Nonlinear vibration of functionally graded sandwich shallow spherical caps resting on elastic foundations by using first-order shear deformation theory in thermal environment

Abstract

A semi-analytical approach to investigate the nonlinear vibration axisymmetric analysis of functionally graded sandwich shallow spherical caps under external pressure resting on elastic foundation in thermal environment is presented. The governing equations are derived by using the first-order shear deformation theory taking into account von Karman geometrical nonlinearity and Pasternak’s two-parameter elastic foundation. The motion equations are determined by Galerkin method and the obtained equation is numerically solved by using Runge–Kutta method. Results of nonlinear dynamic responses show the effects of foundation, material, geometric parameters, and temperature change on the nonlinear vibration of shells.

Keywords

Functionally graded sandwich shallow spherical cap first-order shear deformation theory thermal environment nonlinear vibration

Introduction

Shallow spherical caps (shallow spherical shells) constitute an important portion in many engineering structures, for example, the aircraft, missile and aerospace components. As a result, the problems relating to mechanic behaviors bring major importance in the design of this type of shell structure and have attracted attention of many researchers.

Uemura [1] studied axisymmetric nonlinear buckling of isotropic shallow spherical caps with initial deformation and subject to uniform pressure is analyzed based on the finite-deformation theory. Nonlinear vibration of orthotropic axisymmetric shallow spherical caps was investigated by Varadan and Pandalai [2]. Numerical solutions based on a two-term mode shape, uniform pressure, and free vibrations are obtained. Xu [3] presented nonlinear buckling of axisymmetrically laminated spherical caps, a Fourier–Bessel series solution is proposed with four types of boundary conditions. Sathyamoorthy [4] investigated nonlinear vibrations of shallow spherical caps by using first-order shear deformation theory (FSDT). Dynamic behavior of shell is obtained by using Galerkin method and the Runge–Kutta method. Based on Timoshenko–Mindlin kinematic hypothesis, the shallow shell theory is extended to include the transverse shear deformation; Cheung [5] studied nonlinear static and dynamic behavior for moderate thick laminated, annular, and spherical caps. Nie [6] proposed the asymptotic iteration method to analyze nonlinear buckling of isotropic shallow spherical caps with various boundary conditions and on nonlinear elastic foundation subjected to external pressure. Static and dynamic nonlinear axisymmetric analysis of thick shallow spherical and conical orthotropic caps was studied by Dube et al. [7] using Galerkin method and the FSDT. Eslami et al. [8] studied non-shallow thin spherical shell by using deep and shallow shell theories with Sanders’s nonlinear kinematic relations. Shahsiah and Eslami [9] investigated the buckling behavior of imperfect isotropic shallow spherical caps under thermal and mechanical loads based on Donnell–Mushtari–Velasov theory. By using the updated iteration method, Li et al. [10] investigated nonlinear stability of isotropic shallow spherical caps including the effects of transverse shear deformation with two types of boundary conditions. Thermoelastic response of symmetric simply supported spherical shells subjected to external pressure for mechanical loading and uniform temperature rise and radial temperature difference for thermal loadings is presented by Islam et al. [11]. Wen et al. [12] studied autofrettage and shakedown of an internally pressurized thick-walled spherical shell based on two strain gradient plasticity solutions.

Functionally graded materials (FGMs) are advanced composite material, consisting of two different constituent materials, have properties varying smoothly through the thickness and become popular in engineering design. Many author focused on the static and dynamic behavior of many type of FGM structures. However, a not many reports of FGM spheres are obtained in open literature.

Shahsiah et al. [13,14] analyzed the thermal instability of FGM shallow and deep spherical caps, respectively. Based on shear deformation theory and geometric nonlinearity in von Karman sense, Ganapathi [15] and Prakash et al. [16] investigated the nonlinear dynamic stability of axisymmetric FGM shallow spherical caps. The dynamic buckling pressure is obtained by using the sudden jump in the maximum average deflection criterion. Fu et al. [17] reported nonlinear transient response of functionally graded shallow spherical caps subjected to time-dependent thermomechanical load. Based on the classical shell theory and the Sanders nonlinear kinematics equations, axisymmetric nonlinear buckling of FGM shallow spherical caps surface-bonded piezoelectric actuators subjected to thermo-electro-mechanical loads was investigated by Boroujerd and Eslami [18].

Nonlinear buckling and vibration of axisymmetric and un-axisymmetric functionally graded thin shallow spherical caps under uniform external pressure including temperature effects are analyzed by Bich et al. [19 –22] and Duc et al. [23,24]. In these papers, classical shell theory, stress function method, and Galerkin method were used to investigate nonlinear behavior of shell. By using FSDT, nonlinear axisymmetric thermomechanical static stability of FGM shallow spherical caps resting on elastic foundations were studied by Tung [25,26] and Anh and Duc [27].

Advanced sandwich plate and shell structures have been mentioned in many studies in recent years and these structures may be placed in an elastic medium described by different elastic foundation models [28 –33].

In this paper, functionally graded sandwich materials are combination of many layers of FGM and isotropic materials. The nonlinear vibration axisymmetric analysis of shear deformable functionally graded sandwich shallow spherical caps resting on Pasternak’s elastic foundation is investigated by a semi-analytical approach. Governing equations of spherical caps are established by using the FSDT. Then, these equations are specialized for axisymmetrically deformed shallow spherical caps taking into account geometric nonlinearity and initial geometrical imperfection. By using the Galerkin method, the motion equation of dynamic response is obtained. The nonlinear dynamic responses are found by using Runge–Kutta method. The results show that the foundation, volume–fractions exponent, initial imperfection, and temperature change strongly influence to the vibration behavior of shells.

Functionally graded sandwich shallow spherical caps

Consider a functionally graded sandwich shallow spherical cap with radius of curvature $R$ , base radius $a$ _, and thickness $h$ as shown in Figure 1. The shell is made from a mixture of ceramics and metals, and is defined in coordinate system $(ϕ, θ, z)$ whose origin is located on the middle surface of the shell, $ϕ$ and $θ$ are in the meridional and circumferential directions, respectively, and z is perpendicular to the middle surface and points inwards $- \frac{h}{2} \leq z \leq \frac{h}{2}$ . The shell is subjected to external pressure and rested on the two-parameter Pasternak elastic foundation.

Figure 1.

Geometry and coordinate system of a FGM sandwich spherical cap.

This paper used a Sigmoid law distribution FGM and assumed that the effective properties of FGM sandwich spherical caps, such as the modulus of elasticity $E$ , volumetric mass density $ρ$ , change in the thickness direction $z$ and can be determined by the rule of mixture as

FGM—Isotropic material—FGM models

With model 1A, Material composition of upper layer is smoothly varied along the thickness direction from ceramic-rich surface ( $z = - h / 2$ ) to metal-rich surface ( $z = - h / 2 + h_{t}$ ), material composition of lower layer is smoothly varied along the thickness direction from metal-rich surface ( $z = h / 2 - h_{b}$ ) to ceramic-rich surface ( $z = h / 2$ ) and the core layer is made from full metal as shown in Figure 2(a).

Contrary, with model 1B, Material composition of upper layer is smoothly varied along the thickness direction from metal-rich surface ( $z = - h / 2$ ) to ceramic-rich surface ( $z = - h / 2 + h_{t}$ ), material composition of lower layer is smoothly varied from ceramic-rich surface ( $z = h / 2 - h_{b}$ ) to metal -rich surface ( $z = h / 2$ ) and the core layer is made from full ceramic as shown in Figure 2(b).

Effective properties $\Pr$ of shell are as: Elastic modulus $E$ , thermal extension coefficient $α$ , mass density $ρ$ can be defined by extended Sigmoid power law

[\Pr] = {\begin{array}{l} [\Pr_{i}] + [\Pr_{j i}] {(\frac{2 z + h}{2 h_{t}})}^{k_{t}}, - \frac{h}{2} \leq z \leq - \frac{h}{2} + h_{t}, \\ [\Pr_{j}], - \frac{h}{2} + h_{t} \leq z \leq \frac{h}{2} - h_{b}, \\ [\Pr_{i}] + [\Pr_{j i}] {(\frac{- 2 z + h}{2 h_{b}})}^{k_{b}}, \frac{h}{2} - h_{b} \leq z \leq \frac{h}{2} \end{array}

(1)

where,

i = c, j = m

with model 1A,

i = m, j = c

with model 1B.

$[\Pr] = [E, α, ρ]$ are temperature dependent effective properties

\Pr_{j i} (T, z) = \Pr_{j} (T, z) - \Pr_{i} (T, z)

$k_{t}$ and $k_{b}$ are volume fraction exponents of upper and lower layer, respectively.

The subscripts m and c refer to metal and ceramic constituents, respectively.

b. Isotropic material—FGM—Isotropic material

With model 2A, upper and lower layers are full ceramic and full metal, respectively. Core layer is functionally graded material. Material composition of FGM layer is smoothly varied along the thickness direction from ceramic-rich surface ( $z = - h / 2 + h_{t}$ ) to metal-rich surface ( $z = h / 2 - h_{b}$ ) (see Figure 2(c)).

With model 2B, upper and lower layers are full metal and full ceramic, respectively. Core layer is functionally graded material. Material composition of FGM layer is smoothly varied along the thickness direction from metal-rich surface ( $z = - h / 2 + h_{t}$ ) to ceramic-rich surface ( $z = h / 2 - h_{b}$ ) (see Figure 2(d)).

Effective properties $\Pr$ of shell are as: Elastic modulus $E$ , thermal extension coefficient $α_{s h}$ , volumetric mass density $ρ_{s h}$ can be defined by extended Sigmoid power law

[\Pr] = {\begin{array}{l} [\Pr_{i}], - \frac{h}{2} \leq z \leq - \frac{h}{2} + h_{t}, \\ [\Pr_{i}] + [\Pr_{j i}] {(\frac{2 z + h - 2 h_{t}}{2 h_{c}})}^{k}, - \frac{h}{2} + h_{t} \leq z \leq \frac{h}{2} - h_{b}, \\ [\Pr_{j}], \frac{h}{2} - h_{b} \leq z \leq \frac{h}{2} \end{array}

(2)

where,

i = c, j = m

with model 2A, and

i = m, j = c

with model 2B.

In this paper, the material properties dependent are not only space-dependent, but temperature-dependent. The material properties of FGM’s constituents are usually expressed as [21 –25]

\Pr_{i} (T) = P_{0} (P_{- 1} T^{- 1} + P_{0} T^{0} + P_{1} T^{1} + P_{2} T^{2} + P_{3} T^{3}), i = c, m

(3)

where

T = T_{0} + Δ T

(K), with

T_{0} = 300 K

(room temperature),

P_{- 1}, P_{0}, P_{1}, P_{2}

and

P_{3}

are the coefficients of temperature and are unique to the constituent material. Elastic foundation is assumed to be temperature-independent.

Governing equations

The FSDT is used in this paper to obtain the equation of motion of nonlinear dynamic response of FGM sandwich spherical caps. For a shallow spherical cap, it is convenient to introduce a variable r referred to as the radius of parallel circle and defined by $r = R \sin ϕ$ . Moreover, due to shallowness of the shell it is approximately assumed that $\cos ϕ = 1$ and $R d ϕ = d r$ .

The shallow spherical cap is assumed to be axisymmetric deformation and displacement components in $ϕ, θ, z$ directions, respectively, at a distance z from the middle surface are represented as

\begin{array}{l} \bar{u} (r, z) = u (r) + z ψ (r), \\ \bar{v} (r, z) = 0, \\ \bar{w} (r, z) = w (r) + w^{*} (r) \end{array}

(4)

where

ψ (r)

is the rotation of a normal to the middle surface,

w^{*} (r)

is an imperfection function representing initial small deviation of shell surface from perfect spherical configuration.

The nonzero strain components of the shallow spherical cap are defined as [9,25,26]

{\begin{cases} ε_{r} \\ ε_{θ} \\ ε_{r z} \end{cases}} = {\begin{cases} ε_{r}^{0} + z χ_{r} \\ ε_{θ}^{0} + z χ_{θ} \\ ψ + w_{, r} + w_{, r}^{*} \end{cases}}

(5)

where

ε_{r}^{0}, ε_{θ}^{0}

are the strains at the middle surface,

χ_{r}, χ_{θ}

are curvature constituents which are related to the displacements and rotation in the form

{\begin{matrix} ε_{r}^{0} \\ ε_{θ}^{0} \end{matrix}} = {\begin{matrix} u_{, r} - \frac{w}{R} + \frac{1}{2} w_{, r}^{2} + w_{, r} w_{, r}^{*} \\ \frac{u}{r} - \frac{w}{R} \end{matrix}}, {\begin{matrix} χ_{r} \\ χ_{θ} \end{matrix}} = {\begin{matrix} ψ_{, r} \\ \frac{ψ}{r} \end{matrix}}

(6)

Hookian law for sandwich FGM taking into account the temperature effect [19,20] is

{\begin{matrix} σ_{r} \\ σ_{θ} \end{matrix}} = \frac{E (z)}{1 - ν^{2}} {\begin{matrix} 1 & ν \\ ν & 1 \end{matrix}} {\begin{matrix} ε_{r} \\ ε_{θ} \end{matrix}} - \frac{E (z)}{1 - ν} α (z) {\begin{matrix} Δ T \\ Δ T \end{matrix}}, σ_{r z} = \frac{E (z)}{2 (1 + ν)} ε_{r z}

(7)

Environment temperature is assumed to be uniformly raised from initial value $T_{i}$ , at which the shell is thermal stress free, to final one $T_{f}$ and temperature change $Δ T = T_{f} - T_{j}$ is independent to thickness variable.

The force and moment are determined by

\begin{array}{l} (N_{r}, M_{r}) = \int_{- h / 2}^{h / 2} (1, z) σ_{r} d z, \\ (N_{θ}, M_{θ}) = \int_{- h / 2}^{h / 2} (1, z) σ_{θ} d z, \\ Q_{r} = K_{s} \int_{- h / 2}^{h / 2} σ_{r z} d z \end{array}

(8)

Substituting equation (5) into equation (7), and then substituting the resultant equations into equation (8), the force and moment expressions are obtained

{\begin{matrix} N_{r} \\ N_{θ} \\ M_{r} \\ M_{θ} \end{matrix}} = \frac{1}{1 - ν^{2}} {\begin{matrix} E_{1} & ν E_{1} & E_{2} & ν E_{2} \\ ν E_{1} & E_{1} & ν E_{2} & E_{2} \\ E_{2} & ν E_{2} & E_{3} & ν E_{3} \\ ν E_{2} & E_{2} & ν E_{3} & E_{3} \end{matrix}} {\begin{matrix} ε_{r}^{0} \\ ε_{θ}^{0} \\ χ_{r} \\ χ_{θ} \end{matrix}} - \frac{1}{1 - ν} {\begin{matrix} Φ_{1} \\ Φ_{1} \\ Φ_{2} \\ Φ_{2} \end{matrix}}

(9)

Q_{r} = K_{s} \frac{E_{1}}{2 (1 + ν)} (ψ + w_{, r} + w_{, r}^{*})

(10)

where

(E_{1}, E_{2}, E_{3}) = \int_{- h / 2}^{h / 2} (1, z, z^{2}) E (z) d z

$(Φ_{1}, Φ_{2}) = Δ T \int_{- h / 2}^{h / 2} (1, z,) E (z) α (z) d z$ and $K_{s}$ is shear correction factor.

The nonlinear equations of motion of an imperfect shallow spherical cap resting on an elastic foundation based on the FSDT with neglecting the inplane and rotational inertia, lead to [10]

\begin{array}{l} {(r N_{r})}_{, r} - N_{θ} = 0, \\ {(r M_{r})}_{, r} - M_{θ} - r Q_{r} = 0, \\ {(r Q_{r})}_{, r} + \frac{r}{R} (N_{r} + N_{θ}) + {[r N_{r} (w_{, r} + w_{, r}^{*})]}_{, r} \\ + r [q - K_{1} w + K_{2} (w_{, r r} + \frac{1}{r} w_{, r})] = r ρ_{1} \ddot{w} + 2 r ε ρ_{1} \dot{w} \end{array}

(11)

where

q

is uniformly distributed pressure,

K_{1}, K_{2}

are two foundation parameters of Pasternak model,

ε

is linear damping coefficient.

ρ_{1} = \int_{- h / 2}^{h / 2} ρ (z, T) d z

Substituting equation (6) into extension force, moment, and shear force expressions (9) and (10) then substituting the resultant into equation (11) and neglecting the higher order infinitesimals, the motion equation system is rewritten by

\begin{array}{l} L_{1} \equiv \frac{E_{1}}{1 - ν^{2}} [r u_{, r r} + u_{, r} - \frac{u}{r} - (1 + ν) \frac{r}{R} w_{, r} + \frac{1}{2} (1 - ν) w_{, r}^{2} + (1 - ν) w_{, r} w_{, r}^{*} \\ + r (w_{, r} w_{, r r} + w_{, r r} w_{, r}^{*} + w_{, r} w_{, r r}^{*})] + \frac{E_{2}}{1 - ν^{2}} (r ψ_{, r r} + ψ_{, r} - \frac{ψ}{r}) = 0 \end{array}

(12)

\begin{array}{l} L_{2} \equiv \frac{E_{2}}{1 - ν^{2}} [r u_{, r r} + u_{, r} - \frac{u}{r} - (1 + ν) \frac{r}{R} w_{, r} + \frac{1}{2} (1 - ν) w_{, r}^{2} + (1 - ν) w_{, r} w_{, r}^{*} \\ + r (w_{, r} w_{, r r} + + w_{, r r} w_{, r}^{*} + w_{, r} w_{, r r}^{*})] + \frac{E_{3}}{1 - ν^{2}} (r ψ_{, r r} + ψ_{, r} - \frac{ψ}{r}) \\ - \frac{K_{s} E_{1}}{2 (1 + ν)} (r ψ + r w_{, r} + r w_{, r}^{*}) = 0 \end{array}

(13)

\begin{array}{l} L_{3} \equiv \frac{K_{s} E_{1}}{2 (1 + ν)} [ψ + r ψ_{, r} + w_{, r} + w_{, r}^{*} + r (w_{, r r} + w_{, r r}^{*})] + \frac{E_{2}}{R (1 - ν)} (r ψ_{, r} + ψ) + \\ + \frac{E_{1}}{R (1 - ν)} (r u_{, r} + u - \frac{2}{R} r w + \frac{r}{2} w_{, r}^{2} + r w_{, r} w_{, r}^{*}) + \\ + \frac{E_{1}}{(1 - ν^{2})} [(1 + ν) u_{, r} (w_{, r} + w_{, r}^{*}) - \frac{1 + ν}{R} (w + r w_{, r}) (w_{, r} + w_{, r}^{*}) + \\ - (1 + ν) \frac{r}{R} w (w_{, r r} + w_{, r r}^{*}) + r u_{, r r} (w_{, r} + w_{, r}^{*}) + (r u_{, r} + ν u) (w_{, r r} + w_{, r r}^{*}) + \\ + \frac{1}{2} w_{, r}^{2} (w_{, r} + w_{, r}^{*}) + \frac{3}{2} r w_{, r}^{2} (w_{, r r} + w_{, r r}^{*}) + w_{, r}^{2} w_{, r}^{*} + 3 r w_{, r} w_{, r r} w_{, r}^{*}] + \\ + \frac{E_{2}}{(1 - ν^{2})} {(w_{, r} + w_{, r}^{*}) [(1 + ν) ψ_{, r} + r ψ_{, r r}] + (w_{, r r} + w_{, r r}^{*}) (r ψ_{, r} + ν ψ)} + r q \\ - K_{1} r w + K_{2} (r w_{, r r} + w_{, r}) - \frac{2 r Φ_{1}}{R (1 - ν)} - \frac{Φ_{1}}{1 - ν} [w_{, r} + w_{, r}^{*} + r (w_{, r r} + w_{, r r}^{*})] \\ = r ρ_{1} \ddot{w} + 2 r ε ρ_{1} \dot{w} \end{array}

(14)

Equations (12)–(14) are the basic equations used to investigate the nonlinear vibration axisymmetric stability of functionally graded shallow imperfect spherical caps in thermal environment.

Solution problems

In this paper, the FGM sandwich spherical caps are assumed to be clamped and immovable along the periphery, subjected to external pressure uniformly distributed on the outer surface of the shells and axisymmetric displacement condition. The boundary conditions are written by

\begin{array}{l} ψ = 0, a t r = 0, \\ w = 0, ψ = 0, u = 0, a t r = a \end{array}

(15)

The following approximate solution for the displacement component and rotations is assumed [25,26]

\begin{matrix} u = U (t) \frac{r (a - r)}{a^{2}}, \\ ψ = Ψ (t) \frac{r (a^{2} - r^{2})}{a^{3}}, \\ w = W (t) \frac{{(a^{2} - r^{2})}^{2}}{a^{4}}, \\ w^{*} = ξ h \frac{{(a^{2} - r^{2})}^{2}}{a^{4}} \end{matrix}

(16)

where

U (t)

W (t)

_, and

Ψ (t)

are time-dependent total amplitudes of displacement constituents and rotation,

ξ

is a small coefficient representing imperfection size.

Substituting solution (16) into motion equations (12)–(14) and applying the Galerkin method in full surface area of shell, i.e.

\int_{0}^{a} \int_{0}^{2 π} L_{1} \frac{r (a - r)}{a^{2}} r d θ d r = 0

(17)

\int_{0}^{a} \int_{0}^{2 π} L_{2} \frac{r (a^{2} - r^{2})}{a^{3}} r d θ d r = 0

(18)

\int_{0}^{a} \int_{0}^{2 π} L_{3} \frac{{(a^{2} - r^{2})}^{2}}{a^{4}} r d θ d r = 0

(19)

leads to

a_{11} U + a_{12} Ψ + a_{13} W + a_{14} W (W + 2 ξ h) = 0

(20)

a_{21} U + a_{22} Ψ + a_{23} W + a_{24} (W + ξ h) + a_{25} W (W + 2 ξ h) = 0

(21)

\begin{array}{l} a_{31} U + a_{32} Ψ + a_{33} U (W + ξ h) + a_{34} Ψ (W + ξ h) + a_{35} W \\ + a_{36} (W + ξ h) + a_{37} W (W + ξ h) + a_{38} W^{2} (W + ξ h) \\ + a_{39} W^{2} ξ h + a_{310} W (W + 2 ξ h) + \frac{8}{105} a^{3} q - \frac{16}{105} a^{3} \frac{Φ_{1}}{(1 - ν) R} \\ + \frac{64}{315} a (W + ξ h) \frac{Φ_{1}}{(1 - ν)} = ρ_{1} \frac{128}{3465} a^{3} \ddot{W} + ρ_{1} \frac{256}{3465} a^{3} ε \dot{W} \end{array}

(22)

where the coefficients

a_{i j}

are given in detail in Appendix 1.

$U$ and $Ψ$ expressions are obtained from equations (20) and (21) are

\begin{array}{l} U = \frac{a_{12} a_{23} - a_{22} a_{13}}{a_{11} a_{22} - a_{12} a_{21}} W + \frac{a_{12} a_{24}}{a_{11} a_{22} - a_{12} a_{21}} (W + ξ h) \\ + \frac{a_{12} a_{25} - a_{22} a_{14}}{a_{11} a_{22} - a_{12} a_{21}} W (W + 2 ξ h) \end{array}

(23)

\begin{array}{l} Ψ = \frac{a_{21} a_{13} - a_{11} a_{23}}{a_{11} a_{22} - a_{12} a_{21}} W - \frac{a_{11} a_{24}}{a_{11} a_{22} - a_{12} a_{21}} (W + ξ h) \\ + \frac{a_{21} a_{14} - a_{11} a_{25}}{a_{11} a_{22} - a_{12} a_{21}} W (W + 2 ξ h) \end{array}

(24)

Substituting equations (23) and (24) into equation (22), leads to

\begin{array}{l} (a_{31} b_{11} + a_{32} b_{21} + a_{35}) W + (a_{31} b_{12} + a_{32} b_{22} + a_{36}) (W + ξ h) \\ + (a_{31} b_{13} + a_{32} b_{23} + a_{310}) W (W + 2 ξ h) + (a_{33} b_{11} + a_{34} b_{21} + a_{37}) W (W + ξ h) \\ + a_{38} W^{2} (W + ξ h) + a_{39} W^{2} ξ h + (a_{33} b_{12} + a_{34} b_{22}) {(W + ξ h)}^{2} \\ + (a_{33} b_{13} + a_{34} b_{23}) W (W + ξ h) (W + 2 ξ h) + \frac{8}{105} a^{3} q \\ - \frac{16}{105} a^{3} \frac{Φ_{1}}{(1 - ν) R} + \frac{64}{315} a (W + ξ h) \frac{Φ_{1}}{(1 - ν)} = ρ_{1} \frac{128}{3465} a^{3} \ddot{W} + ρ_{1} \frac{256}{3465} a^{3} ε \dot{W} \end{array}

(25)

where the coefficients

b_{i j}

are given in detail in Appendix 2.

Equation (25) is second-order differential equation with time dependent to investigate the nonlinear vibration of axisymmetric FGM sandwich spherical caps resting on two-parameter elastic foundation and subjected to external pressure and temperature.

In the case of free, linear, and without damping vibration, equation (25) reduces to

\begin{array}{l} ρ_{1} \frac{128}{3465} a^{3} \ddot{W} - (a_{31} b_{11} + a_{32} b_{21} + a_{35}) W \\ - (a_{31} b_{12} + a_{32} b_{22} + a_{36}) W - \frac{64 a}{315} \frac{Φ_{1}}{(1 - ν)} W = 0 \end{array}

(26)

From equation (26), the explicit expression of fundamental frequency of shell is obtained

ω_{0} = \sqrt{- [(a_{31} b_{11} + a_{32} b_{21} + a_{35}) + (a_{31} b_{12} + a_{32} b_{22} + a_{36}) + \frac{64 a}{315} \frac{Φ_{1}}{(1 - ν)}] \frac{3465}{128 ρ_{1} a^{3}}}

(27)

Consider a sandwich FGM spherical cap subjected to harmonic external pressure $q = Q \sin Ω t$ in thermal environment. Equation (25) can be rewritten by

\begin{array}{l} - ρ_{1} \frac{128}{3465} a^{3} \ddot{W} - ρ_{1} \frac{256}{3465} a^{3} ε \dot{W} + (a_{31} b_{11} + a_{32} b_{21} + a_{35}) W + \\ + (a_{31} b_{12} + a_{32} b_{22} + a_{36}) (W + ξ h) + (a_{31} b_{13} + a_{32} b_{23} + a_{310}) W (W + 2 ξ h) + \\ + (a_{33} b_{11} + a_{34} b_{21} + a_{37}) W (W + ξ h) + + a_{38} W^{2} (W + ξ h) + a_{39} W^{2} ξ h + \\ + (a_{33} b_{12} + a_{34} b_{22}) {(W + ξ h)}^{2} + (a_{33} b_{13} + a_{34} b_{23}) W (W + ξ h) (W + 2 ξ h) + \\ - \frac{16}{105} a^{3} \frac{Φ_{1}}{(1 - ν) R} + \frac{64}{315} a (W + ξ h) \frac{Φ_{1}}{(1 - ν)} = - \frac{8}{105} a^{3} Q \sin Ω t \end{array}

(28)

Equation (28) is solved by Runge–Kutta method to obtained the nonlinear forced vibration of shell with the initial condition $W (0) = 0$ and $\dot{W} (0) = 0$ .

In the case of linear force vibration, equation (28) is reduced by neglecting nonlinear terms, leads to

\begin{array}{l} - ρ_{1} \frac{128}{3465} a^{3} \ddot{W} - ρ_{1} \frac{256}{3465} a^{3} ε \dot{W} + (a_{31} b_{11} + a_{32} b_{21} + a_{35}) W \\ + (a_{31} b_{12} + a_{32} b_{22} + a_{36}) (W + ξ h) - \frac{16}{105} a^{3} \frac{Φ_{1}}{(1 - ν) R} + \\ + \frac{64}{315} a (W + ξ h) \frac{Φ_{1}}{(1 - ν)} = - \frac{8}{105} a^{3} Q \sin Ω t \end{array}

(29)

For free vibration case, $q (t) = 0$ , the initial deflection amplitude of shell is $W (0) \neq 0$ or the initial velocity of deflection amplitude $\dot{W} (0) \neq 0$ . Equation (28) leads to

\begin{array}{l} - ρ_{1} \frac{128}{3465} a^{3} \ddot{W} - ρ_{1} \frac{256}{3465} a^{3} ε \dot{W} + (a_{31} b_{11} + a_{32} b_{21} + a_{35}) W + \\ + (a_{31} b_{12} + a_{32} b_{22} + a_{36}) (W + ξ h) + (a_{31} b_{13} + a_{32} b_{23} + a_{310}) W (W + 2 ξ h) + \\ + (a_{33} b_{11} + a_{34} b_{21} + a_{37}) W (W + ξ h) + a_{38} W^{2} (W + ξ h) + a_{39} W^{2} ξ h + \\ + (a_{33} b_{12} + a_{34} b_{22}) {(W + ξ h)}^{2} + (a_{33} b_{13} + a_{34} b_{23}) W (W + ξ h) (W + 2 ξ h) + \\ - \frac{16}{105} a^{3} \frac{Φ_{1}}{(1 - ν) R} + \frac{64}{315} a (W + ξ h) \frac{Φ_{1}}{(1 - ν)} = 0 \end{array}

(30)

Equation (30) is solved by using fourth-order Runge–Kutta method to obtain the nonlinear free vibration response of sandwich FGM spherical caps.

Similarly for linear free vibration case, equation (29) becomes

\begin{array}{l} - ρ_{1} \frac{128}{3465} a^{3} \ddot{W} - ρ_{1} \frac{256}{3465} a^{3} ε \dot{W} + (a_{31} b_{11} + a_{32} b_{21} + a_{35}) W + \\ + (a_{31} b_{12} + a_{32} b_{22} + a_{36}) (W + ξ h) - \frac{16}{105} a^{3} \frac{Φ_{1}}{(1 - ν) R} + \\ + \frac{64}{315} a (W + ξ h) \frac{Φ_{1}}{(1 - ν)} = 0 \end{array}

(31)

Numerical results and discussions

To validate this proposed approach, the fundamental frequency parameters given by the present analysis are compared with the results given by Varadan and Pandalai [2] and Sathyamoorthy [4] in Table 1. As can be observed, a good agreement is obtained in this comparison study.where $H = a^{2} / 2 R$ .

Table 1.

Comparison of linear frequency parameter $ω / λ,$ $λ = {h / a (E / ρ a^{2})}^{1 / 2}$ with results of Varadan and Pandalai [2] and Sathyamoorthy [4].

$H / h$	Varadan and Pandalai [2]		Sathyamoorthy [4]	Present
$H / h$	(One term)	(Two term)	Sathyamoorthy [4]	Present
0	2.98	2.95	2.97	2.93
2	6.20	6.08	6.14	6.51
5	13.92	12.72	13.08	12.45

Fundamental frequencies

In this section, the sandwich FGM spherical cap is considered with geometrical parameters as: $k_{t} = k_{b},$ $h_{t} = h_{b}$ , $a = 1 m$ , $R = 3 m,$ $h = 0.05 m,$ $k_{t} = 1$ , $h_{t} = 0.1 h,$ $ξ = 0.01$ , $q = 10^{5} \sin (100 t)$ N/m², $Δ T = 300 K$ . The Pasternak elastic foundation parameters are $K_{1} = 5 \times 10^{7} N / m^{3}$ , $K_{2} = 2 \times 10^{5} N / m$ . Shear correction is used $K_{s} = \frac{5}{6}$ and Poisson ration is chosen to be 0.3. Used material constituents for investigation are Si₃N₄ and SUS304, the material properties are shown in Table 2.

Table 2.

Temperature coefficients for constituent of Si₃N₄ and SUS304 [25].

Materials	Propeties	$P_{0}$	$P_{1}$	$P_{2}$	$P_{3}$
Si₃N₄	$E (P a)$	3.48e⁺¹¹	−3.070e⁻⁴	2.16e⁻⁷	−8.946e⁻¹¹
	$α (1 / K)$	5.87e⁻⁶	9.095e⁻⁴	0	0
	$ρ (k g / m^{3})$	2370	0	0	0
SUS304	$E (P a)$	2.01e⁺¹¹	3.078e⁻⁴	−6.53e⁻⁷	0
	$α (1 / K)$	12.33e⁻⁶	8.086e⁻⁴	0	0
	$ρ (k g / m^{3})$	8166	0	0	0

Table 3 investigated fundamental frequency of four models of the sandwich FGM spherical cap with difference volume fraction exponent. As can be seem, fundamental frequency of model 1B spherical caps is the largest. Contrary, fundamental frequency of model 1A spherical caps is the smallest. Furthermore, volume fraction exponent strong influence to the fundamental frequency of shell. Fundamental frequency of shell increases when volume fraction exponent increases with models 1A and 2A, the opposite phenomenon are obtained with models 1B and 2B.

Table 3.

Fundamental frequency of four material distribution models of shell (×10³ rad/s) ( $a = 1 m, R = 3 m, h = 0.05 m, h_{t} = 0.1 h, Δ T = 0$ ).

$k$	Model 1A	Model 1B	Model 2A	Model 2B
0.2	2.7368	5.8926	3.1161	4.7059
1	2.8610	5.4192	3.7549	3.7530
5	2.9916	5.0263	4.7039	3.1165
10	3.0222	4.9460	4.9997	2.9964

Effect of temperature change on the fundamental frequency of Sandwich FGM shallow spherical cap with model 1A is presented in Table 4. Clearly, when $a / R$ ratio decreases, fundamental frequency also decreases.

Table 4.

Effect of temperature change on the fundamental frequency of sandwich functionally graded material (FGM) shallow spherical cap (×10³ rad/s) ( $a = 1 m, h = 0.05 m, h_{t} = 0.1 h, k_{t} = 1$ ).

$a / R$	$Δ T = 0 K$	$Δ T = 100 K$	$Δ T = 200 K$	$Δ T = 300 K$	$Δ T = 400 K$
0.5	4.1681	4.0529	3.9059	3.7274	3.5180
0.4	3.3793	3.2502	3.0898	2.8984	2.6764
0.3	2.6055	2.4504	2.2599	2.0320	1.7630
0.2	1.8658	1.6575	1.3912	1.0380	0.4603

Table 5 shows effect of $h_{t} / h$ and elastic foundation on the fundamental frequency of shell (model 1A). The obtained results valid that when $h_{t} / h$ ratio increases (the thickness of FGM layers increase) fundamental frequency increases, and it reaches the greatest value at $h_{t} / h = 0.5$ (symmetric sigmoid FGM). Table 5 also shows fundamental frequency of shell rested on elastic foundation is larger than one of without elastic foundation shell.

Table 5.

Effect of thickness of FGM layer ( $h_{t} / h$ ratio) on the fundamental frequency of sandwich Functionally graded material (FGM) shallow spherical cap (×10³ rad/s) ( $a = 1 m, R = 3 m, h = 0.05 m, k_{t} = 1$ ).

$h_{t} / h$	$Δ T = 0 K$			$Δ T = 300 K$
$h_{t} / h$	Without elastic foundation	Winkler elastic foundation	Pasternak elastic foundation	Without elastic foundation	Winkler elastic foundation	Pasternak elastic foundation
0.1	2.8374	2.8605	2.8610	2.2984	2.3269	2.3275
0.2	3.0336	3.0570	3.0575	2.4956	2.5240	2.5246
0.3	3.2466	3.2704	3.2710	2.7059	2.7345	2.7351
0.4	3.4820	3.5064	3.5070	2.9349	2.9639	2.9646
0.5	3.7472	3.7724	3.7730	3.1898	3.2194	3.2201

Dynamic response behavior

Linear and nonlinear free vibration

Figures 3 and 4 investigate dynamic responses of sandwich shallow spherical caps of undamped free vibration in two cases: linear and nonlinear vibration theories with initial deflection $W (0)$ . As can be observed, amplitudes of linear theory are larger than ones of nonlinear theories. Furthermore, difference between linear and nonlinear vibration theories is proportional to the initial amplitude.

Figures 5 and 6 present deflection–velocity curve of sandwich shallow spherical caps of undamped and damped free vibration, respectively. Figure 5 shows that deflection–velocity curve is a closed circular curve with linear vibration and is a closed uncircular curve with nonlinear vibration. It means that undamped vibration is harmonic with two cases of nonlinear and linear vibration. Contrary, deflection–velocity curve in Figure 6 of damped vibration is unclosed curve and is inharmonic response.

Linear and nonlinear forced vibration

Comparison of dynamic response of nonlinear forced and free vibration of shell is presented in Figure 7. Unlike the free vibration, dynamic response of forced vibration is an irregular curve. Similar free vibration case, amplitudes of linear theory are larger than ones of nonlinear theories. Figure 8 also shows the harmonic beat phenomenon when excitation frequency is near to fundamental frequency of shell. As can be seen, amplitude and length of beat of linear theory are larger than one of nonlinear theory.

Effect of temperature change on the dynamic response of sandwich FGM spherical cap is studied in Figure 9. Clearly, vibration amplitude increases when temperature change increases. Effect of imperfection on the dynamic response of sandwich FGM spherical cap is presented in Figure 10 with $ξ = 0.03$ . The result shows that imperfection strongly influences to the dynamic response of shell.

Comparisons of dynamic response of shell of four models are investigated in Figures 11 and 12 (1A and 2A in Figure 11, 1B and 2B in Figure 12). The obtained results show that amplitude of model 1A is larger than one of model 2A and amplitude of model 2B is larger than one of model 1B.

Figure 2.

Material models of sandwich functionally graded spherical caps. (a) Model 1A, (b) Model 1B, (c) Model 2A, (d) Model 2B.

Figure 3.

Dynamic responses of sandwich shallow spherical caps of undamped free vibration ( $ε = 0$ , $Δ T = 0 K$ , $W (0) = 0.1 m$ ).

Figure 4.

Dynamic responses of sandwich shallow spherical caps of undamped free vibration ( $ε = 0$ , $Δ T = 0 K$ , $W (0) = 0.05 m$ ).

Figure 5.

Deflection–velocity curve of sandwich shallow spherical caps of undamped free vibration ( $W (0) = 0.1 m$ , $Δ T = 0 K$ , $ε = 0$ ).

Figure 6.

Deflection–velocity curve of sandwich shallow spherical caps of nonlinear damped free vibration ( $W (0) = 0.1 m$ , $Δ T = 0 K$ , $ε = 10$ ).

Figure 7.

Comparison of dynamic response of nonlinear forced and free vibration of shell ( $ε = 10$ , $Δ T = 0$ , $q = 10^{7} \sin (1000 t)$ ).

Figure 8.

Comparison of dynamic response of linear and nonlinear forced vibration theories of shell ( $ε = 0$ , $Δ T = 0$ , $q = 10^{5} \sin (3000 t)$ ).

Figure 9.

Effect of temperature change on the dynamic response of sandwich FGM spherical cap ( $ε = 3$ ).

Figure 10.

Effect of imperfection on the dynamic response of sandwich FGM spherical cap ( $ξ = 0.03$ , $Δ T = 0 K$ ).

Figure 11.

Comparison of dynamic response of shell of models 1A and 2A.

Figure 12.

Comparison of dynamic response of shell of models 1B and 2B.

Figure 13.

Effect of FGM thickness ( $h_{t} / h$ ratio) on the dynamic response of sandwich FGM spherical cap ( $Δ T = 200 K$ ).

Figure 14.

Effect of volume fraction exponent on the dynamic response of sandwich FGM spherical cap ( $k_{t} = k$ ).

Effect of FGM thickness, volume fraction exponent, curvature ( $a / R$ ratio) and damping coefficient on the nonlinear vibration of sandwich FGM shallow spherical caps are investigated in Figures 13, 14, 15 and 16, respectively. As can be seen, the FGM thickness, volume fraction exponent and curvature of shell strongly influence to the dynamic response of shell. Contrary, damping coefficient lightly influences to the dynamic behavior of shell.

Figure 15.

Effect of $a / R$ ratio on the nonlinear vibration of sandwich FGM shell ( $Δ T = 0 K$ ).

Figure 16.

Effect of damping coefficient on the nonlinear vibration of sandwich FGM shell ( $Δ T = 0 K$ ).

When the excitation frequency is near to fundamental frequency, the harmonic beat phenomenon is observed in Figure 17. The vibration amplitude rapidly increases when the excitation frequency approaches the fundamental frequency. Strong effect of excitation amplitude on the dynamic response is shown in Figure 18.

Figure 17.

Effect of excitation frequency on the harmonic beat phenomenon of sandwich FGM spherical cap ( $ε = 0$ , $Δ T = 0 K$ , $Q = 10^{5} N / m^{2}$ ).

Figure 18.

Effect of excitation amplitude on the harmonic beat phenomenon of sandwich FGM spherical cap ( $ε = 0$ , $Δ T = 0 K$ , $Ω = 2500$ rad/s).

Deflection–velocity curve of a beat period of nonlinear forced damped and undamped vibration is presented in Figures 19 and 20, respectively. Clearly, vibration is harmonic after each period of beat with undamped case and is inharmonic with damped case.

Figure 19.

Deflection–velocity curve of sandwich shallow spherical caps of undamped forced vibration ( $q = 10^{6} \sin 2500 t$ (N/m²), $ε = 0$ , $Δ T = 0 K$ ).

Figure 20.

Deflection–velocity curve of sandwich shallow spherical caps of damped forced vibration ( $q = 10^{6} \sin 2500 t$ (N/m²), $ε = 20$ , $Δ T = 0 K$ ).

When the excitation frequencies are far from fundamental frequency (Figures 21 and 22), the complex deflection–velocity curves are obtained. When the excitation amplitude is very far from fundamental frequency of shell (Figures 23 and 24), the deflection–velocity curves become disturbed curves.

Figure 21.

Deflection–velocity curve of sandwich shallow spherical caps of undamped forced vibration ( $q = 10^{6} \sin 1000 t$ (N/m²), $ε = 0$ , $Δ T = 0 K$ ).

Figure 22.

Deflection–velocity curve of sandwich shallow spherical caps of undamped forced vibration ( $q = 10^{5} \sin 50000 t$ (N/m²), $ε = 0$ , $Δ T = 0 K$ ).

Figure 23.

Deflection–velocity curve of sandwich shallow spherical caps of undamped forced vibration ( $q = 10^{8} \sin 2500 t$ (N/m²), $ε = 0$ , $Δ T = 0 K$ ).

Figure 24.

Deflection–velocity curve of sandwich shallow spherical caps of undamped forced vibration ( $q = 10^{8} \sin 1000 t$ (N/m²), $ε = 0$ , $Δ T = 0 K$ ).

Conclusion

A semi-analytical approach to investigate the nonlinear vibration axisymmetric of four FGM sandwich models spherical caps subjected to uniform external pressure is proposed in this paper. The spherical caps are assumed to be axisymmetric deformation in the circumferential direction and the FSDT is used to obtain the equation system of motion of shell with geometrical nonlinearity in von-Karman sense and initial geometrical imperfection. This equation system is solved by using Galerkin method and fourth-order Runge–Kutta method is applied to investigate linear and nonlinear vibration behavior of structure in numerical form. Effect of excitation force, geometrical nonlinearity, geometrical imperfection, and temperature change on the dynamic response and deflection–velocity curve were investigated in details.

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 Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02–2017.11.

Appendix 1

References

Uemura

Axisymmetrical buckling of an initially deformed shallow spherical shell under external pressure. Int J Nonlinear Mech 1971; 6: 177–192.

Varadan

Pandalai

KAV.

Nonlinear flexural oscillations of orthotropic shallow spherical shells. Comput Struct 1978; 9: 417–425.

CS.

Buckling and postbuckling of symmetrically laminated moderately thick spherical caps. Int J Solids Struct 1991; 28: 1171–1184.

Sathyamoorthy

Vibrations of moderately thick shallow spherical shells at large amplitudes. J Sound Vibrat 1994; 172: 63–70.

Cheung

YK.

Nonlinear static and dynamic analysis for laminated, annular, spherical caps of moderate thickness. Nonlinear Dyn 1995; 8: 251–268.

Nie

GH.

Asymptotic buckling analysis of imperfect shallow spherical shells on non-linear elastic foundation. Int J Mech Sci 2001; 43: 543–555.

Dube

Joshi

Dumir

PC.

Nonlinear analysis of thick shallow spherical and conical orthotropic caps using Galerkin’s method. Appl Math Modelling 2001; 25: 755–773.

Eslami

Ghorbani

Shakeri

Thermoelastic buckling of thin spherical shells. J Therm Stresses 2001; 24: 1177–1198.

Shahsiah

Eslami

MR.

Thermal and mechanical instability of an imperfect shallow spherical cap. J Therm Stresses 2003; 26: 723–737.

10.

Liu

Tang

Buckling of shallow spherical shells including the effects of transverse shear deformation. Int J Mech Sci 2003; 45: 1519–1529.

11.

Islam

Kar

Kanoria

Thermoelastic response in a symmetric spherical shell. Acta Mech 2014; 225: 2841–2864.

12.

Wen

Gao

Xuan

, et al. Autofrettage and shakedown analyses of an internally pressurized thick-walled spherical shell based on two strain gradient plasticity solutions. Acta Mech 2017; 228: 89–105.

13.

Shahsiah

Eslami

Naj

Thermal instability of functionally graded shallow spherical shell. J Therm

Stresses 2006; 29: 771–790.

14.

Shahsiah

Eslami

Boroujerdy

MS.

Thermal instability of functionally graded deep spherical shell. Arch Appl Mech 2011; 81: 1455–1471.

15.

Ganapathi

Dynamic stability characteristics of functionally graded materials shallow spherical shells. Composite Struct 2007; 79: 338–343.

16.

Prakash

Sundararajan

Ganapathi

On the nonlinear axisymmetric dynamic buckling behavior of clamped functionally graded spherical caps. J Sound Vibrat 2007; 299: 36–43.

17.

Mao

Nonlinear transient response of functionally graded shallow spherical shells subjected to mechanical load and unsteady temperature field. Acta Mechanica Solida Sinica 2014; 27: 496–508.

18.

Boroujerd

Eslami

MR.

Axisymmetric snap-through behavior of Piezo-FGM shallow clamped spherical shells under thermo-electro-mechanical loading. Int J Pressure Vessels Piping 2014; 120–121: 19–26.

19.

Bich

Tung

HV.

Non-linear axisymmetric response of functionally graded shallow spherical shells under uniform external pressure including temperature effects. Int J Nonlinear Mech 2011; 46: 1195–1204.

20.

Bich

Dung

Hoa

LK.

Nonlinear static and dynamic buckling analysis of functionally graded shallow spherical shells including temperature effects. Composite Struct 2012; 94: 2952–2960.

21.

Bich

Dung

Nam

VH.

Nonlinear axisymmetric dynamic buckling and vibration of functionally graded shallow spherical shells under external pressure including temperature effects resting on elastic foundation. In: Vietnam national conference on mechanics of deformable solids, Ho Chi Minh City, 2013, pp. 101–110.

22.

Bich

Dung

Nam

VH.

Nonlinear axisymmetric dynamic analysis of shallow spherical shells with functionally graded coatings under external pressure including temperature effects and resting on elastic foundation. In: Vietnam national conference on mechanic engineering, Hanoi, 2014, pp. 25–30.

23.

Duc

Anh

NTT

Cong

PH.

Nonlinear axisymmetric response of FGM shallow spherical shells on elastic foundations under uniform external pressure and temperature. Eur J Mech A/Solids 2014; 45: 80–89.

24.

Duc

Quang

Anh

VTT.

The nonlinear dynamic and vibration of the S-FGM shallow spherical shells resting on an elastic foundations including temperature effects. Int J Mech Sci 2017; 123: 54–63.

25.

Tung

HV.

Nonlinear thermomechanical stability of shear deformable FGM shallow spherical shells resting on elastic foundations with temperature dependent properties. Compos Struct 2014; 114: 107–116.

26.

Tung

HV.

Nonlinear axisymmetric response of FGM shallow spherical shells with tangential edge constraints and resting on elastic foundations. Compos Struct 2016; 149: 231–238.

27.

Anh

VTT

Duc

ND.

Nonlinear response of shear deformable S-FGM shallow spherical shell with ceramic-metal-ceramic layers resting on elastic foundation in thermal environment. J Mech Adv Mater Struct 2016; 23: 926–934.

28.

Kolahchi

Keshtegar

Fakhar

MH.

Optimization of dynamic buckling for sandwich nanocomposite plates with sensor and actuator layer based on sinusoidal-visco-piezoelasticity theories using Grey Wolf algorithm. J Sandwich Struct Mater. 2020; 22: 3–27.

29.

Kolahchi

A comparative study on the bending, vibration and buckling of viscoelastic sandwich nano-plates based on different nonlocal theories using DC, HDQ and DQ methods. Aerospace Sci Technol 2017; 66: 235–248.

30.

Kolahchia

Zareib

Hajmohammadc

, et al. Visco-nonlocal-reﬁned zigzag theories for dynamic buckling of laminated nanoplates using differential cubature-Bolotin methods. Thin-Walled Struct 2017; 113: 162–169.

31.

Meksi

Benyoucef

Mahmoudi

, et al. An analytical solution for bending, buckling and vibration responses of FGM sandwich plates. J Sandwich Struct Mater. 2019; 21: 727–757.

32.

Dong

Dung

DV.

A third-order shear deformation theory for nonlinear vibration analysis of stiffened functionally graded material sandwich doubly curved shallow shells with four material models. J Sandwich Struct Mater. 2019; 21: 1316–1356.

33.

Tung

HV.

Nonlinear thermomechanical response of pressure-loaded doubly curved functionally graded material sandwich panels in thermal environments including tangential edge constraints. J Sandwich Struct Mater. 2018; 20: 974–1008.