G} / h_{G}} \\ γ_{22}^{G} = \frac{E_{22}^{G} / E^{m} - 1}{E_{22}^{G} / E^{m} + 2 b_{G} / h_{G}} \\ γ_{12}^{G} = \frac{G_{12}^{G} / G^{m} - 1}{G_{12}^{G} / G^{m}} \end{array}

where $a_{G}$ , $b_{G}$ , and $h_{G}$ are the length, width, and effective thickness of the graphene sheet, respectively. $(E^{m}, G^{m})$ and $(E_{11}^{G}, E_{22}^{G}, G_{12}^{G})$ are the orthotropic Young’s moduli and shear moduli of the matrix and the graphene sheet, respectively. $V_{G}$ and $V_{m}$ are the graphene and matrix volume fractions, respectively, with the condition $V_{G} + V_{m} = 1$ . In equation (4), the graphene efficiency parameters $η_{j} (j = 1, 2, 3)$ are determined according to the graphene volume fraction $V_{G}$ in Table 1.

Table 1.

The graphene efficiency parameters of GRC structures.³⁵

T (K)	V_G	$η_{1}$	$η_{2}$	$η_{3}$
300	0.03	2.929	2.855	11.842
	0.05	3.068	2.962	15.944
	0.07	3.013	2.966	23.575
	0.09	2.647	2.609	32.816
	0.11	2.311	2.260	33.125
400	0.03	2.977	2.896	13.928
	0.05	3.128	3.023	15.229
	0.07	3.060	3.027	22.588
	0.09	2.701	2.603	28.869
	0.11	2.405	2.337	29.527
500	0.03	3.388	3.382	16.712
	0.05	3.544	3.414	16.018
	0.07	3.462	3.339	23.428
	0.09	3.058	2.936	29.754
	0.11	2.736	2.665	30.773

GRC: graphene-reinforced composite.

Thermal expansion coefficients of GRC layer in the x and y directions are taken according to the Schapery model, as

\begin{array}{l} α_{11} = \frac{V_{G} E_{11}^{G} α_{11}^{G} + V_{m} E^{m} α^{m}}{V_{G} E_{11}^{G} + V_{m} E^{m}} \\ α_{22} = (1 + ν_{12}^{G}) V_{G} α_{22}^{G} + (1 + ν^{m}) V_{m} α^{m} - ν_{12} α_{11} \end{array}

where $α_{11}^{G}$ , $α_{22}^{G}$ , and $α^{m}$ are the thermal expansion coefficients and $ν_{12}^{G}$ and $ν^{m}$ are the Poisson’s ratios of graphene sheet and matrix, respectively. The material properties are assumed to be $ν^{m} = 0.34$ , $α^{m} = 45 (1 + 0.0005 Δ T) \times 10^{- 6} K^{- 1}$ , and $E^{m} = (3.52 - 0.0034 T) GPa$ , in which $T = T_{0} + Δ T$ and $T_{0} = 300 K$ (room temperature). The Poisson’s ratio of GRCs is strongly influenced by environment temperature and is determined by

ν_{12} = V_{G} ν_{12}^{G} + V_{m} ν^{m}

The orthotropic elastic constants and thermal expansions of graphene sheet are determined by

\begin{array}{l} ν_{12}^{G} = 0.177 \\ E_{11}^{G} = (2.16637 - 0.00193 T + 2.93701 \times 10^{- 6} T^{2} - 1.51775 \times 10^{- 9} T^{3}) TPa \\ E_{22}^{G} = (2.16868 - 0.00193 T + 2.85954 \times 10^{- 6} T^{2} - 1.45145 \times 10^{- 9} T^{3}) TPa \\ G_{12}^{G} = (0.53514 + 8.24436 \times 10^{- 4} T - 1.2932 \times 10^{- 6} T^{2} + 5.78507 \times 10^{- 10} T^{3}) TPa \\ α_{11}^{G} = (- 3.83788 + 0.01416 T - 1.63355 \times 10^{- 5} T^{2} + 6.33589 \times 10^{- 9} T^{3}) \times 10^{- 6} K^{- 1} \\ α_{22}^{G} = (- 3.73997 + 0.01296 T - 1.35033 \times 10^{- 5} T^{2} + 4.60392 \times 10^{- 9} T^{3}) \times 10^{- 6} K^{- 1} \end{array}

Fundamental equations

In the present approach, the Donnell shell theory and Stein and McElman assumption⁴² are applied to establish the governing equations for nonlinear buckling and postbuckling of FG-GRC laminated toroidal shell segments subjected to uniform external pressure in thermal environment

[\begin{array}{l} ε_{x} \\ ε_{y} \\ γ_{x y} \end{array}] = [\begin{array}{l} ε_{x}^{0} \\ ε_{y}^{0} \\ γ_{x y}^{0} \end{array}] - z [\begin{array}{l} χ_{x} \\ χ_{y} \\ 2 χ_{x y} \end{array}]

where $ε_{x}^{0}$ and $ε_{y}^{0}$ are normal strains, $γ_{x y}^{0}$ is the shear strain at the midplane of the shell, and

[\begin{array}{l} ε_{x}^{0} \\ ε_{y}^{0} \\ γ_{x y}^{0} \end{array}] = [\begin{array}{l} u_{, x} - w / a + w_{, x}^{2} / 2 \\ v_{, y} - w / R + w_{, y}^{2} / 2 \\ u_{, y} + v_{, x} + w_{, x} w_{, y} \end{array}], [\begin{array}{l} χ_{x} \\ χ_{y} \\ χ_{x y} \end{array}] = [\begin{array}{l} w_{, x x} \\ w_{, y y} \\ w_{, x y} \end{array}]

where the nonlinearity terms in von Kármán sense and longitudinal curvature term are mentioned. $u = u (x, y)$ , $v = v (x, y)$ , and $w = w (x, y)$ are displacements in x, y, and z directions, respectively.

Note that when longitudinal radius $a \to \infty$ , equation (10) becomes strain–displacement relations of cylindrical shell.

The deformation compatibility equation is received from equation (10), as

ε_{x, y y}^{0} + ε_{y, x x}^{0} - γ_{x y, x y}^{0} = - \frac{1}{R} w_{, x x} - \frac{1}{a} w_{, y y} + w_{, x y}^{2} - w_{, x x} w_{, y y}

Stress–strain relations of Hooke’s law for FG-GRC laminated layer including thermal effects are written by

[\begin{array}{l} σ_{x} \\ σ_{y} \\ σ_{x y} \end{array}] = [\begin{matrix} Q_{11} & Q_{12} & 0 \\ Q_{12} & Q_{22} & 0 \\ 0 & 0 & Q_{66} \end{matrix}] ([\begin{array}{l} ε_{x} \\ ε_{y} \\ γ_{x y} \end{array}] - [\begin{array}{l} α_{11} \\ α_{22} \\ 0 \end{array}] Δ T)

where $Q_{11} = \frac{E_{11}}{1 - ν_{12} ν_{21}}, Q_{22} = \frac{E_{22}}{1 - ν_{12} ν_{21}}, Q_{12} = \frac{ν_{21} E_{11}}{1 - ν_{12} ν_{21}}, Q_{66} = G_{12}$ , and the expressions of $E_{11}, E_{22}, G_{12}, ν_{12}, ν_{21}$ are determined from equations (4), (5), (7), and (8), and $Δ T$ is the uniform temperature rise of environment with respect to a reference temperature.

The force and moment effects of FG-GRC laminated toroidal shell segments are determined by

(N_{x}, N_{y}, N_{x y}) = \int_{\frac{- h}{2}}^{\frac{h}{2}} (σ_{x}, σ_{y}, σ_{x y}) d z, (M_{x}, M_{y}, M_{x y}) = \int_{\frac{- h}{2}}^{\frac{h}{2}} (σ_{x}, σ_{y}, σ_{x y}) z d z

Substituting equations (equation (9) into equation (12) taking into account equation (6), (7), and (10), the integrals in 13) become

[\begin{array}{l} N_{x} \\ N_{y} \\ N_{x y} \\ M_{x} \\ M_{y} \\ M_{x y} \end{array}] = [\begin{matrix} A_{11} & A_{12} & 0 & B_{11} & B_{12} & 0 \\ A_{12} & A_{22} & 0 & B_{12} & B_{22} & 0 \\ 0 & 0 & A_{66} & 0 & 0 & B_{66} \\ B_{11} & B_{12} & 0 & D_{11} & D_{12} & 0 \\ B_{12} & B_{22} & 0 & D_{12} & D_{22} & 0 \\ 0 & 0 & B_{66} & 0 & 0 & D_{66} \end{matrix}] [\begin{array}{l} ε_{x}^{0} \\ ε_{y}^{0} \\ γ_{x y}^{0} \\ - χ_{x} \\ - χ_{y} \\ - 2 χ_{x y} \end{array}] - [\begin{array}{l} φ_{1 x}^{T} \\ φ_{1 y}^{T} \\ 0 \\ φ_{2 x}^{T} \\ φ_{2 y}^{T} \\ 0 \end{array}]

where $A_{i j}, B_{i j}, D_{i j}$ are components of stiffness matrix of the shell

(A_{i j}, B_{i j}, D_{i j}) = \int_{- \frac{h}{2}}^{\frac{h}{2}} Q_{i j} (1, z, z^{2})

The components of stiffness matrix of the shell are calculated by numerical method and $φ_{1 x}^{T}$ , $φ_{1 y}^{T}$ are determined by

φ_{1 x}^{T} = \int_{Π} [Q_{11} α_{11} + Q_{12} α_{22}] Δ T d z, φ_{1 y}^{T} = \int_{Π} [Q_{12} α_{11} + Q_{22} α_{22}] Δ T d z

with $Π$ is the thickness domain of layer.

The normal and shear–strain expressions at the midplane of the shell are reversely determined from equation (14), as

\begin{array}{l} ε_{x}^{0} = A_{22}^{*} N_{x} - A_{12}^{*} N_{y} + B_{11}^{*} χ_{x} + B_{12}^{*} χ_{y} + A_{22}^{*} φ_{1 x}^{T} - A_{12}^{*} φ_{1 y}^{T} \\ ε_{y}^{0} = A_{11}^{*} N_{y} - A_{12}^{*} N_{x} + B_{21}^{*} χ_{x} + B_{22}^{*} χ_{y} - A_{12}^{*} φ_{1 x}^{T} + A_{11}^{*} φ_{1 y}^{T} \\ γ_{x y}^{0} = N_{x y} A_{66}^{*} + 2 B_{66}^{*} χ_{x y} \end{array}

where the expressions of $A_{i j}^{*}, B_{i j}^{*}$ are presented in Appendix 1.

Substituting equation (15) into moment expressions in equation (14), the moment expressions are rewritten by

\begin{array}{l} M_{x} = B_{11}^{*} N_{x} + B_{21}^{*} N_{y} - D_{11}^{*} χ_{x} - D_{12}^{*} χ_{y} + B_{11}^{*} φ_{1 x}^{T} + B_{21}^{*} φ_{1 y}^{T} - φ_{2 x}^{T} \\ M_{y} = B_{12}^{*} N_{x} + B_{22}^{*} N_{y} - D_{21}^{*} χ_{x} - D_{22}^{*} χ_{y} + B_{12}^{*} φ_{1 x}^{T} + B_{22}^{*} φ_{1 y}^{T} - φ_{2 y}^{T} \\ M_{x y} = B_{66}^{*} N_{x y} - 2 D_{66}^{*} χ_{x y} \end{array}

where the expressions of $D_{i j}^{*}$ are presented in Appendix 1.

The equilibrium equation system of FG-GRC laminated toroidal shell segments including Pasternak’s elastic foundations (equation (3)) based on the Donnell shell theory with Stein and McElman assumption⁴² (equations (1) and (2)) is presented as

\begin{array}{l} N_{x, x} + N_{x y, y} = 0, N_{x y, x} + N_{y, y} = 0, \\ M_{x, x x} + 2 M_{x y, x y} + M_{y, y y} + N_{x} w_{, x x} + 2 N_{x y} w_{, x y} + N_{y} w_{, y y} + \frac{N_{y}}{R} + \frac{N_{x}}{a} - K_{1} w \\ + K_{2} (w_{, x x} + w_{, y y}) + q_{0} = 0 \end{array}

Assuming that the Airy’s stress function $φ (x, y)$ may be determined which satisfies the conditions

N_{x} = φ_{, y y}, N_{y} = φ_{, x x}, N_{x y} = - φ_{, x y}

Substituting the proposed Airy’s stress function (equation (18)) into the first two equations of equation (17), these equations are completely satisfied.

Then, substituting equations (equation (16) and equations ((18) into the third equation of 17) and combining 9) and (10), equilibrium equations are rewritten by

\begin{array}{l} D_{11}^{*} w_{, x x x x} + (D_{12}^{*} + D_{21}^{*} + 4 D_{66}^{*}) w_{, x x y y} + D_{22}^{*} w_{, y y y y} + B_{21}^{*} φ_{, x x x x} \\ - (B_{11}^{*} + B_{22}^{*} - 2 B_{66}^{*}) φ_{, x x y y} - B_{12}^{*} φ_{, y y y y} - \frac{1}{R} φ_{, x x} - \frac{1}{a} φ_{,}_{y y} - φ_{, y y} w_{, x x} \\ - φ_{, x x} w_{, y y} + 2 φ_{, x y} w_{, x y} + K_{1} w - K_{2} (w_{, x x} + w_{, y y}) = q_{0} \end{array}

Now, substituting equation (15) into the deformation compatibility equation (11) taking into account equation (18), equation (11) becomes

\begin{array}{l} A_{11}^{*} φ_{, x x x x} + (A_{66}^{*} - 2 A_{12}^{*}) φ_{, x x y y} + A_{22}^{*} φ_{, y y y y} + B_{21}^{*} w_{, x x x x} + \\ + (B_{11}^{*} + B_{22}^{*} - 2 B_{66}^{*}) w_{, x x y y} + B_{12}^{*} w_{, y y y y} - w_{, x y}^{2} + w_{, x x} w_{, y y} + \frac{1}{R} w_{, x x} + \frac{1}{a} w_{, y y} = 0 \end{array}

The nonlinear buckling and postbuckling behavior of FG-GRC laminated toroidal shell segments subjected to external pressure surrounded by two-parameter Pasternak’s elastic foundation in thermal environment is obtained using the equilibrium and compatibility equations (19) and (20).

Solution of problem

In this article, FG-GRC laminated toroidal shell segment subjected to external pressure with simply supported at the edges is considered. Thus, the boundary conditions considered in the current study are

w = 0,_{}^{}_{}^{} M_{x} = 0,_{}^{}_{}^{} N_{x} = 0,_{}^{}_{}^{} N_{x y} = 0,_{}^{}_{}^{} a t_{}^{}_{}^{} x = 0;_{}^{} L

Then, the deflection of shell satisfying the boundary conditions (equation (21)) on the approximate sense is chosen in three-term form^12,13,19

w = f_{0} + f_{1} sin \frac{m π x}{L} sin \frac{n y}{R} + f_{2} {sin}^{2} \frac{m π x}{L}

where f ₀ is the prebuckling deflection amplitude, f ₁ and f ₂ are linear and nonlinear postbuckling deflection amplitudes, respectively, and m and n are buckling modes (number of half wave) in longitudinal and circumferential directions, respectively.

Substituting the solution form of deflection (equation (22)) into equation (20), the form of stress function may be expressed as

\begin{array}{l} φ = φ_{1} cos \frac{2 m π x}{L} + φ_{2} cos \frac{2 n y}{R} - φ_{3} sin \frac{m π x}{L} sin \frac{n y}{R} + φ_{4} sin \frac{3 m π x}{L} sin \frac{n y}{R} - σ_{0 y} h \frac{x^{2}}{2} \end{array}

in which the expressions of $φ_{i} (i = 1 \div 4)$ can be found in Appendix 1.

The deflection and stress function forms are introduced into equation (19). The Galerkin procedure is applied in the ranges $0 \leq y \leq 2 π R$ and $0 \leq x \leq L$ leads to

\frac{σ_{0 y} h}{R} = q_{0} - \frac{K_{1} (f_{2} + 2 f_{0})}{2}

\begin{array}{l} (D + \frac{B^{2}}{A}) f_{1} + (\frac{m^{4} π^{4}}{16 A_{22}^{*}} + \frac{n^{4} λ^{4}}{16 A_{11}^{*}}) f_{1}^{3} + [\frac{2 B m^{2} n^{2} π^{2} λ^{2}}{A} - \frac{n^{2} λ^{2} (λ L - 4 B_{21}^{*} m^{2} π^{2})}{4 A_{11}^{*}}] f_{1} f_{2} \\ + m^{4} n^{4} π^{4} λ^{4} (\frac{1}{A} + \frac{1}{F}) f_{1} f_{2}^{2} - σ_{0 y} h n^{2} L^{2} λ^{2} f_{1} + L^{4} K_{1} f_{1} + L^{2} K_{2} f_{1} [{(λ n)}^{2} + {(m π)}^{2}] = 0 \end{array}

\begin{array}{l} {[4 B_{21}^{*} {(\frac{m π}{L})}^{4} - \frac{1}{R} {(\frac{m π}{L})}^{2}] \frac{n^{2} λ^{2}}{16 A_{11}^{*} m^{2} π^{2}} + \frac{B}{2 A} {(\frac{m π}{L})}^{2} {(\frac{n}{R})}^{2}} f_{1}^{2} \\ + \frac{1}{2} m^{2} n^{2} π^{2} λ^{2} {(\frac{m π}{L})}^{2} {(\frac{n}{R})}^{2} (\frac{1}{A} - \frac{1}{F}) f_{1}^{2} f_{2} \\ + {4 D_{11}^{*} {(\frac{m π}{L})}^{4} - [4 B_{21}^{*} {(\frac{m π}{L})}^{4} - \frac{1}{R} {(\frac{m π}{L})}^{2}] \frac{λ L - 4 B_{21}^{*} m^{2} π^{2}}{4 A_{11}^{*} m^{2} π^{2}}} f_{2} \\ + \frac{σ_{0 y} h}{R} - q_{0} + K_{1} (f_{2} \frac{3}{4} + f_{0}) + K_{2} f_{2} {(\frac{m π}{L})}^{2} = 0 \end{array}

where the expression of D is shown in Appendix 1.

For all shell of revolution, the circumferential closed condition must be added in average sense, as

\int_{0}^{2 π R} \int_{0}^{L} v_{, y} d x d y = \int_{0}^{2 π R} \int_{0}^{L} (ε_{y}^{0} + \frac{w}{R} - \frac{1}{2} w_{, x}^{2}) d x d y = 0

By combining equations (equation (15), (22), and (23), the integral (27)) leads to

- 2 A_{11}^{*} σ_{0 y} h + \frac{(f_{2} + 2 f_{0})}{R} - {(\frac{n}{2 R})}^{2} f_{1}^{2} - 2 (A_{12}^{*} φ_{1 x}^{T} - A_{11}^{*} φ_{1 y}^{T}) = 0

Substituting $σ_{0 y}$ in equations (equation (24) into equation (25) and equations ((26) and the circumferential closed condition (28)), equilibrium 24) to (26) are rewritten by

f_{0} = \frac{P_{32}}{2 P_{31}} f_{1}^{2} + \frac{P_{33}}{2 P_{31}} q_{0} - \frac{1}{2} f_{2} - \frac{P_{34}}{2 P_{31}}

P_{11} + P_{12} f_{0} + P_{13} f_{1}^{2} + P_{14} f_{2} + P_{15} f_{2}^{2} - P_{16} q_{0} = 0

f_{1}^{2} = - \frac{P_{23} f_{2}}{P_{21} + P_{22} f_{2}}

where the expressions of $P_{i j}$ are presented in Appendix 1.

Substituting f ₀ and $f_{1}^{2}$ from equations (equation (29) and (31) into 30) leads to

\begin{array}{l} q_{0} = [P_{11} - \frac{P_{12} P_{32} P_{23} f_{2}}{2 P_{31} (P_{21} + P_{22} f_{2})} + (P_{14} - \frac{P_{12}}{2}) f_{2} - \frac{P_{12} P_{34}}{2 P_{31}} - \\ - P_{13} \frac{P_{23} f_{2}}{P_{21} + P_{22} f_{2}} + P_{15} f_{2}^{2}] {(P_{16} - \frac{P_{12} P_{33}}{2 P_{31}})}^{- 1} \end{array}

When $f_{2} \to 0$ , the upper external pressure buckling load of FG-GRC laminated toroidal shell segments is received from equation (32) as follows

q_{0 upper} = \frac{2 P_{11} P_{31} - P_{12} P_{34}}{2 P_{31} P_{16} - P_{12} P_{33}}

The critical external pressure buckling load $q_{0}^{cr}$ is the minimal value of upper external pressure buckling loads $q_{0upper}$ in equation (33) versus $(m, n)$ .

From equation (22), the maximal deflection of the shells is determined by

W_{max} = f_{0} + f_{1} + f_{2}

locates at $x = \frac{i L}{2 m}$ , $y = \frac{j π R}{2 n}$ , where $i, j$ are odd integer numbers.

Substituting equations (equation (29) and (31) into 34), the maximal deflection is rewritten respected to f ₂

W_{max} = - \frac{P_{32} P_{23} f_{2}}{2 P_{31} (P_{21} + P_{22} f_{2})} + q_{0} \frac{P_{33}}{2 P_{31}} - \frac{P_{34}}{2 P_{31}} + \frac{f_{2}}{2} + {(- \frac{P_{23} f_{2}}{P_{21} + P_{22} f_{2}})}^{1 / 2}

As can be seen, $q_{0} - W_{max} / h$ relation is very difficult to express in explicit form, therefore, the postbuckling curve $q_{0} - W_{max} / h$ of FG-GRC laminated toroidal shell segments can be obtained by combining equations (32) and (35).

Numerical results and discussion

Comparison results

The validation of present approach is discussed by the comparisons of critical external pressure buckling load $q_{0}^{cr}$ (MPa) of FG-GRC laminated cylindrical shells ( $a \to \infty$ ) with the results of Shen and Xiang³⁵ and Kiani⁴¹ as presented in Tables 2 and 3. Very good agreements in these comparisons are observed.

Table 2.

Comparisons of critical buckling external pressure load $q_{0}^{cr}$ (MPa) of GRC laminated cylindrical shells with different geometric dimensions ( $R / h = 20$ , $h = 2 mm$ , and $T = 300 K$ ).

$L^{2} / R h$	Direction arrangement	Shen and Xiang³⁵		Present
$L^{2} / R h$	Direction arrangement	UD	FG-X	UD	FG-X
100	(0)₁₀	20.321 (1;4)^a	22.162 (1;4)	20.885 (1;4)	23.087 (1;4)
	(0/90/0/90/0)_S	20.384 (1;4)	22.226 (1;4)	20.849 (1;4)	23.048 (1;4)
	(0/90)_5T	20.418 (1;4)	22.294 (1;4)	20.988 (1;4)	23.232 (1;4)
300	(0)₁₀	11.173 (1;3)	12.082 (1;3)	11.335 (1;3)	12.346 (1;3)
	(0/90/0/90/0)_S	11.207 (1;3)	12.117 (1;3)	11.691 (1;3)	12.705 (1;3)
	(0/90)_5T	11.226 (1;3)	12.158 (1;3)	11.390 (1;3)	12.425 (1;3)
500	(0)₁₀	9.525 (1;3)	10.426 (1;3)	9.678 (1;3)	10.671 (1;3)
	(0/90/0/90/0)_S	9.566 (1;3)	10.470 (1;3)	9.719 (1;3)	10.717 (1;3)
	(0/90)_5T	9.585 (1;3)	10.509 (1;3)	9.739 (1;3)	10.757 (1;3)

GRC: graphene-reinforced composite.

^a The numbers in brackets indicate the buckling mode $(m; n)$ .

Table 3.

Comparisons of critical buckling external pressure load $q_{0}^{cr}$ (MPa) of GRC laminated cylindrical shells with the results estimated from the work of Kiani⁴¹ ( $R / h = 20$ , $h = 2 mm$ ).

$L$	$T (K)$	Source	UD	FG-X	FG-V	FG-Λ	FG-O
$\sqrt{100 R h}$	300	Present	20.885 (1;4)	23.087 (1;4)	16.449 (1;4)	16.964 (1;4)	14.616 (1;4)
	300	Kiani⁴¹	20.300 (1;4)	22.350 (1;4)	16.090 (1;4)	16.596 (1;4)	14.509 (1;4)
	400	Present	18.579 (1;4)	20.381 (1;4)	14.834 (1;4)	14.973 (1;4)	13.007 (1;4)
	400	Kiani⁴¹	18.024 (1;4)	19.680 (1;4)	14.484 (1;4)	14.622 (1;4)	12.750 (1;4)
	500	Present	17.484 (1;4)	19.635 (1;4)	14.556 (1;4)	14.557 (1;4)	12.608 (1;4)
	500	Kiani⁴¹	16.920 (1;4)	18.901 (1;4)	14.173 (1;4)	14.175 (1;4)	12.334 (1;4)
$\sqrt{300 R h}$	300	Present	11.335 (1;3)	12.346 (1;3)	8.976 (1;3)	9.198 (1;3)	8.027 (1;3)
	300	Kiani⁴¹	11.142 (1;3)	12.107 (1;3)	8.854 (1;3)	9.075 (1;3)	7.936 (1;3)
	400	Present	10.116 (1;3)	10.970 (1;3)	8.089 (1;3)	8.164 (1;3)	7.156 (1;3)
	400	Kiani⁴¹	9.933 (1;3)	10.743 (1;3)	7.974 (1;3)	8.048 (1;3)	7.070 (1;3)
	500	Present	9.559 (1;3)	10.614 (1;3)	7.948 (1;3)	7.966 (1;3)	6.953 (1;3)
	500	Kiani⁴¹	9.374 (1;3)	10.376 (1;3)	7.821 (1;3)	7.840 (1;3)	6.862 (1;3)
$\sqrt{500 R h}$	300	Present	9.678 (1;3)	10.671 (1;3)	7.548 (1;3)	7.647 (1;3)	6.603 (1;3)
	300	Kiani⁴¹	9.477 (1;3)	10.428 (1;3)	7.418 (1;3)	7.517 (1;3)	6.504 (1;3)
	400	Present	8.670 (1;3)	9.528 (1;3)	6.797 (1;3)	6.838 (1;3)	5.901 (1;3)
	400	Kiani⁴¹	8.481 (1;3)	9.297 (1;3)	6.675 (1;3)	6.716 (1;3)	5.808 (1;3)
	500	Present	8.207 (1;3)	9.255 (1;3)	6.698 (1;3)	6.717 (1;3)	5.756 (1;3)
	500	Kiani⁴¹	8.015 (1;3)	9.013 (1;3)	6.565 (1;3)	6.584 (1;3)	5.658 (1;3)

GRC: graphene-reinforced composite.

Material and geometrical parameters

In this section, the material and geometrical parameters of the considered FG-GRC laminated toroidal shell segments are given by $L = 0.27 m$ , $R / h = 60$ , $a = 4 R$ , $K_{1} = 2.5 \times 10^{7} {N m}^{-}^{3}$ , and $K_{2} = 2.5 \times 10^{5} {N m}^{- 1}$ . The total thickness of FG-GRC laminated toroidal shell segments is $h = 2 mm$ , which includes 10 layers with 0.2 mm for each layer. The geometrical parameters of the graphene sheets are taken in the report of Shen and Xiang³⁵ as follows: $a_{G} = 14.76 nm$ , $b_{G} = 14.77 nm$ , and $h_{G} = 0.188 nm$ . The performance parameters of each GRC layer are found in Table 1.

Effect of longitudinal curvature

The effect of longitudinal curvature on the critical external pressure buckling load $q_{0}^{cr}$ (MPa) of FG-GRC shell is presented in Table 4. Three types of shell, namely, convex toroidal shell segment, concave toroidal shell segment, and cylindrical shell, are considered. The results indicate that the critical external pressure buckling loads of convex toroidal shell segment are larger than one of cylindrical shell and those are larger than one of concave toroidal shell segment. As can be observed, the (0)₁₀ direction arrangement shells have the lowest critical buckling load, while the (0/90)_5T direction arrangement shells have the highest critical buckling load among the three considered laminate arrangements. Table 4 also shows that the critical buckling of FG-X type is the largest and FG-O is the smallest. Difference between FG-X shells and FG-O shells can be up to about 19.91% with convex toroidal shell segment, 13.83% with cylindrical shell, and 6.44% with concave toroidal shell segment.

Table 4.

Effects of longitudinal curvature on the critical external pressure buckling load $q_{0}^{cr}$ (MPa) of FG-GRC shell ( $L = 0.27 m$ , $T = 300 K$ , $R / h = 60$ , $h = 2 mm$ , $K_{1} = 2.5 \times 10^{7} N m^{-}^{3}$ , $K_{2} = 2.5 \times 10^{5} N m^{- 1}$ ).

	Direction arrangement	UD	FG-X	FG-V	FG-Λ	FG-O
Convex toroidal shell segment ( $a = 4 R$ )	(0)₁₀	6.016 (1;8)	6.126 (1;7) (+19.91%)^a	5.334 (1;8)	5.393 (1;8)	5.109 (1;8)
	(0/90/0/90/0)_S	6.019 (1;8)	6.121 (1;7) (+19.64%)	5.339 (1;8)	5.395 (1;8)	5.116 (1;8)
	(0/90)_5T	6.024 (1;8)	6.129 (1;7) (+19.87%)	5.347 (1;8)	5.399 (1;8)	5.113 (1;8)
Concave toroidal shell segment ( $a = - 4 R$ )	(0)₁₀	3.412 (1;3)	3.470 (1;3) (+6.41%)	3.323 (1;3)	3.302 (1;3)	3.261 (1;3)
	(0/90/0/90/0)_S	3.413 (1;3)	3.472 (1;3) (+6.41%)	3.324 (1;3)	3.304 (1;3)	3.263 (1;3)
	(0/90)_5T	3.414 (1;3)	3.473 (1;3) (+6.44%)	3.326 (1;3)	3.305 (1;3)	3.263 (1;3)
Cylindrical shell ( $a = \infty$ )	(0)₁₀	3.611 (1;5)	3.713 (1;5) (+13.76%)	3.360 (1;5)	3.388 (1;5)	3.264 (1;5)
	(0/90/0/90/0)_S	3.614 (1;5)	3.717 (1;5) (+13.70%)	3.365 (1;5)	3.391 (1;5)	3.269 (1;5)
	(0/90)_5T	3.617 (1;5)	3.721 (1;5) (+13.83%)	3.368 (1;5)	3.394 (1;5)	3.269 (1;5)

FG-GRC: functionally graded graphene-reinforced composite.

^a Difference between FG-X shells and FG-O shells.

Effect of thermal environment

Tables 5 and 6 show the effects of thermal environment $(T)$ on the critical external pressure buckling load of FG-GRC toroidal shell segments surrounded by elastic foundation. Clearly, when the temperature increases, the elastic moduli of the GRCs are reduced due to the temperature-dependent material properties of both graphene sheets and the matrix. Obviously, the critical buckling load is decreased for the GRC laminated cylindrical shells when the temperature rises. In addition, graphene direction arrangement of toroidal shell segment lightly influences the critical external pressure buckling load; conversely, thermal environment and graphene distribution type strongly influence the critical external pressure buckling load of shells. Critical buckling load of shell decreases when the thermal environment increases. Critical buckling load of toroidal shell segments is largest with FG-X graphene distribution type and smallest with FG-O graphene distribution type. In addition, the critical external pressure buckling load of FG-GRC convex toroidal shell segment is greater than one of concave toroidal shell segment.

Table 5.

Effects of thermal environment on the critical external pressure buckling load $q_{0}^{cr}$ (MPa) of FG-GRC convex toroidal shell segments ( $L = 0.27 m$ , $R / h = 60$ , $h = 2 mm$ , $a = 4 R$ , $K_{1} = 2.5 \times 10^{7} N m^{-}^{3}$ , $K_{2} = 2.5 \times 10^{5} N m^{-}^{1}$ ).

T (K)	Direction arrangement	UD	FG-X	FG-V	FG-Λ	FG-O
300	(0)₁₀	6.016 (1;8)	6.126 (1;7)	5.334 (1;8)	5.393 (1;8)	5.109 (1;8)
	(0/90/0/90/0)_S	6.019 (1;8)	6.121 (1;7)	5.339 (1;8)	5.395 (1;8)	5.116 (1;8)
	(0/90)_5T	6.024 (1;8)	6.129 (1;7)	5.347 (1;8)	5.399 (1;8)	5.113 (1;8)
400	(0)₁₀	5.613 (1;8)	5.710 (1;7)	5.024 (1;8)	5.052 (1;8)	4.803 (1;8)
	(0/90/0/90/0)_S	5.615 (1;8)	5.709 (1;7)	5.025 (1;8)	5.054 (1;8)	4.805 (1;8)
	(0/90)_5T	5.617 (1;8)	5.714 (1;7)	5.032 (1;8)	5.062 (1;8)	4.803 (1;8)
500	(0)₁₀	5.454 (1;8)	5.605 (1;7)	4.971 (1;8)	4.987 (1;8)	4.734 (1;8)
	(0/90/0/90/0)_S	5.461 (1;8)	5.604 (1;7)	4.967 (1;8)	4.986 (1;8)	4.737 (1;8)
	(0/90)_5T	5.471 (1;8)	5.610 (1;7)	4.970 (1;8)	4.991 (1;8)	4.731 (1;8)

FG-GRC: functionally graded graphene-reinforced composite.

Table 6.

Effects of thermal environment on the critical external pressure buckling load $q_{0}^{cr}$ (MPa) of FG-GRC concave toroidal shell segments ( $L = 0.27 m$ , $R / h = 60$ , $h = 2 mm$ , $a = - 4 R$ , $K_{1} = 2.5 \times 10^{7} N m^{-}^{3}$ , $K_{2} = 2.5 \times 10^{5} N m^{-}^{1}$ ).

T (K)	Direction arrangement	UD	FG-X	FG-V	FG-Λ	FG-O
300	(0)₁₀	3.412 (1;3)	3.470 (1;3)	3.323 (1;3)	3.302 (1;3)	3.261 (1;3)
	(0/90/0/90/0)_S	3.413 (1;3)	3.472 (1;3)	3.324 (1;3)	3.304 (1;3)	3.263 (1;3)
	(0/90)_5T	3.414 (1;3)	3.473 (1;3)	3.326 (1;3)	3.305 (1;3)	3.263 (1;3)
400	(0)₁₀	3.350 (1;3)	3.397 (1;3)	3.267 (1;3)	3.262 (1;3)	3.218 (1;3)
	(0/90/0/90/0)_S	3.351 (1;3)	3.399 (1;3)	3.268 (1;3)	3.263 (1;3)	3.219 (1;3)
	(0/90)_5T	3.351 (1;3)	3.400 (1;3)	3.269 (1;3)	3.264 (1;3)	3.219 (1;3)
500	(0)₁₀	3.320 (1;3)	3.374 (1;3)	3.252 (1;3)	3.253 (1;3)	3.205 (1;3)
	(0/90/0/90/0)_S	3.323 (1;3)	3.377 (1;3)	3.253 (1;3)	3.254 (1;3)	3.207 (1;3)
	(0/90)_5T	3.324 (1;3)	3.378 (1;3)	3.253 (1;3)	3.254 (1;3)	3.207 (1;3)

FG-GRC: functionally graded graphene-reinforced composite.

The effects of thermal temperature on the $q_{0} - W_{max} / h$ postbuckling curve with the critical buckling mode of UD and FG-X type of (0/90)_5T convex and concave toroidal shell segments and cylindrical shells are investigated in Figures 4 to 6. As can be observed, the complex behavior of nonlinear postbuckling curves is obtained. Results indicate that the postbuckling curve of FG-X toroidal shell segments is larger than one of UD toroidal shell segments, and the postbuckling curve of FG-GRC toroidal shell segments is upper with lesser thermal environment, respectively. Snap-through phenomenon is clearly observed with convex toroidal shell segment and cylindrical shell, while it is not obtained with concave toroidal shell segments in the considered investigation results.

Figure 4.

Effects of thermal environment on $q_{0} - W_{max} / h$ postbuckling curves of FG-GRC convex toroidal shell segment.

Figure 5.

Effects of thermal environment on $q_{0} - W_{max} / h$ postbuckling curves of FG-GRC cylindrical shell.

Figure 6.

Effects of thermal environment on $q_{0} - W_{max} / h$ postbuckling curves of FG-GRC concave toroidal shell segment.

Effect of geometric dimensions R/h

The effect of $R / h$ ratio and graphene distribution type on the critical external pressure buckling load of FG-GRC toroidal shell segments is investigated in Table 7 for convex toroidal shell segment and in Table 8 for concave toroidal shell segment. Clearly, $R / h$ ratio of shells significantly influences the critical external pressure buckling load. In addition, the effect of $R / h$ ratio on the $q_{0} - W_{max} / h$ postbuckling curves is presented in Figures 7 and 8. Due to the complex behavior of FG-GRC toroidal shell segments, the irregular changes on postbuckling curve are obtained when $R / h$ ratio varies.

Figure 7.

Effect of $R / h$ ratio on $q_{0} - W_{max} / h$ postbuckling curve of FG-GRC convex toroidal shell segment.

Figure 8.

Effect of $R / h$ ratio on $q_{0} - W_{max} / h$ postbuckling curve of FG-GRC concave toroidal shell segment.

Table 7.

Effect of $R / h$ ratio on the critical external pressure buckling load $q_{0}^{cr}$ (MPa) of FG-GRC convex toroidal shell segment ( $L = 0.27 m$ , $T = 300 K$ , $h = 0.002 m$ , $a = 4 R$ , $K_{1} = 2.5 \times 10^{7} N m^{-}^{3}$ , $K_{2} = 2.5 \times 10^{5} N m^{-}^{1}$ ).

Direction arrangement	UD	FG-X	FG-V	FG-Λ	FG-O
$R / h = 60$
(0)₁₀	6.016 (1;8)	6.126 (1;7)	5.334 (1;8)	5.393 (1;8)	5.109 (1;8)
(0/90/0/90/0)_S	6.019 (1;8)	6.121 (1;7)	5.339 (1;8)	5.395 (1;8)	5.116 (1;8)
(0/90)_5T	6.024 (1;8)	6.129 (1;7)	5.347 (1;8)	5.399 (1;8)	5.113 (1;8)
$R / h = 70$
(0)₁₀	4.763 (1;8)	4.826 (1;8)	4.246 (1;9)	4.291 (1;9)	4.064 (1;9)
(0/90/0/90/0)_S	4.764 (1;8)	4.824 (1;8)	4.251 (1;9)	4.294 (1;9)	4.071 (1;9)
(0/90)_5T	4.766 (1;8)	4.831 (1;8)	4.257 (1;9)	4.298 (1;9)	4.069 (1;9)
$R / h = 80$
(0)₁₀	3.893 (1;9)	3.964 (1;9)	3.510 (1;10)	3.545 (1;10)	3.357 (1;10)
(0/90/0/90/0)_S	3.894 (1;9)	3.964 (1;9)	3.512 (1;9)	3.549 (1;10)	3.364 (1;10)
(0/90)_5T	3.896 (1;9)	3.970 (1;9)	3.516 (1;9)	3.552 (1;10)	3.363 (1;10)

FG-GRC: functionally graded graphene-reinforced composite.

Table 8.

Effect of $R / h$ ratio on the critical external pressure buckling load $q_{0}^{cr}$ (MPa) of FG-GRC convex toroidal shell segment ( $L = 0.27 m$ , $T = 300 K$ , $h = 0.002 m$ , $a = - 4 R$ , $K_{1} = 2.5 \times 10^{7} N m^{-}^{3}$ , $K_{2} = 2.5 \times 10^{5} N m^{-}^{1}$ ).

Direction arrangement	UD	FG-X	FG-V	FG-Λ	FG-O
$R / h = 60$
(0)₁₀	3.413 (1;3)	3.470 (1;3)	3.323 (1;3)	3.302 (1;3)	3.261 (1;3)
(0/90/0/90/0)_S	3.413 (1;3)	3.472 (1;3)	3.324 (1;3)	3.304 (1;3)	3.263 (1;3)
(0/90)_5T	3.414 (1;3)	3.473 (1;3)	3.326 (1;3)	3.305 (1;3)	3.263 (1;3)
$R / h = 70$
(0)₁₀	3.103 (1;4)	3.146 (1;4)	3.001 (1;4)	2.967 (1;4)	2.931 (1;4)
(0/90/0/90/0)_S	3.103 (1;4)	3.146 (1;4)	3.000 (1;4)	2.968 (1;4)	2.932 (1;4)
(0/90)_5T	3.104 (1;4)	3.147 (1;4)	3.002 (1;4)	2.968 (1;4)	2.932 (1;4)
$R / h = 80$
(0)₁₀	2.530 (1;4)	2.576 (1;4)	2.464 (1;4)	2.452 (1;4)	2.419 (1;4)
(0/90/0/90/0)_S	2.531 (1;4)	2.578 (1;4)	2.465 (1;4)	2.453 (1;4)	2.421 (1;4)
(0/90)_5T	2.532 (1;4)	2.578 (1;4)	2.466 (1;4)	2.454 (1;4)	2.421 (1;4)

FG-GRC: functionally graded graphene-reinforced composite.

Effect of elastic foundation stiffness

Tables 9 and 10 and Figures 9 and 10 show the effects of Pasternak’s elastic foundation stiffness on the critical external pressure buckling load and postbuckling curve of FG-GRC toroidal shell segments, respectively. These tables and figures show a beneficial influence of elastic foundation on the buckling and postbuckling behavior of shells. Specifically, the capability of load carrying is enhanced and the intensity of snap-through behavior is reduced due to the presence of elastic foundations. In all numerical investigation, the snap-through phenomenon does not exist with concave toroidal shell segment.

Table 9.

Effect of elastic foundation on the critical external pressure buckling load $q_{0}^{cr}$ (MPa) of FG-GRC convex toroidal shell segment ( $L = 0.27 m$ , $R / h = 60$ , $h = 0.002 m$ , $a = 4 R$ , $T = 300 K$ ).

Direction arrangement	UD	FG-X	FG-V	FG-Λ	FG-O
$K_{1} = 0 N m^{-}^{3}$ , $K_{2} = 0 N m^{- 1}$
(0)₁₀	3.808 (1;8)	3.884 (1;7)	3.127 (1;8)	3.186 (1;8)	2.903 (1;8)
(0/90/0/90/0)_S	3.812 (1;8)	3.878 (1;7)	3.132 (1;8)	3.188 (1;8)	2.909 (1;8)
(0/90)_5T	3.816 (1;8)	3.887 (1;7)	3.140 (1;8)	3.193 (1;8)	2.907 (1;8)
$K_{1} = 1 \times 10^{7} N m^{-}^{3}$ , $K_{2} = 1 \times 10^{5} N m^{-}^{1}$
(0)₁₀	4.698 (1;8)	4.779 (1;7)	4.008 (1;8)	4.067 (1;8)	3.784 (1;8)
(0/90/0/90/0)_S	4.694 (1;8)	4.774 (1;7)	4.013 (1;8)	4.069 (1;8)	3.791 (1;8)
(0/90)_5T	4.698 (1;8)	4.782 (1;7)	4.021 (1;8)	4.074 (1;8)	3.788 (1;8)
$K_{1} = 2.5 \times 10^{7} N m^{-}^{3}$ , $K_{2} = 2.5 \times 10^{5} N m^{-}^{1}$
(0)₁₀	6.016 (1;8)	6.126 (1;7)	5.334 (1;8)	5.393 (1;8)	5.109 (1;8)
(0/90/0/90/0)_S	6.019 (1;8)	6.121 (1;7)	5.339 (1;8)	5.395 (1;8)	5.116 (1;8)
(0/90)_5T	6.024 (1;8)	6.129 (1;7)	5.347 (1;8)	5.399 (1;8)	5.113 (1;8)

FG-GRC: functionally graded graphene-reinforced composite.

Table 10.

Effect of elastic foundation on the critical external pressure buckling load $q_{0}^{cr}$ (MPa) of FG-GRC concave toroidal shell segment ( $L = 0.27 m$ , $R / h = 60$ , $h = 0.002 m$ , $a = - 4 R$ , $T = 300 K$ ).

Direction arrangement	UD	FG-X	FG-V	FG-Λ	FG-O
$K_{1} = 0 N m^{-}^{3}$ , $K_{2} = 0 N m^{- 1}$
(0)₁₀	0.536 (1;3)	0.594 (1;3)	0.447 (1;3)	0.426 (1;3)	0.385 (1;3)
(0/90/0/90/0)_S	0.538 (1;3)	0.595 (1;3)	0.448 (1;3)	0.428 (1;3)	0.387 (1;3)
(0/90)_5T	0.538 (1;3)	0.596 (1;3)	0.450 (1;3)	0.428 (1;3)	0.387 (1;3)
$K_{1} = 1 \times 10^{7} N m^{-}^{3}$ , $K_{2} = 1 \times 10^{5} N m^{-}^{1}$
(0)₁₀	1.685 (1;3)	1.742 (1;3)	1.596 (1;3)	1.575 (1;3)	1.533 (1;3)
(0/90/0/90/0)_S	1.686 (1;3)	1.744 (1;3)	1.597 (1;3)	1.577 (1;3)	1.536 (1;3)
(0/90)_5T	1.687 (1;3)	1.745 (1;3)	1.598 (1;3)	1.577 (1;3)	1.536 (1;3)
$K_{1} = 2.5 \times 10^{7} N m^{-}^{3}$ , $K_{2} = 2.5 \times 10^{5} N m^{-}^{1}$
(0)₁₀	3.412 (1;3)	3.470 (1;3)	3.323 (1;3)	3.302 (1;3)	3.261 (1;3)
(0/90/0/90/0)_S	3.413 (1;3)	3.472 (1;3)	3.324 (1;3)	3.304 (1;3)	3.263 (1;3)
(0/90)_5T	3.414 (1;3)	3.473 (1;3)	3.326 (1;3)	3.305 (1;3)	3.263 (1;3)

FG-GRC: functionally graded graphene-reinforced composite.

Figure 9.

Effect of elastic foundation on $q_{0} - W_{max} / h$ postbuckling curve of FG-GRC convex toroidal shell segment.

Figure 10.

Effect of elastic foundation on $q_{0} - W_{max} / h$ postbuckling curve of FG-GRC concave toroidal shell segment.

Effects of GRC direction arrangement

Figures 11 and 12 present the $q_{0} - W_{max} / h$ postbuckling curves of (0/90)_5T, (0/90/0/90/0)_S, and (0)₁₀ FG-GRC toroidal shell segments with two investigated types of graphene distribution, FG-X and UD types. As can be observed, the postbuckling curve of (0)₁₀ FG-GRC toroidal shell segment is lower than those of (0/90/0/90/0)_S and (0/90)_5T FG-GRC toroidal shell segments with small deflection; however, the postbuckling curve of (0)₁₀ FG-GRC toroidal shell segment is higher than those of (0/90/0/90/0)_S and (0/90)_5T FG-GRC toroidal shell segments when the deflection is sufficiently large.

Figure 11.

Effects of direction arrangement on $q_{0} - W_{max} / h$ postbuckling curve of FG-GRC convex toroidal shell segment.

Figure 12.

Effects of direction arrangement on $q_{0} - W_{max} / h$ postbuckling curve of FG-GRC concave toroidal shell segment.

Effects of graphene reinforcement type

Figures 13 and 14 investigate the effects of graphene reinforcement type (UD, FG-X, FG-O, FG-V, and FG-Λ) on $q_{0} - W_{max} / h$ postbuckling curve of shell. The small differences between the postbuckling curves of FG-V and FG-Λ graphene reinforcement types are obtained in these figures.

Figure 13.

Effects of graphene reinforcement type on $q_{0} - W_{max} / h$ postbuckling curve of FG-GRC convex toroidal shell segment.

Figure 14.

Effects of graphene reinforcement type on $q_{0} - W_{max} / h$ postbuckling curve of FG-GRC concave toroidal shell segment.

Conclusions

The nonlinear buckling and postbuckling behavior of FG-GRC convex and concave toroidal shell segments exposed to thermal environments and subjected to uniform external pressure has been investigated in this article. An analytical approach with three-term solution form of deflection and Galerkin procedure is established using Donnell shell theory and von Kármán nonlinearity terms. Results show some important remarks:

Direction arrangement of graphene layers lightly influences the critical buckling of toroidal shell segments.

FG-X type is the most effective on critical buckling load of FG-GRC toroidal shell segments in all graphene reinforcement types. Reversely, critical buckling load of FG-O shell is the smallest. Difference between critical buckling load of FG-X shells and one of FG-O shells can be up to about 19.91% with convex toroidal shell segment, 13.83% with cylindrical shell, and 6.44% with concave toroidal shell segment.

Critical buckling load of FG-GRC convex toroidal shell segment is greater than one of convex toroidal shell segment.

Snap-through phenomenon is clearly observed with cylindrical shell and convex toroidal shell segment, while this phenomenon is not obtained with concave shells.

As a final remark, elastic foundation and thermal environment strongly influence the nonlinear buckling behavior of FG-GRC toroidal shell segments subjected to external pressure.

It can be seen that the FG-GRC toroidal shell segment is the complex structure, and the present analytical approach is the effective solution to analyze its thermomechanical behavior in practical applications.

Footnotes

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Vu Hoai Nam

Appendix 1

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