Thermal Stresses Vibration of Thick FGM Conical Shells by Using TSDT

Abstract

The technical study of the presented manuscript is to investigate the thermal vibration of thick functionally graded material (FGM) conical shells with fully-homogeneous equations coupled in third-order shear-deformation theory (TSDT). The method in the generalized-differential quadrature (GDQ) approach is used to calculate the dynamic numerical data of FGM conical shells subjected to thermal-vibration only. Some parametric effects of minor middle-surface radius, environment temperature, and FGM power-law index on thermal stress and displacement of thick FGM conical shells are investigated with the frequency approach of the fully homogeneous equation. The novelties and main contributions of the present paper are that the thermal vibration GDQ study is original in thick FGM conical shells and contains some contributions to science and physics, by using the higher-order analysis of the TSDT displacement model and GDQ numerical results to obtain more accurate data in the thermal analyses of displacements and stresses for the thick FGM conical shells.

1. Introduction

The research on the effects of shear deformation on functionally graded material (FGM) conical shells is presented. Rahmani et al. (2020) [1] used Hamilton’s energy principle and the Galerkin method to study the vibration behavior of porous FGM conical sandwich shells under thermal surroundings. The variation of frequencies of numerical results is presented. Javani et al. (2019) [2] applied the displacement model in first-order shear-deformation theory (FSDT) and geometric nonlinearity to study the thermally induced vibration behavior of FGM conical shells. The numerical solutions of hybrid generalized differential quadrature (GDQ) and Crank–Nicolson method for the nonlinear axis-symmetric response under rapid heating are presented. Sofiyev (2019) [3] presented an exhaustive review of the literature to study the buckling and vibrations for FGM conical shells. Chan et al. (2018) [4] presented the FSDT of displacements and the Galerkin method to study thermal buckling investigations for a stiffened elastic foundation on FGM. The analytical solutions are presented and investigated. It would be interesting to apply a fully homogeneous equation for the frequency value of vibrations and include the third-order shear-deformation theory (TSDT) term for displacements on the thick-walled FGM conical shells.

The shear effect on the displacement study by considering the higher-order theory of shear deformation in thermal analysis of structures was presented as follows. Żur et al. (2020) [5] presented the free vibration study for the FGM nanoplates with piezomagnetic layers by using the nonlocal modified higher-order sinusoidal shear deformation theory (SSDT) and Navier solution technique. Mohammadi et al. (2019) [6] presented the thermo-elastic response study for the FGM carbon-nanotube cylindrical vessels with the TSDT. Arefi and Rabczuk (2019) [7] presented the electro-elastic bending study for the piezoelectric-nano shell by using the nonlocal higher-order shear deformation theory. Arefi et al. (2019) [8] presented the thermal and mechanical buckling study for the FGM micro plate with graphene nanoplatelets (GNPs) by using the TSDT, SSDT and modified strain gradient theory (MSGT). Sladek et al. (2019) [9] presented the crack study for the quasicrystal materials by using the finite element method (FEM). Mohammad-Rezaei Bidgoli and Arefi (2021) [10] presented the free vibration study for the FGM micro plate with GNPs by using the MSGT. Arefi et al. (2021) [11] presented the thermo-mechanical loads study for the cylindrical shell with GNPs by using the two-dimensional (2D) FSDT. Arefi et al. (2020) [12] presented the deflection study for the FGM porous micro plate with GNPs by using the TSDT.

More important topics were presented for the conical shells as follows. Sofiyev (2018) [13] investigated the free-vibration study for the laminated conical shells by using the FSDT. Avey and Yusufoglu (2020) [14] presented the large-amplitude vibration study for the carbon-nano-tube-based double-curved shallow shell (CNTBDCSS) based on superposition, Galerkin’s method and semi-inverse approach. It’s fundamental to investigate the thermal-time responses in displacement and stress of TSDT coupled with a non-advanced shear factor approach for FGM conical shells. In the traditional study for the shear correction coefficient, it was considered as a constant value, e.g., 5/6. Parametric effects of minor middle-surface radius, temperature, and power-law FGM index on stress and displacement of thick FGM conical shells are investigated with the frequency approach of the fully homogeneous equation. The novelties and main contributions of the present paper are that the thermal vibration GDQ study is original in thick FGM conical shells and contains some contributions, e.g., significantly new fundamental insights of displacement and stress values to the science and physics.

2. Formulation

For a two-material FGM conical shell with a point ($x$,$\theta$,$z$) of curve coordinates, is shown in Figure 1 with FGM constituent material 1 (FGM 1) thickness $h_1$ and FGM constituent material 2 (FGM 2) thickness $h_2$, respectively. L is the length of FGM conical shells, expression $h^{*}=h_1+h_2$ denotes total thickness of shell, r denotes minor radius on $x$ = 0, R denotes major radius on $x$ = L, $\beta$ denotes the angle of semi-vertex. $z$ is the thickness coordinate and states the z-origin at the mid-surface. ${\mathrm{R}}_{\mathrm{n}}$ is the power law index of FGMs. The FGM 1 is a metal, e.g., stainless steel, and the FGM 2 is a ceramic, e.g., silicon nitride, usually used for the FGM analysis. There are some material properties of FGM conical shells, e.g., Young’s modulus, Poisson’s ratios, densities, thermal-expansions, thermal-conductivities, specific-heats. The property of FGMs in ${\mathrm{R}}_{\mathrm{n}}$ power-law function $E_{fgm}=\left(E_2-E_1\right)({\displaystyle \frac{z+h^{*}/2}{h^{*}}}{)}^{R_n}+E_1$ is assumed for Young’s modulus. In which ${\mathrm{R}}_{\mathrm{n}}$ value is a power-index number, e.g., it can be given in positive rational values ${\mathrm{R}}_{\mathrm{n}}=$ 0.1, 0.2, …, 1, 2, …, 10, etc. The other properties of FGMs are applied in simple average forms, e.g., ${\nu }_{fgm}=\left({\nu }_2+{\nu }_1\right)/2$, where $E_{fgm}$, $E_1$ and $E_2$ are the Young’s modulus; ${\nu }_{fgm}$, ${\nu }_1$ and ${\nu }_2$ are the Poisson’s ratios, respectively, in the FGM conical shell, FGM 1 and FGM 2. The terms of constituent materials $E_1$, $E_2$, ${\nu }_1$ and ${\nu }_2$, etc., are expressed in corresponding to the term $P_i$. For $P_i$ value of FGMs can be found in function of environment-temperature T by the expression $P_i=P_0{P}_{-1}{T}^{-1}+P_0+P_0{P}_{1}T+P_0{P}_2{T}^2+P_0{P}_3{T}^3$, where $P_{-1}$, $P_0$, $P_1$, $P_2$, $P_3$ are coefficients, thus $E_1$, $E_2$, ${\nu }_1$, ${\nu }_2$, etc., can be obtained and are dependent on the given $T$ value [15], in which $T$ does not vary through the thickness, e.g., $T$ = 100 K used for room-temperature.

2.1. Displacements

The nonlinear displacements $u$, $v$ and $w$ are usually functions of time, curve coordinates ($x$,$\theta$,$z$), shear rotations and the nonlinear coefficient $c_1$ term. The coefficient parameter $c_1$ be considered in varied forms and in function of the total thickness of FGM. Nonlinear displacements $u$, $v$ and $w$ are dependent on time and $x$, $\theta$, $z$ of thick FGM conical shells assumed and applied in the nonlinear coefficient $c_1$ term with third order of z for TSDT equations as follows [15],

$$u=u_0\left(x,\theta ,t\right)+z{\varphi }_x\left(x,\theta ,t\right){-c}_1{z}^3({\varphi }_x+{\displaystyle \frac{\partial w}{\partial x}}),$$

(1)

$$v=v_0\left(x,\theta ,t\right)+z{\varphi }_{\theta }\left(x,\theta ,t\right){-c}_1{z}^3({\varphi }_{\theta }+{\displaystyle \frac{\partial w}{R^{*}\partial \theta }}),$$

(2)

$$w=w\left(x,\theta ,t\right),$$

(3)

where $u_0$ and $v_0$ denote displacement in $x$ and $\theta$ directions, $w$ denotes displacement in $z$ direction. ${\varphi }_x$ and ${\varphi }_{\theta }$ denote shear rotations. $R^{*}$ is the middle-surface radius at ($x$,$\theta$,$z$) of FGM conical shells, thus variable $R^{*}=r+x\mathrm{s}\mathrm{i}\mathrm{n}\beta$, in which $0\le x\le L$ for the given value of $r$ or $\beta$. When the value of $\beta =0$, then it becomes a circular cylindrical shell. $t$ denotes time. Coefficient term of $c_1=4/(3{h^{*}}^2)$ used for TSDT. This is a preliminary physical assumption in $c_1$ expression used for $z^3$ term in displacements $u$ and $v$ only due to thick-walled structure effect, e.g., L/$h^{*}\le 10$.

2.2. Stresses

Normal stress ${\sigma }_x$, ${\sigma }_{\theta }$ and shear stress ${\sigma }_{x\theta }$, ${\sigma }_{\theta z}$, ${\sigma }_{xz}$ in (k)th constituent layer of FGM conical shells under thermal loading parameter $∆T$ denoted for temperature difference, can be obtained in terms of stiffness ${\overline{Q}}_{ij}$, subscripts i, j = 1, 2, 4, 5, 6 and strains (${\epsilon }_x$, ${\epsilon }_{\theta }$, ${\epsilon }_{x\theta }$, ${\epsilon }_{\theta z}$ and ${\epsilon }_{xz}$) [15]. The value for $∆T$ can be assumed as a function of $x$, $\theta$ and $t$. Also assumed in linear and uncoupled effects on $∆T$ versus z without heat-generation. Thus, heat-conduction can be expressed in frequency $\gamma$ of simplified sinusoidal-heat flux, then used for the vibration of thermal loadings.

2.3. Thermal Loads

The vibration of time sinusoidal displacement, shear rotations can be assumed and used in terms of two-dimensional natural frequency ${\omega }_{mn}$ with respect to subscripts m and n mode shape. The temperature-expression for $∆T$ is assumed and used for thermal loads as follows [15],

$${∆T=\overline{T}}_1{\displaystyle \frac{z}{h^{*}}}\mathrm{s}\mathrm{i}\mathrm{n}(\pi x/L)\mathrm{s}\mathrm{i}\mathrm{n}(\pi \theta )\mathrm{s}\mathrm{i}\mathrm{n}(\gamma t),$$

(4)

where $\gamma$ denotes heat-flux frequency, its value can be calculated by a simplified heat-conduction expression. ${\overline{T}}_1$ is the temperature of heat flux.

2.4. Dynamic Equations

Dynamic equations in terms of TSDT for an infinitesimal volume element of thick FGM conical shells can be applied from FGM cylindrical shells [15] by reasonably assuming that the middle-surface radius $R^{*}=r+x\mathrm{s}\mathrm{i}\mathrm{n}\beta$ at any point ($x$,$\theta$,$z$) is used and applied further as follows, in which $0\le x\le L$ for the given value of $r$ or $\beta$,

$${\displaystyle \frac{\partial N_{xx}}{\partial x}}+{\displaystyle \frac{1}{R^{*}}}{\displaystyle \frac{\partial N_{x\theta }}{\partial \theta }}=I_0{\displaystyle \frac{{\partial }^2{u}_0}{\partial t^2}}+J_1{\displaystyle \frac{{\partial }^2{\varphi }_x}{\partial t^2}}-c_1{I}_3{\displaystyle \frac{{\partial }^2}{\partial t^2}}\left({\displaystyle \frac{\partial w}{\partial x}}\right),$$

(5)

$${\displaystyle \frac{\partial N_{x\theta }}{\partial x}}+{\displaystyle \frac{1}{R^{*}}}{\displaystyle \frac{\partial N_{\theta \theta }}{\partial \theta }}=I_0{\displaystyle \frac{{\partial }^2{v}_0}{\partial t^2}}+J_1{\displaystyle \frac{{\partial }^2{\varphi }_{\theta }}{\partial t^2}}-c_1{I}_3{\displaystyle \frac{{\partial }^2}{\partial t^2}}\left({\displaystyle \frac{\partial w}{R^{*}\partial \theta }}\right),$$

(6)

$${\displaystyle \frac{\partial {\overline{Q}}_x}{\partial x}}+{\displaystyle \frac{1}{R^{*}}}{\displaystyle \frac{\partial {\overline{Q}}_{\theta }}{\partial \theta }}+c_1\left({\displaystyle \frac{{\partial }^2{P}_{xx}}{\partial x^2}}+{\displaystyle \frac{2}{R^{*}}}{\displaystyle \frac{{\partial }^2{P}_{x\theta }}{\partial x\partial \theta }}+{\displaystyle \frac{1}{{R^{*}}^2}}{\displaystyle \frac{{\partial }^2{P}_{\theta \theta }}{\partial {\theta }^2}}\right)+q=I_0{\displaystyle \frac{{\partial }^{2}w}{\partial t^2}}-{c_1}^2{I}_6{\displaystyle \frac{{\partial }^2}{\partial t^2}}({\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}+{\displaystyle \frac{1}{{R^{*}}^2}}{\displaystyle \frac{{\partial }^{2}w}{\partial {\theta }^2}})+c_1[I_3{\displaystyle \frac{{\partial }^2}{\partial t^2}}\left({\displaystyle \frac{\partial u_0}{\partial x}}+{\displaystyle \frac{1}{R^{*}}}{\displaystyle \frac{\partial v_0}{\partial \theta }}\right)+J_4{\displaystyle \frac{{\partial }^2}{\partial t^2}}\left({\displaystyle \frac{\partial {\varphi }_x}{\partial x}}+{\displaystyle \frac{1}{R^{*}}}{\displaystyle \frac{\partial {\varphi }_{\theta }}{\partial \theta }}\right)],$$

(7)

$${\displaystyle \frac{\partial {\overline{M}}_{xx}}{\partial x}}+{\displaystyle \frac{1}{R^{*}}}{\displaystyle \frac{\partial {\overline{M}}_{x\theta }}{\partial \theta }}-{\overline{Q}}_x={\displaystyle \frac{{\partial }^2}{\partial t^2}}\left(J_1{u}_0+K_2{\varphi }_x{-c}_1{J}_4{\displaystyle \frac{\partial w}{\partial x}}\right),$$

(8)

$${\displaystyle \frac{\partial {\overline{M}}_{x\theta }}{\partial x}}+{\displaystyle \frac{1}{R^{*}}}{\displaystyle \frac{\partial {\overline{M}}_{\theta \theta }}{\partial \theta }}-{\overline{Q}}_{\theta }={\displaystyle \frac{{\partial }^2}{\partial t^2}}\left(J_1{v}_0+K_2{\varphi }_{\theta }{-c}_1{J}_4{\displaystyle \frac{1}{R^{*}}}{\displaystyle \frac{\partial w}{\partial \theta }}\right),$$

(9)

where ${\overline{M}}_{\alpha \beta }=M_{\alpha \beta }-c_1{P}_{\alpha \beta }$, ${\overline{Q}}_{\alpha }=Q_{\alpha }-3c_1{R}_{\alpha }$, $(\alpha ,\beta =x,\theta )$.

$\left\{\begin{array}{c}\\ \\ \end{array}\right.\left.\begin{array}{c}{N}_{xx}\\ N_{\theta \theta }\\ N_{x\theta }\end{array}\right\}={\displaystyle {\int }_{-\frac{h^{*}}{2}}^{\frac{h^{*}}{2}}\left\{\begin{array}{c}\\ \\ \end{array}\right.}\left.\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{\theta \theta }\\ {\sigma }_{x\theta }\end{array}\right\}dz$, $\left\{\begin{array}{c}\\ \\ \end{array}\right.\left.\begin{array}{c}{M}_{xx}\\ M_{\theta \theta }\\ M_{x\theta }\end{array}\right\}={\displaystyle {\int }_{-\frac{h^{*}}{2}}^{\frac{h^{*}}{2}}\left\{\begin{array}{c}\\ \\ \end{array}\right.}\left.\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{\theta \theta }\\ {\sigma }_{x\theta }\end{array}\right\}zdz$, $\left\{\begin{array}{c}\\ \\ \end{array}\right.\left.\begin{array}{c}{P}_{xx}\\ P_{\theta \theta }\\ P_{x\theta }\end{array}\right\}={\displaystyle {\int }_{-\frac{h^{*}}{2}}^{\frac{h^{*}}{2}}\left\{\begin{array}{c}\\ \\ \end{array}\right.}\left.\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{\theta \theta }\\ {\sigma }_{x\theta }\end{array}\right\}{z}^{3}dz$, $\left\{\begin{array}{c}\\ \end{array}\right.\left.\begin{array}{c}{R}_{\theta }\\ R_x\end{array}\right\}={\displaystyle {\int }_{-\frac{h^{*}}{2}}^{\frac{h^{*}}{2}}\left\{\begin{array}{c}\\ \end{array}\right.}\left.\begin{array}{c}{\sigma }_{\theta z}\\ {\sigma }_{xz}\end{array}\right\}{z}^{2}dz$, $\left\{\begin{array}{c}\\ \end{array}\right.\left.\begin{array}{c}{Q}_{\theta }\\ Q_x\end{array}\right\}={\displaystyle {\int }_{-\frac{h^{*}}{2}}^{\frac{h^{*}}{2}}\left\{\begin{array}{c}\\ \end{array}\right.}\left.\begin{array}{c}{\sigma }_{\theta z}\\ {\sigma }_{xz}\end{array}\right\}dz$.

$I_i={\displaystyle \sum _{k=1}^{N^{*}}{\displaystyle {\int }_k^{k+1}{\rho }^{(k)}}}{z}^{i}dz$, where i = 0, 1, 2, …, 6, and $N^{*}$ denotes the total layer of materials, ${\rho }^{(k)}$ is the density of (k)th constituent materials. $J_i=I_i-c_1{I}_{i+2}$ with subscript i = 1, 4. $K_2=I_2-2c_1{I}_4+{c_1}^2{I}_6$. The role of nonlinear $c_1$ terms in the governing equations is to investigate the extra effects of nonlinear displacements, shear rotations and stress integrations on the dynamic equations of motion with respect to the linear case, e.g., FSDT and $c_1=0$. By using the higher-order analysis of the TSDT displacement model and GDQ numerical results to obtain more accurate data in the thermal analyses for stress and displacement.

The assumption expressions of ${\displaystyle \frac{\partial v_0}{\partial z}}={\displaystyle \frac{-v_0}{R^{*}}}$, ${\displaystyle \frac{\partial u_0}{\partial z}}={\displaystyle \frac{-u_0}{R^{*}}}$ and ${\displaystyle \frac{\partial w}{\partial z}}={\displaystyle \frac{\partial {\phi }_x}{\partial z}}={\displaystyle \frac{\partial {\phi }_{\theta }}{\partial z}}=0$ can be assumed in relations of strain-displacement and used for the nonlinear Equations (1)–(3). By substituting equations of stresses and strain-displacement relations into Equations (5)–(9), the dynamic-differential equation with TSDT can be expressed in partial derivatives of displacement, shear rotation, and inertia term under thermal load and mechanical load. The equations of thermal loads and integrations of stiffness ${\overline{Q}}_{ij}$ are used with $q$ is an external mechanical-pressure load. $k_{\alpha }$ denotes the non-advanced shear factor and can be defined in the expression $k_{\alpha }=FGMZS/\left(h^{*}FGMZI\right)$, in which $FGMZS$ and $FGMZI$ are parameters in functions of $E_1$, $E_2$, $h^{*}$ and ${\mathrm{R}}_{\mathrm{n}}$ for $c_1$ = 0 used simply [15]. Computational non-advanced $k_{\alpha }$ value can be obtained and ${\overline{Q}}_{ij}$ in a simple expression for a thick FGM conical shell in terms of $z/R$.

2.5. Numerical Method

The numerical method of differential quadrature (DQ) first introduced by Bert et al. [16]. The implementation of the numerical method for GDQ was provided by Shu and Du in 1997 [17], and studied by Hong (2025) [15]. For an arbitrarily typical grid-point, dynamic-discrete GDQ equations in terms of general matrix form can be obtained as follows,

$$[A]\left\{W^{*}\right\}=\left\{B\right\}$$

(10)

in which $[A]$ denotes the $N^{**}$ by $N^{**}$ coefficient matrix with $N^{**}=(N-2)\times (M-2)\times 5$ in terms of stiffness-coefficient ($A_{i^s{j}^s}$, $B_{i^s{j}^s}$, $D_{i^s{j}^s}$, $E_{i^s{j}^s}$, $F_{i^s{j}^s}$, $H_{i^s{j}^s}$) and weighting-parameter ($A_{i,l}^{(m)},B_{j,m}^{(m)}$). $\left\{W^{*}\right\}$ is a $N^{**}$ th-order unknown column vector for the components of displacements, shear rotations. And $\left\{B\right\}$ denotes $N^{**}$ th-order row vector of external-load. The software program Lahey-Fujitsu Fortran (LFPro7.8.1_VS2015, Microsoft Company, Redmond, DC, USA) can be used to implement Equation (10).

3. Numerical Results

The incremental contributions of this conical shell paper are listed as follows when connected with previous works for geometric nonlinearity of conical shells and the effect of radius variation to cylindrical shells [15] and plates [18]. The geometric nonlinearity with third order of z in the TSDT model for displacement v includes a varied radius parameter $R^{*}$. The effect of radius variation in $R^{*}=r+x\mathrm{s}\mathrm{i}\mathrm{n}\beta$ is expressed on the variation of minor radius r then the results of deformations and stresses in the conical shell can be investigated. It’s fundamental for assuming time sinusoidal form of $u_0$, $v_0$, $w$, ${\phi }_x$ and ${\phi }_{\theta }$ under free vibration to obtain ${\omega }_{mn}$, also the approach of fully-homogeneous equation including the nonlinear $c_1$ terms can be applied. A similar procedure of obtaining ${\omega }_{mn}$ can also be referred to by Hong (2021) [19]. When the ${\omega }_{mn}$ and $\gamma$ were found and next used in time-sinusoidal displacements, shear-rotations, thermal-vibration temperatures for the calculation of dynamic GDQ discrete matrix Equation (10). Non-advanced $k_{\alpha }$ calculations with FGM constituent material of stacking $(0^{°}/0^{°})$ shells used simply supported boundary conditions on four sides and $q=0$. The SUS304 used for FGM 1, the Si₃N₄ used for FGM 2, and $L/R=1$, $h^{*}$ = 1.2 mm, $h_1$ = $h_2$ = 0.6 mm used for the numerical GDQ computations, thus $c_1=4/\left(3{h^{*}}^2\right)$ = 0.925925/mm² for $h^{*}$ = 1.2 mm value chosen. The coordinates $x_i$ and ${\theta }_j$ used as follows to compute the GDQ data,

$$x_i=0.5[1-\cos \left({\displaystyle \frac{i-1}{N-1}}\pi \right)]L,i=1,2,\dots ,N$$

(11)

$${\theta }_j=0.5[1-\cos \left({\displaystyle \frac{j-1}{M-1}}\pi \right)]R,j=1,2,\dots ,M$$

(12)

where N and M denote the grid number in the x and $\theta$ directions.

3.1. Dynamic Convergence

The dynamic thermal vibration of the GDQ computation for nonlinear TSDT thick FGM plates was presented by Hong (2019) [18], and for the thermal vibration case of thick FGM cylindrical shells was presented by Hong (2025) [15]; a similar GDQ procedure of obtaining deflections and stresses can also be referred to for the conical shells. The dynamic convergence study of center displacements $w(L/2,2\pi /2)$ (mm) in thermal-vibration with $c_1$ = 0.925925/mm² and $c_1$ = 0 are investigated. For thick FGM conical shells L/$h^{*}$ = 10 under applied heat flux $\gamma$ = 0.2618004/s and L/$h^{*}$ = 5 under $\gamma$ = 0.2618019/s at $t$ = 6 s, T = 100 K, ${\overline{T}}_1$ = 1 K vs. r = R/8, R/4, R/2, 3R/4 and R are presented. Vary values of ${\omega }_{11}$ are computed with the vibration frequency approach of fully homogeneous equation for $k_{\alpha }$ and ${\mathrm{R}}_{\mathrm{n}}$ in the nonlinear case $c_1$ = 0.925925/mm² and linear case $c_1$ = 0 of L/$h^{*}$ = 5 vs. r = 3R/4. The error-accuracy of displacements equals to 4.85 × 10⁻⁴ found for $c_1$= 0.925925/mm²,${\mathrm{R}}_{\mathrm{n}}$ = 2, L/$h^{*}$ = 10 vs. r = 3R/4 in

Table 1. Dynamic TSDT displacement convergence of fully homogeneous equation vs. r = 3R/4.

${\mathit{c}}_1$(1/mm2)	$\mathit{L}/{\mathit{h}}^{\mathit{*}}$	GDQ Grids	$\mathit{w}(\mathit{L}/2,2\mathit{\pi }/2)$$\left(\mathbf{mm}\right)\mathbf{at}\mathit{t}$ = 6 s
		$\mathit{N}\times \mathit{M}$	${\mathbf{R}}_{\mathbf{n}}$ = 0.5	${\mathbf{R}}_{\mathbf{n}}$ = 1	${\mathbf{R}}_{\mathbf{n}}$ = 2
0.925925	10	7 × 7	0.050928	0.051106	0.051598
		9 × 9	0.050838	0.051046	0.051550
		11 × 11	0.050832	0.051047	0.051537
		13 × 13	0.050821	0.051034	0.051534
		15 × 15	0.050824	0.051050	0.051533
		17 × 17	0.050808	0.050592	0.051508
	5	7 × 7	0.005936	0.006000	0.006200
		9 × 9	0.005891	0.005962	0.006167
		11 × 11	0.005885	0.005957	0.006151
		13 × 13	0.005822	0.005902	0.006110
		15 × 15	0.005808	0.005887	0.006096
		17 × 17	0.005682	0.005781	0.006004
0	10	7 × 7	−0.062602	−0.054677	−0.095378
		9 × 9	0.042434	0.042418	0.042327
		11 × 11	0.039775	0.041446	0.042427
		13 × 13	0.042628	0.042614	0.042349
		15 × 15	0.042767	0.042717	0.042301
		17 × 17	0.042469	0.042443	0.042328
	5	7 × 7	0.006319	0.007558	0.010122
		9 × 9	0.006566	0.007779	0.010368
		11 × 11	0.006470	0.007669	0.010222
		13 × 13	0.006569	0.007782	0.010373
		15 × 15	0.006518	0.007724	0.010293
		17 × 17	0.006569	0.007782	0.010371

. Thus the grids number N = M = 17 used in dynamic convergence good status and provide further enough grid point in GDQ calculation for time responses of displacement. The time responses of inter-laminar stresses ${\sigma }_x$, ${\sigma }_{\theta }$, ${\sigma }_{x\theta }$, ${\sigma }_{\theta z}$, ${\sigma }_{xz}$ also can be obtained.

3.2. Time Responses

The displacement and stress in time response values are obtained for thick-walled FGM conical shells under thermal vibration with $\gamma$ and ${\omega }_{11}$ values. The values of $\gamma$ decreasing from $\gamma$ = 15.707960/s, $t$ = 0.1 s to $\gamma$ = 0.523601/s, $t$ = 3.0 s used for L/$h^{*}$ = 5, also from $\gamma$ = 15.707963/s, $t$ = 0.1 s to $\gamma$ = 0.523599/s, $t$ = 3.0 s used in L/$h^{*}$ = 10 case. The Figure 2 displays $w(L/2,2\pi /2)$ (mm) versus $t$(s) for nonlinear TSDT $c_1$ = 0.925925/mm² and linear $c_1$ = 0 in FGM conical shells. The numerical data is presented for thick L/$h^{*}$ = 5 and 10, respectively, ${\mathrm{R}}_{\mathrm{n}}=1$, $k_{\alpha }$= 0.149001, T = 100 K, ${\overline{T}}_1$ = 1 K vs. r = 3R/4 and $t$ = 0.1–3.0 s. In Figure 2a, $w(L/2,2\pi /2)$ maximum value 0.093789 mm obtained at $t$ = 0.2 s, $c_1$ = 0.925925/mm² for L/$h^{*}$ = 5. In Figure 2b, an another $w(L/2,2\pi /2)$ maximum value 21.041734 mm obtained at $t$ = 0.1 s, $c_1$ = 0/mm² for L/$h^{*}$ = 10. The tendencies of $w(L/2,2\pi /2)$ are found in decreasing vs. time for $c_1$ = 0/mm², L/$h^{*}$ = 10. The amplitudes of $w(L/2,2\pi /2)$ in $c_1$ = 0/mm² are larger than that in $c_1$ = 0.925925/mm² for L/$h^{*}$ = 10. The physical mechanism of large displacements at $t$ = 3.0 s such as the linear model in FSDT with $c_1$ = 0 ignoring shear deformation leading to underestimated stiffness.

The Figure 3 displays the normal-stress ${\sigma }_x$, shear-stress ${\sigma }_{xz}$ versus $t$(s) based on the numerical data used in Figure 2, T = 100 K, ${\overline{T}}_1$ = 1 K, r = 3R/4 for nonlinear TSDT $c_1$ = 0.925925/mm² in FGM conical shells. The stress values vary with the thickness-direction of conical shells. Figure 3a,b display the stress ${\sigma }_x$(GPa) time-responses of center-position on $z$ = −0.5$h^{*}$ and ${\mathrm{R}}_{\mathrm{n}}$ = 1 value chosen as basic linear data, thick-walled L/$h^{*}$ = 5 and 10, respectively. The ${\sigma }_x$ maximum value 1.6361 × 10⁻⁵ GPa i.e., 16361 Pa is reasonable under low room-temperature 100 K and temperature 1 K of heat-flux for thermal load only obtained at $t$ = 0.1 s for L/$h^{*}$ = 5. The tendencies of ${\sigma }_x$ are found in decreasing vs. time in L/$h^{*}$ = 5. In Figure 3c,d, the ${\sigma }_{xz}$(GPa) vs. time displayed on $z$ = 0.0$h^{*}$ for ${\mathrm{R}}_{\mathrm{n}}$ = 1, L/$h^{*}$ = 5, 10. The ${\sigma }_{xz}$ maximum absolute-value in |−5.8430 × 10⁻⁴ GPa| = 5.8430 × 10⁻⁴ GPa of stress occurs at $t$ = 0.3 s for L/$h^{*}$ = 10. In the thermal load only applied for the study, the shear stress is the dominated one in the thick FGM conical shells.

3.3. Compared Results

In Figure 4 shows the compared values of $w(L/2,2\pi /2)$ (mm) versus T (100 K, 600 K and 1000 K) and ${\mathrm{R}}_{\mathrm{n}}$ (0.1 to 10) with ${\overline{T}}_1$ = 1 K, r = 3R/4, $c_1$ = 0.925925/mm² at $t$ = 0.1 s in FGM conical shells for L/$h^{*}$ = 5, 10. In Figure 4a, the curves of $w(L/2,2\pi /2)$ vs. T and ${\mathrm{R}}_{\mathrm{n}}$ displayed for L/$h^{*}$ = 5, $w(L/2,2\pi /2)$ maximum value 0.515556 mm obtained on T = 1000 K, ${\mathrm{R}}_{\mathrm{n}}$ = 0.1. The tendencies of $w(L/2,2\pi /2)$ are found in increasing vs. T and ${\mathrm{R}}_{\mathrm{n}}$. The $w(L/2,2\pi /2)$ in L/$h^{*}$ = 5 can’t withstand on 1000 K, thus excessive deformation would occur. In Figure 4b, the curves of $w(L/2,2\pi /2)$ vs. T and ${\mathrm{R}}_{\mathrm{n}}$ displayed for L/$h^{*}$ = 10, $w(L/2,2\pi /2)$ maximum value 0.070256 mm obtained on T = 1000 K, ${\mathrm{R}}_{\mathrm{n}}$ = 0.2. The tendencies of $w(L/2,2\pi /2)$ are found also in increasing vs. T and ${\mathrm{R}}_{\mathrm{n}}$. In L/$h^{*}$ = 10, the $w(L/2,2\pi /2)$ also can’t withstand on 1000 K, thus excessive deformation would occur.

In Figure 5 shows the compared values of ${\sigma }_x$(GPa) at center $z$ = −0.5$h^{*}$ vs. T (100 K, 600 K and 1000 K), ${\mathrm{R}}_{\mathrm{n}}$ with ${\overline{T}}_1$ = 1 K, r = 3R/4, $c_1$ = 0.925925/mm² at $t$ = 0.1 s in FGM conical shells for L/$h^{*}$ = 5 and 10, respectively. In Figure 5a, the curves of ${\sigma }_x$ vs. T and ${\mathrm{R}}_{\mathrm{n}}$ displayed for L/$h^{*}$ = 5, ${\sigma }_x$ maximum value 2.1292 × 10⁻⁵ GPa obtained on T = 600 K for ${\mathrm{R}}_{\mathrm{n}}$ = 1. In L/$h^{*}$ = 5, the stress ${\sigma }_x$ can withstand on temperature 1000 K of environment, thus excessive stress would not occur. In Figure 5b, the curves of ${\sigma }_x$ vs. T and ${\mathrm{R}}_{\mathrm{n}}$ displayed for L/$h^{*}$ = 10. The tendencies of ${\sigma }_x$ vs. T and ${\mathrm{R}}_{\mathrm{n}}$ are found in increasing firstly and then in decreasing secondly. The ${\sigma }_x$ maximum value 1.7036 × 10⁻⁵ GPa obtained on 600 K. In L/$h^{*}$ = 10, at $t$=0.1 s, the ${\sigma }_x$ can also withstand on 1000 K, thus excessive stress would not occur.

In Figure 6 shows the values of $w(L/2,2\pi /2)$ (mm) versus T (100 K, 600 K and 1000 K) with r = R/8, R/4, R/2, 3R/4, R and ${\mathrm{R}}_{\mathrm{n}}$ = 0.1, ${\overline{T}}_1$ = 1 K, $c_1$=0.925925/mm² at $t$ = 0.1 s in FGM conical shells for L/$h^{*}$ = 5, 10. Figure 6a shows the L/$h^{*}$ = 5 case for values of $w(L/2,2\pi /2)$ vs. T and r, maximum value 6.022941 mm obtained on r = R/2, and T = 100 K. The tendencies of $w(L/2,2\pi /2)$ vs. T are found in decreasing firstly, then in decreasing secondly, on r = R/2. The $w(L/2,2\pi /2)$ in r = R/2 can withstand higher temperatures of 1000 K; thus, excessive deformation would not occur. Figure 6b displays the L/$h^{*}$ = 10 case for values of $w(L/2,2\pi /2)$ vs. T and r, another maximum absolute value in |−0.100733 mm| = 0.100733 mm of deflection obtained on r = R/8, and T = 600 K. The tendencies of $w(L/2,2\pi /2)$ are found in increasing versus T except that r = R/8. In r = R/8, at $t$ = 0.1 s, the $w(L/2,2\pi /2)$ can also withstand 1000 K, thus excessive deformation would not occur.

In Figure 7 shows the values of ${\sigma }_{xz}$(GPa) on center position of middle surface $z$ = 0.0$h^{*}$ versus T (100 K, 600 K and 1000 K) for r (R/8, R/4, R/2, 3R/4 and R) with ${\mathrm{R}}_{\mathrm{n}}$ = 0.1, ${\overline{T}}_1$ = 1 K, $c_1$ = 0.925925/mm² at $t$ = 0.1 s in FGM conical-shells for L/$h^{*}$ = 5, 10. Figure 7a displays the L/$h^{*}$ = 5 case for values of ${\sigma }_{xz}$ versus T and r, maximum absolute-value in |−1.0672 × 10⁻² GPa| = 1.0672 × 10⁻² GPa of stress obtained on r = R/2, and T = 100 K. The absolute value tendencies of ${\sigma }_{xz}$ vs. T are found in increasing order first, then in decreasing order secondly, on r = R/8. For another r = R/2, the absolute value tendencies of ${\sigma }_{xz}$ vs. T are found in decreasing order firstly, then in decreasing order secondly. The ${\sigma }_{xz}$ in r = R/8 and r = R/2 can withstand 1000 K, thus excessive stress would not occur. Figure 7b displays the values of ${\sigma }_{xz}$ versus T and r for L/$h^{*}$ = 10, maximum absolute value |−6.7393 × 10⁻² GPa| = 6.7393 × 10⁻² GPa of stress obtained on r = R/8, and T = 600 K. The absolute value tendencies of ${\sigma }_{xz}$ vs. T are found in increasing order first, then in decreasing order secondly, on r = R/8. The ${\sigma }_{xz}$ in r = R/8 can withstand higher temperatures of 1000 K, thus excessive stress would not occur.

4. Discussions

The GDQ method of this paper is solid and used to provide new insights into the displacement/stress response laws of FGM conical shells in a thermal-thermal coupling environment. The formula in the GDQ method can be expressed in general matrix form Equation (10) with a high-level language to implement and find the unknown values. The FGM conical shells topic of this paper has engineering application value, the method selection is reasonable, and the numerical analysis is detailed. There are some more important discussions and technical points of view about stresses in the thermal load, which are only applicable to the study. The shear stress is usually the dominant one in the thick FGM conical shells presented in Figure 7. But in the thick-walled FGM circular-cylindrical shell study, the normal stress is usually the dominant one presented by Hong (2025) [15]. In the thick FGM plates study, the normal stress is usually the dominant one presented by Hong (2019) [18]. In the thick-walled FGM spherical-shell study, the normal stress is usually the dominant one. In the future, there are some technical expressions in the application that need this new finding. The simulation method of FGM conical shell is readily adaptable to other laminated or curved structures and is especially valuable for high-temperature applications in aerospace and energy systems. The interesting references to the relevant literature in the fields of quasi three-dimensional (3D) higher-order shear-deformation theory (HSDT) and ply-orientation in multilayer shells [20], shell and internal-fluid interface model [21] would be applied and studied in the future, if possible.

5. Conclusions

By using the higher-order TSDT displacement model and GDQ numerical method to obtain more accurate dynamic data in the thermal analyses of displacement and stress on thick-walled FGM conical shells. By investigating the parameter effects of minor middle-surface radius, temperature, and power-law FGM index on stress and displacement used in numerical computations. The amplitudes of $w(L/2,2\pi /2)$ time response in $c_1$ = 0/mm² are larger than those in $c_1$ = 0.925925/mm² for L/$h^{*}$ = 10. In the thermal load only applied for the present study, the shear stress ${\sigma }_{xz}$ is the dominated one in the thick-walled FGM conical shells. The normal stress ${\sigma }_x$ on $z$ = −0.5$h^{*}$, r = 3R/4 in both L/$h^{*}$ = 5 and 10 can withstand higher temperatures of 1000 K at $t$ = 0.1 s, thus excessive stress would not occur. The values of shear stress ${\sigma }_{xz}$ on r = R/8 in both L/$h^{*}$ = 5 and 10 can withstand higher temperatures of 1000 K at $t$ = 0.1 s, and excessive stress would not occur. The thermal vibration GDQ data can be obtained successfully for thick-walled FGM conical shells by using the fully homogeneous equation and the TSDT approach.