A new semi-analytical approach for nonlinear electro-thermo-mechanical dynamic responses of FG-GPLRC shallow spherical caps and circular plates with porous core

Abstract

The main aim of this paper is to analyze the nonlinear electro-thermo-mechanical dynamic buckling and vibration of third-order shear deformable circular plates and spherical caps with functionally graded graphene platelet reinforced composite (FG-GPLRC) coatings, piezoelectric layers, and porous core. The circular plates and spherical caps are assumed to be rested on the Pasternak visco-elastic foundation, and subjected to dynamic external pressure in the thermal environment. The total potential energy expression of structures is established using the Lagrange function. The Euler-Lagrange equations and Rayleigh dispassion functions are applied to obtain the motion equation of structures. This motion equation can be solved using the Runge-Kutta method to obtain the dynamic responses of circular plates and spherical caps. Significant discussions of the different effects of graphene distribution, graphene volume fraction, piezoelectric layers, porous core, and foundation parameters are presented through the investigated examples.

Introduction

Circular plate and spherical cap are two types of structures popularly designed in ocean engineering, civil engineering, aerospace, and spacecraft engineering. Due to its popularity in the application, many authors studied the electro-thermo-mechanical problems of these structures with different isotropic and advanced materials.

For isotropic materials, the linear mechanical and thermal buckling responses of shallow and deep spherical shells were investigated using Sanders’s shell theory, Euler equations, and the Galerkin method. ¹ First-order shear deformation shell theory (FSDST) and an iteration method are applied to study the nonlinear buckling behavior of moderately thick spherical caps. ² Nonlinear vibration responses of spherical caps were mentioned for moderately thick caps ³ and orthotropic caps. ⁴ Birman and Simitses ⁵ considered the orthotropic properties and investigated the vibration behavior of stiffened shallow spherical thin shells by the closed solution form.

The original idea of classical functionally graded material (FGM) created a revolution in advanced material technology. The mechanical and thermal load-carrying capacities of FGM structures were studied and validated in many different publications. The axisymmetrically mechanical buckling and free vibration behavior of FGM circular plates under in-plane peripheral load was mentioned taking into account the thermal environment and applying the classical plate theory (CPT). ⁶ By applying the differential quadrature method, the two-parameter elastic foundation model, and the first-order shear deformation plate theory (FSDPT), the FGM circular plates with variable thickness were performed for thermal buckling problems. ⁷ The snap-through phenomenon of FGM circular plates in the thermal postbuckling state was investigated for static and dynamic buckling problems. ⁸ Active control for thermal vibration of FGM circular plates with temperature-dependent and piezoelectric layers was investigated using FSDPT. ⁹ The analytical solutions to axisymmetric bending problems for FGM circular plates were presented using the CPT, FSDPT, and the third-order shear deformation plate theory (TSDPT). ¹⁰ By using the implicit layerwise shear correction factors, the dynamic behavior of FGM circular plates was investigated based on the zigzag-elasticity formulations. ¹¹ The stability and hygro-thermal wave propagation of sandwich FGM plates were mentioned using the quasi-3D TSDPT.^12,13 Nonlinear low-velocity impact responses of FGM spherical caps were investigated using the Chebyshev collocation and Newmark methods. ¹⁴ Nonlinear vibration and buckling behavior of FGM spherical caps under mechanical loads and temperature change was studied^15,16 using the first-order shear deformation shell theory (FSDST) and third-order shear deformation shell theory (TSDST), Galerkin and Ritz energy methods, respectively. The thermo-mechanical postbuckling behavior of FGM plates with circular cut-outs was studied using the TSDPT taking into account the plate-foundation interaction. ¹⁷ The FGM annular spherical thin shell segments stiffened by isotropic stiffeners were considered for linear buckling problems using the closed-form solution. ¹⁸ By using the FSDST and penalty method, the vibration of spinning FGM spherical–cylindrical–conical shells was performed considering the Coriolis force, the centrifugal force, and the initial hoop tension. ¹⁹ The flexibility of FGM can be improved if the constituent materials are varied in two dimensions in space. The vibration and bending responses of bi-directional FGM plates and doubly curved shell panels were investigated using the finite element method.^20,21

Graphene platelet (GPL) and graphene sheet are two excellent candidates for reinforcement into isotropic matrices, creating nanocomposite materials with outstanding thermo-mechanical properties.^22,23 Functionally graded GPLs reinforced composite (FG-GPLRC) is manufactured by reinforcing the isotropic matrix with GPLs by the continuous distribution laws through the structure thickness. The great properties of FG-GPLRC can be expressed in the significant effects on the behavior of many beam, plate, and shell structures. Nonlinear deflection, stability, and vibration behavior of FG-GPLRC and porous FG-GPLRC rectangular plates were investigated using the Navier solution, ²⁴ differential quadrature method,^25,26 and variational differential quadrature-finite element method. ²⁷ Free vibration and dynamic responses of FG-GPLRC rectangular plates were studied using quasi-3D plate theory,^28,29 and FSDPT. ³⁰ The free vibration behavior of FG-GPLRC point-supported skew plates was studied using the FSDPT. ³¹ FG-GPLRC doubly curved shells with arbitrary edge supports were considered in the vibration characteristic problem. ³² The FG-GPLRC cylindrical shells were also mentioned in many problems of dynamic analysis, ³³ free vibration and buckling, ³⁴ and thermo-elastic bending. ³⁵ Thermal buckling, vibration, deformation, and stress analysis of FG-GPLRC annular plates was studied using the FSDPT, ³⁶ TSDPT, ³⁷ and under magneto-electro-elastic loads. ³⁸ Linear and nonlinear free vibration behavior of FG-GPLRC circular plates was investigated using the meshfree method, ³⁹ and differential quadrature method. ⁴⁰ Thermal buckling, mechanical buckling, free vibration, and forced vibration of FG-GPLRC spherical caps were studied by applying the Ritz energy method,^41,42 the analytical solution, ⁴³ and the Galerkin method. ⁴⁴ Stiffened FG-GPLRC spherical thin shells were also considered in the nonlinear thermo-mechanical buckling problems by improving a smeared stiffener technique. ⁴⁵

Porous materials are suitably used as the core layer for multilayer structures with properties of lightweight, sound, and heat resistance. Many researches on the thermo-mechanical behavior of sandwich beams, plates, and shells with porous cores were performed, and the effects of porous cores were investigated and validated.^46–54 The effects of porosity distributions and coefficients were investigated and discussed in detail.

Piezoelectric layers may be attached to the structures as sensors or actuators for the control system. Its effects on the wave propagation, and mechanical and thermal buckling behavior of plates and shells were shown in many works.^55–58

For the first time, the semi-analytical approach of the dynamic responses and vibration behavior for FG-GPLRC spherical caps and circular plates with porous core and piezoelectric layers is mentioned in this paper. The mechanical, thermal, and electrical responses of the structures are considered simultaneously and a simple and effective algorithm is established. By using Reddy’s TSDST, ⁵⁸ and von Karman nonlinearities, the basic formulas of the structures are established. The foundation interaction is simulated by the visco-elastic foundation model, and the thermal environment is considered. The motion equation can be obtained using the Euler-Lagrange equations with the viscous damping of the foundation modeled by the Rayleigh dispassion function. Then, the Runge-Kutta method is used to determine the dynamic behavior of considered structures. Remarkable investigations and discussions for the numerical results on the effects of graphene distribution, graphene volume fraction, piezoelectric layers, porous core, and foundation parameters are presented through the investigated examples.

Coordinate system, material, and geometrical properties of shallow spherical caps and circular plates

As shown in Figure 1, the FG-GPLRC spherical caps with porous core and piezoelectric layers are considered, with the thickness $h$ , the radius of the shell’s curvature $R_1$ , base radius $a_0$ , and the spherical coordinate system $\left(\phi ,\theta ,z\right)$ . The shell caps and plates are rested on the viscoelastic foundation with Winkler parameter $K_1$ (N/m³), Pasternak parameter $K_2$ (N/m), and viscous damping coefficient $\mathrm{\mu }$ . For shallow caps, by simplifying the coordinate system to the polar coordinate system $\left(r,\theta ,z\right)$ with $r=R_1\hspace{0.17em}\sin \hspace{0.17em}\mathrm{\phi }$ , a simpler problem can be obtained. For the circular plates, the governing equations are obtained by applying the infinity for the main radius $R_1\to \infty$ .

Figure 1.

Geometrical parameters and Coordinate system of FG-GPLRC spherical caps and circular plates with porous core and piezoelectric layers.

For the FG-GPLRC layers, the UD, X, O, V, and A types of GPL distributions are considered. Piezoelectric layers, FG-GPLRC layers, and porous core are combined into five models as PE/UD/PC/UD/PE, PE/X/PC/X/PE, PE/O/PC/O/PE, PE/V/PC/A/PE, and PE/A/PC/V/PE.

The Halpin-Tsai model is used to determine the elastic modulus as follows ³⁷

$E^{*}=\frac{3{\overline{E}}_m\left(1+{\mathrm{\psi }}_1{\mathrm{\phi }}_1{\overline{V}}_{GPL}\right)}{8\left(1-{\mathrm{\phi }}_1{\overline{V}}_{GPL}\right)}+\frac{5{\overline{E}}_m\left(1+{\mathrm{\psi }}_2{\mathrm{\phi }}_2{\overline{V}}_{GPL}\right)}{8\left(1-{\mathrm{\phi }}_2{\overline{V}}_{GPL}\right)},$

(1)

where

${\mathrm{\phi }}_1=\frac{{\overline{E}}_{GPL}-{\overline{E}}_m}{{\overline{E}}_{GPL}+{\mathrm{\psi }}_1{\overline{E}}_m},{\mathrm{\phi }}_2=\frac{{\overline{E}}_{GPL}-{\overline{E}}_m}{{\overline{E}}_{GPL}+{\mathrm{\psi }}_2{\overline{E}}_m},{\mathrm{\psi }}_1=\frac{2a_{GPL}}{t_{GPL}},{\mathrm{\psi }}_2=\frac{2b_{GPL}}{t_{GPL}},$

(2)

with ${\overline{E}}_m$ and ${\overline{E}}_{GPL}$ are respectively the elastic moduli of the matrix and GPL. $a_{GPL},b_{GPL},$ and $t_{GPL}$ are the denotes of the length, width and thickness of the GPLs, respectively. ${\overline{V}}_{GPL}$ is the volume fraction of the GPLs $\left({\overline{V}}_m+{\overline{V}}_{GPL}=1\right)$ , defined as

${\overline{V}}_{GPL}=\frac{{\overline{\mathrm{\rho }}}_m{\overline{W}}_{GPL}}{{\overline{\mathrm{\rho }}}_m{\overline{W}}_{GPL}+{\overline{\mathrm{\rho }}}_{GPL}\left(1-{\overline{W}}_{GPL}\right)},$

(3)

where ${\overline{\mathrm{\rho }}}_m$ and ${\overline{\mathrm{\rho }}}_{GPL}$ are the densities of the matrix and the GPLs, respectively, ${\overline{W}}_{GPL}$ is the mass fraction of GPLs which depends on the thickness of the shell with the following functions

$+)\hspace{0.17em}\text{The}\hspace{0.17em}\text{upper}\hspace{0.17em}\text{coating}\hspace{0.17em}\left(\frac{-h}{2}+h_{pe}\le z\le \frac{-h_{pc}}{2}\right)$

$‐\text{UD}\hspace{0.17em}\text{type}:{\overline{W}}_{GPL}=W_{GPL}^{*},$

(4)

$‐\mathrm{X}\hspace{0.17em}\text{type}:{\overline{W}}_{GPL}=\left(\left|\frac{8z-4h_{pe}+2h+2h_{pc}}{h-h_{pc}-2h_{pe}}\right|\right)W_{GPL}^{*},$

(5)

$‐\mathrm{O}\hspace{0.17em}\text{type}:{\overline{W}}_{GPL}=\left(2-\left|\frac{8z-4h_{pe}+2h+2h_{pc}}{h-h_{pc}-2h_{pe}}\right|\right)W_{GPL}^{*},$

(6)

$‐\mathrm{V}\hspace{0.17em}\text{type}:{\overline{W}}_{GPL}=\left(\frac{-4z-2h_{pc}}{h-h_{pc}-2h_{pe}}\right)W_{GPL}^{*},$

(7)

$‐\mathrm{\Lambda }\hspace{0.17em}\text{type}:{\overline{W}}_{GPL}=\left(\frac{4z+2h-4h_{pe}}{h-h_{pc}-2h_{pe}}\right)W_{GPL}^{*},$

(8)

$+)\hspace{0.17em}\text{The}\hspace{0.17em}\text{lower}\hspace{0.17em}\text{coating}\left(\frac{h_{pc}}{2}\le z\le -h_{pe}+\frac{h}{2}\right),$

$‐\text{UD}\hspace{0.17em}\text{type}:{\overline{W}}_{GPL}=W_{GPL}^{*},$

(9)

$‐\mathrm{X}\hspace{0.17em}\text{type}:{\overline{W}}_{GPL}=\left(\left|\frac{8z+4h_{pe}-2h-2h_{pc}}{h-h_{pc}-2h_{pe}}\right|\right)W_{GPL}^{*},$

(10)

$‐\mathrm{O}\hspace{0.17em}\text{type}:{\overline{W}}_{GPL}=\left(2-\left|\frac{8z+4h_{pe}-2h-2h_{pc}}{h-h_{pc}-2h_{pe}}\right|\right)W_{GPL}^{*},$

(11)

$‐\mathrm{V}\hspace{0.17em}\text{type}:{\overline{W}}_{GPL}=\left(\frac{4z-2h_{pc}}{h-h_{pc}-2h_{pe}}\right)W_{GPL}^{*},$

(12)

$‐\mathrm{\Lambda }\hspace{0.17em}\text{type}:{\overline{W}}_{GPL}=\left(\frac{-4z+2h-4h_{pe}}{h-h_{pc}-2h_{pe}}\right)W_{GPL}^{*},$

(13)

where $W_{GPL}^{*}$ is the denote of the total mass fraction of GPLs.

The Poisson’s ratio and thermal expansion coefficient of FG-GPLRC spherical caps are determined according to the mixture rule as

${\mathrm{\nu }}^{*}={\overline{\mathrm{\nu }}}_m{\overline{V}}_m+{\overline{\mathrm{\nu }}}_{GPL}{\overline{V}}_{GPL},{\mathrm{\alpha }}^{*}={\overline{\mathrm{\alpha }}}_m{\overline{V}}_m+{\overline{\mathrm{\alpha }}}_{GPL}{\overline{V}}_{GPL},{\mathrm{\rho }}^{*}={\overline{\mathrm{\rho }}}_m{\overline{V}}_m+{\overline{\mathrm{\rho }}}_{GPL}{\overline{V}}_{GPL}.$

(14)

The piezoelectric layers are assumed to be PZT-5A with $E_0^{pe}=63$ GPa, $G_0^{pe}=24.2$ GPa, ${\mathrm{\nu }}_{pe}=0.3$ , ${\mathrm{\alpha }}_0^{pe}=0.9\times {10}^{-6}$ /K, ${\mathrm{\rho }}_{pe}=7600$ kg/m³, $E_1^{pe}=-0.0005$ , $G_1^{pe}=-0.0002$ , ${\mathrm{\alpha }}_1^{pe}=0.0005$ , ${\overline{D}}_{31}={\overline{D}}_{32}=2.54\times {10}^{-10}$ m/V. ⁴⁰ In this paper, to accommodate the temperature-independent material properties assumption of FG-GPLRC, the piezoelectric material is also assumed to be temperature-independent, leading to

$E_{pe}=E_0^{pe},G_{pe}=G_0^{pe},{\mathrm{\alpha }}_{pe}={\mathrm{\alpha }}_0^{pe}.$

(15)

The core layers are made of the same material as the matrix in this paper. Effective features of the porous core including Young’s modulus, coefficient of thermal expansion, and Poisson ratio can be estimated as⁴⁶

$\{\begin{array}{l}{E}_{pc}={\overline{E}}_m\left[1-e_0\hspace{0.17em}\cos \left(\mathrm{\pi }z/h\right)\right]\\ {\mathrm{\alpha }}_{pc}={\overline{\mathrm{\alpha }}}_m\\ {\mathrm{\rho }}_{pc}={\overline{\mathrm{\rho }}}_m\left[1-\left(1-\sqrt{1-e_0}\right)\cos \left(\mathrm{\pi }z/h\right)\right]\end{array},-\frac{h_{pc}}{2}<z<\frac{h_{pc}}{2},$

(16)

where $e_0$ is the porosity coefficient ( $0\le e_0\le 1$ ).

Hooke’s law is applied to the FG-GPLRC layers and porous core considering the thermal strains, presented as

$\left\{\begin{array}{l}{\mathrm{\sigma }}_r\\ {\mathrm{\sigma }}_{\mathrm{\theta }}\end{array}\right\}=\left[\begin{array}{ll}{Q}_{11}^{*}& Q_{12}^{*}\\ Q_{12}^{*}& Q_{22}^{*}\end{array}\right]\left\{\left\{\begin{array}{l}{\mathrm{\epsilon }}_r\\ {\mathrm{\epsilon }}_{\mathrm{\theta }}\end{array}\right\}-\left\{\begin{array}{l}{\mathrm{\epsilon }}_r^{*}\\ {\mathrm{\epsilon }}_{\mathrm{\theta }}^{*}\end{array}\right\}\right\},{\mathrm{\sigma }}_{rz}=Q_{44}^{*}{\mathrm{\epsilon }}_{rz},$

(17)

where $\left\{\begin{array}{l}{\mathrm{\epsilon }}_r^{*}\\ {\mathrm{\epsilon }}_{\mathrm{\theta }}^{*}\end{array}\right\}=\left\{\begin{array}{l}{\mathrm{\alpha }}^{*}\\ {\mathrm{\alpha }}^{*}\end{array}\right\}\mathrm{\Delta }T,$ and for the piezoelectric layers, presented as

$\left\{\begin{array}{l}{\mathrm{\sigma }}_r\\ {\mathrm{\sigma }}_{\mathrm{\theta }}\end{array}\right\}=\left[\begin{array}{ll}{Q}_{11}^{pe}& Q_{12}^{pe}\\ Q_{12}^{pe}& Q_{22}^{pe}\end{array}\right]\left\{\left\{\begin{array}{l}{\mathrm{\epsilon }}_r\\ {\mathrm{\epsilon }}_{\mathrm{\theta }}\end{array}\right\}-\left\{\begin{array}{l}{\mathrm{\epsilon }}_r^{*}\\ {\mathrm{\epsilon }}_{\mathrm{\theta }}^{*}\end{array}\right\}\right\},{\mathrm{\sigma }}_{rz}=Q_{44}^{pe}{\mathrm{\epsilon }}_{rz},$

(18)

where the temperature difference from the freely thermal state to the thermally deformed state is denoted by $\mathrm{\Delta }T$ , and

$\left\{\begin{array}{l}{\mathrm{\epsilon }}_r^{*}\\ {\mathrm{\epsilon }}_{\mathrm{\theta }}^{*}\end{array}\right\}=\left\{\begin{array}{l}{\mathrm{\alpha }}_{pe}\mathrm{\Delta }T+\frac{{\overline{D}}_{31}{V}_0}{h_{pe}}\\ {\mathrm{\alpha }}_{pe}\mathrm{\Delta }T+\frac{{\overline{D}}_{32}{V}_0}{h_{pe}}\end{array}\right\}$ , and the reduced stiffnesses are determined for GPLRC layers as

$Q_{11}^{*\left(GPLRC\right)}=Q_{22}^{*\left(GPLRC\right)}=\frac{E^{*}}{1-{\left({\mathrm{\nu }}^{*}\right)}^2},Q_{12}^{*\left(GPLRC\right)}=\frac{E^{*}{\mathrm{\nu }}^{*}}{1-{\left({\mathrm{\nu }}^{*}\right)}^2},Q_{44}^{*\left(GPLRC\right)}=\frac{E^{*}}{2+2{\mathrm{\nu }}^{*}},$

(19)

for the porous core layer

$Q_{11}^{*\left(pc\right)}=Q_{22}^{*\left(pc\right)}=\frac{E_{pc}}{1-{\overline{\mathrm{\nu }}}_m^2},Q_{12}^{*\left(pc\right)}=\frac{E_{pc}{\overline{\mathrm{\nu }}}_m}{1-{\overline{\mathrm{\nu }}}_m^2},Q_{44}^{*\left(pc\right)}=\frac{E_{pc}}{2+2{\overline{\mathrm{\nu }}}_m},$

(20)

and for the piezoelectric layers

$Q_{11}^{pe}=Q_{22}^{pe}=\frac{E_{pe}}{1-{\mathrm{\nu }}_{pe}^2},Q_{12}^{pe}=\frac{E_{pe}{\mathrm{\nu }}_{pe}}{1-{\mathrm{\nu }}_{pe}^2},Q_{44}^{pe}=G_{pe}.$

(21)

Based on the TSDST of Reddy with the axisymmetrically deformed assumption, the displacement components at a distance $z$ from the mid-plane of the spherical caps can be expressed as^41,42

$\begin{array}{l}\overline{u}\left(r,z\right)=u\left(r\right)+z{u}_1\left(r\right)-\left[u_1\left(r\right)+w{\left(r\right)}_{,r}\right]\frac{4z^3}{3h^2},\\ \overline{w}\left(r,z\right)=w\left(r\right)+w_0\left(r\right),\end{array}$

(22)

where $u_1\left(r\right)$ and $w_0\left(r\right)$ are the rotation, and the geometrically imperfect deflection of spherical caps. The displacement in the latitudinal direction $\overline{v}\left(r,z\right)=0$ according to the axisymmetrically deformed assumption.

The strain components of the axisymmetric shallow spherical caps at a distance of $z$ from the mid-plane are presented as^41,42

$\left\{\begin{array}{l}{\mathrm{\epsilon }}_r\\ {\mathrm{\epsilon }}_{\mathrm{\theta }}\\ {\mathrm{\epsilon }}_{rz}\end{array}\right\}=\left\{\begin{array}{l}{\overline{\mathrm{\epsilon }}}_r\\ {\overline{\mathrm{\epsilon }}}_{\mathrm{\theta }}\\ {\overline{\mathrm{\epsilon }}}_{rz}\end{array}\right\}+z\left\{\begin{array}{l}{u}_{1,r}\\ \frac{u_1}{r}\\ 0\end{array}\right\}-z^2\mathrm{\lambda }\left\{\begin{array}{l}0\\ 0\\ u_1+w_{,r}\end{array}\right\}-z^3\mathrm{\lambda }\left\{\begin{array}{l}{u}_{1,r}+w_{,rr}\\ \frac{1}{r}{w}_{,r}+\frac{u_1}{r}\\ 0\end{array}\right\},$

(23)

where $\mathrm{\lambda }=4/3h^2$ , ${\overline{\mathrm{\epsilon }}}_r,{\overline{\mathrm{\epsilon }}}_{\mathrm{\theta }},{\overline{\mathrm{\epsilon }}}_{rz}$ are the middle plane strains of shells formulated using the large deflection nonlinearity of von Karman, as

$\begin{array}{l}{\overline{\mathrm{\epsilon }}}_r=u_{,r}-\frac{w}{R_1}+w_{,r}{w}_{0,r}+\frac{1}{2}{w}_{,r}^2,\\ {\overline{\mathrm{\epsilon }}}_{\mathrm{\theta }}=-\frac{w}{R_1}+\frac{u}{r},\\ {\overline{\mathrm{\epsilon }}}_{rz}=w_{,r}+u_1.\end{array}$

(24)

The expressions of internal forces and moments, such as the extension and shear forces, moments and third-order moments and shear forces can be achieved by integrating Hooke’s law, as

$\begin{array}{l}\left\{\begin{array}{l}{N}_r\\ N_{\mathrm{\theta }}\\ M_r\\ M_{\mathrm{\theta }}\\ P_r\\ P_{\mathrm{\theta }}\end{array}\right\}=\left\{\begin{array}{llllll}{b}_{11}& b_{12}& 0& 0& 0& 0\\ b_{12}& b_{22}& 0& 0& 0& 0\\ 0& 0& f_{11}& f_{12}& g_{11}& g_{12}\\ 0& 0& f_{12}& f_{22}& g_{12}& g_{22}\\ 0& 0& g_{11}& g_{12}& l_{11}& l_{12}\\ 0& 0& g_{12}& g_{22}& l_{12}& l_{22}\end{array}\right\}\left\{\begin{array}{c}{\overline{\mathrm{\epsilon }}}_r\\ {\overline{\mathrm{\epsilon }}}_{\mathrm{\theta }}\\ u_{1,r}\\ \frac{u_1}{r}\\ -\mathrm{\lambda }\left(u_{1,r}+w_{,rr}\right)\\ -\mathrm{\lambda }\left(\frac{u_1}{r}+\frac{1}{r}{w}_{,r}\right)\end{array}\right\}-\left\{\begin{array}{l}{\mathrm{\varphi }}_1\\ {\mathrm{\varphi }}_1\\ 0\\ 0\\ 0\\ 0\end{array}\right\},\\ \left\{\begin{array}{l}{Q}_r\\ R_r\end{array}\right\}=\left\{\begin{array}{ll}{m}_{44}& m_{45}\\ m_{45}& m_{55}\end{array}\right\}\left\{\begin{array}{c}{\overline{\mathrm{\epsilon }}}_{rz}\\ -\mathrm{\lambda }\left(u_1+w_{,r}\right)\end{array}\right\},\end{array}$

(25)

where

$\begin{array}{l}\left(b_{ij},f_{ij},g_{ij},l_{ij}\right)={\int }_{-h/2}^{-h/2+h_{pe}}{Q}_{ij}^{pe}\left(1,z^2,z^4,z^6\right)dz\\ +{\int }_{-h/2+h_{pe}}^{-h_{pc}/2}{Q}_{ij}^{*\left(GPLRC\right)}\left(1,z^2,z^4,z^6\right)dz+{\int }_{-h_{pc}/2}^{h_{pc}/2}{Q}_{ij}^{*\left(pc\right)}\left(1,z^2,z^4,z^6\right)dz\\ +{\int }_{h_{pc}/2}^{-h_{pe}+h/2}{Q}_{ij}^{*\left(GPLRC\right)}\left(1,z^2,z^4,z^6\right)dz+{\int }_{-h_{pe}+h/2}^{h/2}{Q}_{ij}^{pe}\left(1,z^2,z^4,z^6\right)dz,\left(\hspace{0.17em}ij=1,2\hspace{0.17em}\right),\end{array}$

$\begin{array}{l}{\mathrm{\varphi }}_1={\int }_{-h/2}^{-h/2+h_{pe}}\left(Q_{11}^{pe}+Q_{12}^{pe}\right){\mathrm{\alpha }}_{pe}\mathrm{\Delta }Tdz\\ +{\int }_{-h/2+h_{pe}}^{-h_{pc}/2}\left(Q_{11}^{*\left(GPLRC\right)}+Q_{12}^{*\left(GPLRC\right)}\right){\mathrm{\alpha }}^{*}\mathrm{\Delta }Tdz+{\int }_{-h_{pc}/2}^{h_{pc}/2}\left(Q_{11}^{*\left(pe\right)}+Q_{12}^{*\left(pe\right)}\right){\mathrm{\alpha }}_{pc}\mathrm{\Delta }Tdz\\ +{\int }_{h_{pc}/2}^{-h_{pe}+h/2}\left(Q_{11}^{*\left(GPLRC\right)}+Q_{12}^{*\left(GPLRC\right)}\right){\mathrm{\alpha }}^{*}\mathrm{\Delta }Tdz+{\int }_{-h_{pe}+h/2}^{h/2}\left(Q_{11}^{pe}+Q_{12}^{pe}\right){\mathrm{\alpha }}_{pe}\mathrm{\Delta }Tdz\\ +{\int }_{-h/2}^{-h/2+h_{pe}}\frac{\left(Q_{11}^{pe}{\overline{D}}_{31}+Q_{12}^{pe}{\overline{D}}_{32}\right)V_0}{h_{pe}}dz+{\int }_{-h_{pe}+h/2}^{h/2}\frac{\left(Q_{11}^{pe}{\overline{D}}_{31}+Q_{12}^{pe}{\overline{D}}_{32}\right)V_0}{h_{pe}}dz.\end{array}$

$\begin{array}{l}\left(m_{44},m_{45},m_{55}\right)={\int }_{-h/2}^{-h/2+h_{pe}}{Q}_{44}^{pe}\left(1,z^2,z^4\right)dz+{\int }_{-h/2+h_{pe}}^{-h_{pc}/2}{Q}_{44}^{*\left(GPLRC\right)}\left(1,z^2,z^4\right)dz\\ +{\int }_{-h_{pc}/2}^{h_{pc}/2}{Q}_{44}^{*\left(pe\right)}\left(1,z^2,z^4\right)dz+{\int }_{h_{pc}/2}^{-h_{pe}+h/2}{Q}_{44}^{*\left(GPLRC\right)}\left(1,z^2,z^4\right)dz+{\int }_{-h_{pe}+h/2}^{h/2}{Q}_{44}^{pe}\left(1,z^2,z^4\right)dz,\end{array}$

The electro-thermo-elastic strain energy of the shell caps and plates, the work done by the external pressure and foundation-structure interaction, and inertia energy are respectively calculated as

${\mathrm{\Theta }}_{\mathrm{int}}=\mathrm{\pi }\underset{-h/2}{\overset{h/2}{{\displaystyle \int }}}\underset{0}{\overset{a_0}{{\displaystyle \int }}}\left[{\mathrm{\sigma }}_r\left({\mathrm{\epsilon }}_r-{\mathrm{\epsilon }}_r^{*}\right)+{\mathrm{\sigma }}_{\mathrm{\theta }}\left({\mathrm{\epsilon }}_{\mathrm{\theta }}-{\mathrm{\epsilon }}_{\mathrm{\theta }}^{*}\right)+{\mathrm{\sigma }}_{rz}{\mathrm{\epsilon }}_{rz}\right]\hspace{0.17em}rdrdz,$

(26)

${\mathrm{\Theta }}_{ext}=2\mathrm{\pi }\underset{0}{\overset{a_0}{{\displaystyle \int }}}qwrdr-\mathrm{\pi }\underset{0}{\overset{a_0}{{\displaystyle \int }}}\left\{\left[K_{1}w-K_2\left(w_{,rr}+\frac{1}{r}{w}_{,r}\right)\right]w\right\}rdr,$

(27)

${\mathrm{\Theta }}_T=\mathrm{\pi }\underset{-h/2}{\overset{h/2}{{\displaystyle \int }}}\underset{0}{\overset{a_0}{{\displaystyle \int }}}\mathrm{\rho }{\overline{w}}^{2}rdrdz.$

(28)

From equations (18) and (19), the total potential energy can be expressed as

${\mathrm{\Theta }}_{Total}={\mathrm{\Theta }}_T+{\mathrm{\Theta }}_{ext}-{\mathrm{\Theta }}_{\mathrm{int}}.$

(29)

Boundary Condition, Approximate Solutions, and Solving Problem

The FG-GPLRC spherical caps and circular plates with porous core and piezoelectric layers are supported by the clamped boundary condition as

$\begin{array}{l}\text{At}\hspace{0.17em}r=0:u=0,w=\text{finite},u_1=0,w_{,r}=0,\\ \text{At}\hspace{0.17em}r=a_0:w=0,w_{,r}=0,u_1=0,u=0\hspace{0.17em}.\end{array}$

(30)

The approximate solutions for the displacements and rotation can be chosen corresponding to the boundary condition (20), as ³

$u=U\frac{a_0-r}{a_0^2}r,u_1=U_1\frac{a_0^2-r^2}{a_0^3}r,w=W\frac{{\left(a_0^2-r^2\right)}^2}{a_0^4},$

(31)

where the amplitudes of the displacement, rotation, and deflection are denoted by $U,U_1,W$ , respectively.

The geometrically imperfect deflection $w_0$ is chosen as the same form of deflection, as

$w_0=\mathrm{\vartheta }h\frac{{\left(a_0^2-r^2\right)}^2}{a_0^4},$

(32)

where $\mathrm{\vartheta }$ is imperfection size which is assumed to be small in comparison with 1.

Utilizing the Rayleigh dissipation function, the potential function of the viscoelastic foundation is presented by

${\overline{D}}_0=\mathrm{\pi }\underset{0}{\overset{R_1}{{\displaystyle \int }}}\mathrm{\mu }{\left(\stackrel{.}{w}\right)}^{2}rdr$

(33)

The Euler–Lagrange equations are applied to obtain the motion equations of considered structures combined with the Rayleigh dissipation function, obtained as

$\begin{array}{l}\frac{d}{dt}\left(\frac{\partial {\mathrm{\Theta }}_{Total}}{\partial \stackrel{.}{W}}\right)-\frac{\partial {\mathrm{\Theta }}_{Total}}{\partial W}+\frac{\partial {\overline{D}}_0}{\partial \stackrel{.}{W}}=0,\\ \frac{d}{dt}\left(\frac{\partial {\mathrm{\Theta }}_{Total}}{\partial \stackrel{.}{U}}\right)-\frac{\partial {\mathrm{\Theta }}_{Total}}{\partial U}=0,\\ \frac{d}{dt}\left(\frac{\partial {\mathrm{\Theta }}_{Total}}{\partial {\stackrel{.}{U}}_1}\right)-\frac{\partial {\mathrm{\Theta }}_{Total}}{\partial U_1}=0,\end{array}$

(34)

leads to

$U{B}_{11}+W{B}_{13}+W\left(W+2\mathrm{\vartheta }h\right)B_{14}=0,$

(35)

$U_1{B}_{22}+W{B}_{23}=0,$

(36)

$\begin{array}{l}U{B}_{31}+U_1{B}_{32}+U\left(W+\mathrm{\vartheta }h\right)B_{33}+W\left(W+4\mathrm{\vartheta }h/3\right)B_{35}+W\left(W+\mathrm{\vartheta }h\right)\left(W+2\mathrm{\vartheta }h\right)B_{36}\\ +W{B}_{38}+\left(W+\mathrm{\vartheta }h\right)B_{39}+B_{310}+q{B}_{311}=\frac{\mathrm{\pi }{a}_0^2}{5}\left(\ddot{W}{B}_{312}+\mathrm{\mu }\stackrel{.}{W}\right),\end{array}$

(37)

where

$B_{11}=-\frac{\mathrm{\pi }\left(2b_{11}+b_{22}\right)}{6\hspace{1em}},B_{13}=\frac{a_0\left(19b_{22}+3b_{11}+22b_{12}\right)\mathrm{\pi }}{R_1},B_{14}=\frac{\mathrm{\pi }\left(46b_{11}-82b_{12}\right)}{315a_0},$

$B_{22}=-\frac{\mathrm{\pi }}{12}\left[\left(9m_{55}{a}_0^2+12l_{11}+4l_{22}\right){\mathrm{\lambda }}^2-\left(6m_{45}{a}_0^2+24g_{11}+8g_{22}\right)\mathrm{\lambda }+a_0^2{m}_{44}+12f_{11}+4f_{22}\right],$

$B_{23}=-\frac{\mathrm{\pi }\left[\left(-3m_{55}{a}_0^2-4l_{11}-4l_{22}/3\right){\mathrm{\lambda }}^2+\left(2m_{45}{a}_0^2+4g_{11}+4g_{22}/3\right)\mathrm{\lambda }-a_0^2{m}_{44}/3\right]}{a_0},$

$B_{31}=\frac{\mathrm{\pi }{a}_0\left(19b_{22}+3b_{11}+22b_{12}\right)}{105R_1},B_{33}=\frac{2\mathrm{\pi }\left(-82b_{12}+46b_{11}\right)}{315a_0},$

$B_{32}=-\frac{\mathrm{\pi }\left[\left(-3a_0^2{m}_{55}-4l_{11}-4l_{22}/3\right){\mathrm{\lambda }}^2+\left(2m_{45}{a}_0^2+4g_{11}+4g_{22}/3\right)\mathrm{\lambda }-m_{44}{a}_0^2/3\right]}{R_1},$

$B_{35}=\frac{4\left(b_{11}+b_{12}\right)\mathrm{\pi }}{5R_1},B_{36}=-\frac{128\mathrm{\pi }{b}_{11}}{105a_0^2},$

$\begin{array}{l}{B}_{38}=\frac{\mathrm{\pi }\left[-K_1{a}_0^4/20+\left(-3{\mathrm{\lambda }}^2{m}_{55}+2m_{45}\mathrm{\lambda }-K_2/3-m_{44}/3\right)a_0^2-4{\mathrm{\lambda }}^2\left(l_{11}+l_{22}/3\right)\right]}{4{a_0}^2}\\ \hspace{0.17em}-\frac{\mathrm{\pi }{a}_0^2\left(b_{11}+2b_{12}+b_{22}\right)}{5R_1^2},\end{array}$

$B_{39}=4\mathrm{\pi }{\mathrm{\varphi }}_1/3,B_{310}=-\frac{a_0^2{\mathrm{\varphi }}_1\mathrm{\pi }}{3R_1},B_{311}=\mathrm{\pi }{a}_0^2/3,$

$B_{312}={\int }_{-h/2}^{-h/2+h_{pe}}{\mathrm{\rho }}_{pe}dz+{\int }_{-h/2+h_{pe}}^{-h_{pc}/2}{\mathrm{\rho }}^{*}dz+{\int }_{-h_{pc}/2}^{h_{pc}/2}{\mathrm{\rho }}_{pc}dz+{\int }_{h_{pc}/2}^{-h_{pe}+h/2}{\mathrm{\rho }}^{*}dz+{\int }_{-h_{pe}+h/2}^{h/2}{\mathrm{\rho }}_{pe}dz.$

From equations (25) and (26), the amplitudes $U$ and $U_1$ can be obtained in the forms, as

$U=C_{11}W+C_{12}W\left(W+2\mathrm{\vartheta }h\right),$

(38)

$U_1=C_{21}W,$

(39)

where

$C_{11}=-\frac{B_{13}{B}_{22}}{B_{11}{B}_{22}},C_{12}=-\frac{B_{14}{B}_{22}}{B_{11}{B}_{22}},C_{21}=-\frac{B_{11}{B}_{23}}{B_{11}{B}_{22}}.$

Substituting $U$ and $U_1$ in equations (28) and (29) into equation (27), leads to

$\begin{array}{l}\left(B_{31}{C}_{11}+B_{32}{C}_{21}+B_{38}\right)W+B_{39}\left(h\mathrm{\vartheta }+W\right)+\left(B_{33}{C}_{11}+B_{34}{C}_{21}\right)W\left(h\mathrm{\vartheta }+W\right)\\ +B_{31}{C}_{12}W\left(2\mathrm{\vartheta }h+W\right)+\left(B_{33}{C}_{12}+B_{36}\right)W\left(2\mathrm{\vartheta }h+W\right)\left(\mathrm{\vartheta }h+W\right)\\ +B_{35}W\left(W+4\mathrm{\vartheta }h/3\right)+B_{310}+B_{311}\hspace{0.17em}q-\frac{\mathrm{\pi }}{5}{a}_0^2\left(\mathrm{\mu }\stackrel{.}{W}+B_{312}\ddot{W}\right)=0.\end{array}$

(40)

The equation of motion (40) is used to analyze the dynamic buckling and vibration of plates and caps. The different functions of the radial load $q$ over time are applied and the dynamic responses of the considered structures can be achieved using the Runge-Kutta method.

For dynamic buckling behavior, a linear function over time $q=\mathrm{\xi }t$ is applied for the external pressure, where $\mathrm{\xi }$ is defined as the loading speed.

The fundamental frequency expression of linear and free vibration of plates and shell caps can be achieved by ignoring the nonlinear terms, applied loads, and damping term, presented as

$\mathrm{\omega }={\left[-\frac{5\left(B_{31}{C}_{11}+B_{32}{C}_{21}+B_{38}+B_{39}\right)}{\mathrm{\pi }{a}_0^2{B}_{312}}\right]}^{\frac{1}{2}}.$

(41)

For the vibration problem, In equation (40), the applied load is chosen in the harmonic form $q=Q\hspace{0.17em}\sin \hspace{0.17em}\mathrm{\Omega }t$ for the case of forced vibration. The vibration amplitude is also assumed to be harmonic form $W=\mathrm{\tau }\hspace{0.17em}\sin \left(\mathrm{\Omega }t\right)$ . Then, the like-Galerkin method is used, which leads to

${\mathrm{\zeta }}^2-\frac{2\mu \zeta }{B_{312}\pi \omega }=\frac{3{\mathrm{\tau }}^2{A}_2}{4}+\frac{8\mathrm{\tau }{A}_1}{3\mathrm{\pi }}+\frac{Q{A}_4}{\mathrm{\tau }{\mathrm{\omega }}^2}+\frac{4A_3}{\pi \tau {\mathrm{\omega }}^2}+1,$

(42)

where the dimensionless frequency is defined as $\mathrm{\zeta }=\mathrm{\Omega }/\mathrm{\omega }$ , and

$A_1=-\frac{5\left(B_{31}{C}_{12}+B_{33}{C}_{11}+B_{35}\right)}{\mathrm{\pi }{a}_0^2{B}_{312}{\mathrm{\omega }}^2},A_2=-\frac{5\left(B_{33}{C}_{12}+B_{36}\right)}{\mathrm{\pi }{a}_0^2{B}_{312}{\mathrm{\omega }}^2},A_3=-\frac{5B_{310}}{\mathrm{\pi }{a}_0^2{B}_{312}},A_4=-\frac{5B_{311}}{\mathrm{\pi }{a}_0^2{B}_{312}}.$

By using the simple calculation process for equation (42), the frequency-amplitude curve $\mathrm{\zeta }-\mathrm{\tau }$ is determined by

$\mathrm{\zeta }=\sqrt{\frac{3{\mathrm{\tau }}^2{A}_2}{4}+\frac{8\mathrm{\tau }{A}_1}{3\mathrm{\pi }}+\frac{4A_3}{\pi \tau {\mathrm{\omega }}^2}+\frac{Q{A}_4}{\mathrm{\tau }{\mathrm{\omega }}^2}+1-{\left(\frac{2\mathrm{\mu }}{B_{312}\pi \omega }\right)}^2}+\frac{2\mathrm{\mu }}{B_{312}\pi \omega }.$

(43)

Validation, Investigation, and Discussion

The dimensionless fundamental frequency of FG-GPLRC circular plates without porous core of present results are compared with those of the numerical results of Javani et al., ⁴⁰ and Chien and Phuc ³⁹ in Table 1. Javani et al. ⁴⁰ used the FSDPT and generalized differential quadrature method, meanwhile, Chien and Phuc ³⁹ used the TSDPT and meshfree approach. In Table 2, the dimensionless fundamental frequency of isotropic spherical caps of present results are compared with those of the results of Sathyamoorthy, ³ Varadan and Pandalai, ⁴ and Phuong et al. ¹⁵ Clearly, the good agreements can be observed in two validation examples.

Table 1.

Validation of dimensionless fundamental frequency $\overline{\mathrm{\omega }}=\mathrm{\omega }{a}_0^2\sqrt{h{\overline{\mathrm{\rho }}}_m/D_m}$ of FG-GPLRC circular plates ( $W_{GPL}^{*}$ = 1%, ${\overline{E}}_m$ = 3 GPa, ${\overline{\mathrm{\rho }}}_m$ = 1200 kg/m³, ${\overline{\mathrm{\nu }}}_m$ = 0.34, $a_0$ = 1m, $D_m={\overline{E}}_m{h}^3/\left[12\left(1-{{\overline{\mathrm{\nu }}}_m}^2\right)\right]$ ).

Type	References	$a_0/h$
		29.4118	10	5
FG-X	Javani et al. 40	24.7333	23.9322	21.7190
	Chien and phuc 39	24.1789	23.1168	20.6648
	Present	25.1197	24.1887	21.6710
UD	Javani et al. 40	21.16378	20.6418	19.1378
	Chien and phuc 39	21.2205	20.5657	19.0329
	Present	21.4086	20.9684	19.6549
FG-O	Javani et al. 40	16.8334	16.5524	15.7053
	Chien and phuc 39	17.7029	17.3051	16.3881
	Present	16.8242	16.6377	16.0492

Table 2.

Validation of dimensionless fundamental frequency $\widehat{\mathrm{\omega }}=\mathrm{\omega }/{\mathrm{\omega }}^{*}$ of isotropic spherical caps ( ${\mathrm{\omega }}^{*}=h/a_0\sqrt{{\overline{E}}_m/\left({\overline{\mathrm{\rho }}}_m{a_0}^2\right)}$ , $H_0=\raisebox{1ex}{$a_0^2$}\!\left/ \!\raisebox{-1ex}{${2R}_1$}\right.$ ).

	$H_0/h$
	0	2	5
Sathyamoorthy 3	2.97	6.14	13.08
Varadan and pandalai 4	2.95	6.08	12.72
Phuong et al. 15	2.93	6.51	12.45
Present	3.06	6.51	13.24

In this paper, the copper matrix FG-GPLRC is chosen for the numerical examples. The porous core is made from copper with the same properties as the copper matrix of FG-GPLRC, and the PZT-5A is chosen for the piezoelectric layers. The material properties of FSG-GPLRC and PZT-5A are respectively referred to Wang et al., ³⁷ and Shen. ⁵⁶

Effects of porosity coefficients and GPL distribution laws on the fundamental frequencies of FG-GPLRC plates and shell caps with piezoelectric layers and porous core are investigated in Table 3. As can be observed, in all investigations, the maximal fundamental frequencies are found with the PE/V/PC/Λ/PE plates and shell caps, and next are PE/X/PC/X/PE, and PE/UD/PC/UD/PE plates and shell caps. Meanwhile, the smallest can be observed for PE/Λ/PC/V/PE structures. The symmetric distribution laws are designed, therefore, the largest stiffnesses are obtained in the case of GPL which is highly concentrated away from the mid-surface, and increased shell stiffness leads to the increased fundamental frequency of the structures. The most interesting phenomenon is that the fundamental frequency increases with the increase of the porosity coefficient $e_0$ . Although the increases in the porosity coefficient reduce the stiffnesses of the structures, the densities of the core layer also decrease, reducing the mass of the structures. Additionally, with only the small curvatures, the fundamental frequencies of the spherical caps are significantly greater than those of the corresponding circular plates.

Table 3.

Fundamental frequencies (rad/s) of FG-GPLRC plates and shell caps with piezoelectric layers and porous core ( $h$ = 16 mm, $h_{pc}$ = 9.6 mm, $h_{GPLRC}$ = 2.4 mm, $h_{pe}$ = 0.8 mm, $R_1=20h$ , $R_0=R_1/0.15$ , $W_{GPL}^{*}$ = 0.5%, $V_0$ = 500V, $\mathrm{\kappa }$ = 0, $\mathrm{\Delta }T$ = 100K, $K_1$ = 15 MN/m³, $K_2$ = 0.5 MN/m).

	$e_0$	0	0.2	0.4	0.6	0.8
Circular plate	PE/UD/PC/UD/PE	1214.30	1293.60	1378.60	1473.70	1590.50
	PE/X/PC/X/PE	1214.70	1294.00	1378.90	1474.00	1590.80
	PE/O/PC/O/PE	1212.00	1291.40	1376.30	1471.40	1588.20
	PE/V/PC/Λ/PE	1238.70	1317.40	1401.90	1496.70	1613.50
	PE/Λ/PC/V/PE	1187.50	1267.40	1352.90	1448.20	1565.10
Spherical cap	PE/UD/PC/UD/PE	2939.30	2930.80	2930.30	2944.20	2991.00
	PE/X/PC/X/PE	2938.00	2929.50	2928.90	2942.70	2989.40
	PE/O/PC/O/PE	2936.90	2928.30	2927.60	2941.40	2988.00
	PE/V/PC/Λ/PE	2948.00	2939.90	2939.70	2954.10	3001.50
	PE/Λ/PC/V/PE	2926.80	2917.80	2916.70	2929.90	2975.70

Figure 2(a) shows the effects of the porosity coefficient $e_0$ on the dynamic responses of the circular plate with the porous core. As can be seen, due to similar reasons in the fundamental frequency cases in Table 2, as the porosity coefficient increases, the vibration amplitude of the plate decreases. Similar to the remarks on the fundamental frequency in Table 3, the increase in porosity fraction reduces the stiffnesses, but it also reduces the density of the plates. Effects of the base radius on the vibration behavior of spherical caps are investigated in Figure 2(b). Clearly, the increase in the base radius sharply increases the shell’s vibration amplitude. The vibration behavior of circular plates and spherical caps is studied in Figure 2(c). The small curvatures of spherical caps are applied, however, the sharp decrease of vibration amplitude of structures can be obtained. The increase of the GPL mass fraction in the face sheets leads to a decrease in the vibration amplitude as in Figure 2(d).

Figure 2.

Effects of material and geometrical parameters on the dynamic responses of plates and shell caps under harmonic loads.

The viscous damping coefficient also strongly affects the vibrational behavior of the plates as in Figure 2(e). After a sufficient number of periods, the vibration response curve becomes noticeably smoother as the viscous damping coefficient increases. Figure 2(f) presents the effects of the porous core thickness on the vibration behavior of spherical caps. As can be observed, the vibration amplitude and tendency of the curves do not change significantly when the core thickness changes.

Figure 3(a) investigates the vibration behavior of FG-GPLRC circular plates with porous core and piezoelectric layers in the cases of forced frequency approximate to the fundamental frequency of the plates. The beat vibrations can be observed in the period and the amplitude of the envelope increases when the forced frequency comes closer to the fundamental frequency of the plates. The increase in the foundation stiffnesses also reduces the shell vibration amplitude as observed in Figure 3(b). Meanwhile, the positive and negative voltage of piezoelectric layers are considered in Figure 3(c). The results show the slight influences of voltage on the vibration behavior of the plates. The adverse effects of temperature on shell vibrations are shown in Figure 3(d). As can be observed, when the temperature change increases, the vibration amplitude largely increases. Due to the thermal expansion, the equilibrium position of the cap is moved to the negative deflection region, leading to the caps vibrating mainly in the negative amplitude region.

Figure 3.

Effects of material and geometrical parameters on the dynamic responses of plates and shell caps under harmonic loads.

The frequency-amplitude curves of plates and shell caps for the nonlinear free and forced vibration are presented in Figure 4(a)–(d). The curves in Figure 4(a) show that there are nonlinear frequency regions in which there is more than one value of the vibration amplitude for each value of frequency. In other words, these frequency regions do not have stable solutions. For nonlinear free vibration, the unstable solution regions appear only with spherical caps and are not observed with circular plates. Meanwhile, this region shifts toward the lower frequency parameter with larger shell curvature. Figure 4(b) presents the frequency-amplitude curves of spherical caps with different porosity coefficients. The results show the unstable solution region of nonlinear free vibration is in the higher frequency parameter as the porosity coefficient increases. The results also show that in the large amplitude region, the frequency-amplitude curve does not change significantly when the porosity coefficient changes.

Figure 4.

Effects of the structure type, porosity coefficient, base radius, and amplitude of forced load on the frequency-amplitude curves of plates and shell caps.

Effects of base radius on the frequency-amplitude curves of spherical caps are investigated in Figure 4(c). In the high-frequency parameter region, the vibration amplitude increases sharply as the base radius increases, while in the low-frequency parameter region, the difference is not large in the investigation cases. Effects of forced amplitude on the frequency-amplitude curve of the circular plate are investigated in Figure 4(d). The results show that the forced vibration curves are asymptotic with the free vibration curve, additionally, the curve of forced vibration diverges from the curve of free vibration as the amplitude of the forced load increases.

Effects of material and geometrical parameters on the dynamic responses of plates and shell caps under linear time-dependent loads are presented in Figure 5(a)–(d). As can be seen in Figure 5(a), the dynamic response curves of spherical caps are upper for a larger base radius. The dynamic buckling behavior can be observed in Figure 5(b) for the spherical caps, however, it cannot be obtained for circular plates. It can also be seen that as the shell curvature increases, the dynamic buckling region is obtained in the larger loads.

Figure 5.

Effects of material and geometrical parameters, and foundation on the dynamic responses of plates and shell caps under linear time-dependent loads.

Figure 5(c) investigates the effects of the GPL mass fraction of two FG-GPLRC face sheets on the dynamic responses of spherical caps. Clearly, the dynamic response curve is upper with the lower GPL mass fraction of the face sheets. As in Figure 5(d), the dynamic response curve gets lower and smoother as the foundation stiffnesses increase. Due to the reaction of the elastic foundation is opposite to the direction of dynamic responses, the amplitude of dynamic responses reduces.

Conclusions

In this paper, the nonlinear dynamic response and vibration behavior of FG-GPLRC circular plates and spherical caps with porous core and piezoelectric layers resting on the Pasternak elastic foundation is considered. The TSDST and the Euler-Lagrange equations are used to formulate the motion equations of considered structures. The equation of motion, fundamental frequency expression, and frequency-amplitude expression are achieved in explicit forms. The results show that the present approach is simple and effective for the nonlinear dynamic response and vibration problems of the considered structures. From the investigations, some interesting remarks are achieved as

- The fundamental frequency increases with the increase of the porosity coefficient, although the increases in the porosity coefficient reduce the stiffnesses of the structures, the densities of the core layer also decrease, reducing the mass of the structures. Additionally, as the porosity coefficient increases, the vibration amplitude of the circular plate decreases.

- The sharp decrease of vibration amplitude of structures is obtained when only the small curvatures of structures are applied.

- For nonlinear free vibration, the unstable solution regions appear only with spherical caps and are not observed with circular plates. Meanwhile, this region shifts toward the lower frequency parameter with larger shell curvature.

- The dynamic buckling behavior can be observed for the spherical caps, however, it cannot be obtained for circular plates.