Advanced Frequency of Thick FGM Spherical Shells by Nonlinear Shear and TSDT

Abstract

An advanced frequency study in thick-walled functionally graded material (FGM) spherical shells is investigated with advanced shear correction. The values of advanced shear correction can be greater than one, be a negative value, and be affected by a nonlinear term of third-order shear deformation theory (TSDT) of displacements, FGM power law index, and temperature. It is novel and interesting to consider using TSDT and advanced shear correction to derive a simple homogeneous equation with reasonable simplifications into a symmetrical sparse matrix subjected to free vibration. The zero determinant of the symmetrical sparse matrix can be expressed to calculate the natural frequency by Newton’s method. The parameter effects of advanced shear correction, a nonlinear TSDT term, temperature, and the FGM power-law index on the natural frequencies of thick-walled FGM spherical shells are presented. The natural-frequency data for the axial and circumferential mode shapes are obtained. This is a new finding, as the assumed simplification in a sparse matrix causes a numerical truncation error; the natural-frequency values of the presented sparse matrix are much greater than those in a full matrix for thick-walled FGM spherical shells.

1. Introduction

Free vibration studies in the materials of spherical shells were investigated for the natural frequency. In 2021, Bagheri et al. [1] presented the free vibration of a functionally graded material (FGM) conical-spherical shell by using the first-order shear deformation theory (FSDT) of displacements and considering various types of boundary conditions on the result of natural frequencies. In 2021, Liu et al. [2] used the three-dimensional (3D) elasticity theory and the state space method to study the free vibrations of functionally graded graphene platelets reinforced composite (FG-GPLRC) spherical shells. In 2021, Tang and Dai [3] used the multiscale method and hygrothermal effects to obtain the analytical results for the nonlinear free vibration of carbon fiber-reinforced polymer (CFRP) spherical shell panels. In 2021, Roy et al. [4] presented the modified higher order zigzag theory (HOZT) of displacements and the experimental modal by Bruel and Kjaer to study the free vibration of laminated composite hybrid and glass fiber reinforced plastic (GFRP) shells. In 2019, Sayyad and Ghugal [5] presented the results of free vibration for laminated spherical shells by using a generalized higher-order shell theory. In 2019, Li et al. [6] presented the free vibration analysis of combined spherical-cylindrical-spherical (CSCS) shells based on the Ritz method with thin FSDT of displacements. In 2019, Li et al. [7] presented the Ritz method and the FSDT of displacements to obtain the free vibration analyses for functionally graded porous spherical shell (FGPSS). In 2016, Fantuzzi et al. [8] presented two-dimensional (2D) computational models and 3D exact shell models for the free vibration of FGM shells. In 2010, Sepiani et al. [9] presented the numerical results of free vibration for FGM shells by using the FSDT of displacements without considering the thermal effect.

Mathematical simulation reviews on thick FGM spherical shell structures are presented. In 2020, Zannon et al. [10] presented a free vibration numerical study for thick-walled FGM spherical shells by using the third-order shear deformation theory (TSDT) model. In 2023, Keibolahi et al. [11] presented a thermal-shock vibration numerical study for deep FGM spherical shells by using the FSDT model. In 2017, Khoa and Tung [12] presented a pressure load study for moderately-thick FGM sandwich spherical shells by using the FSDT model. In 2025, Zhang et al. [13] presented a thermal and mechanical loads static study for thick FGM spherical shells by using a heat conduction model. In 2026, Nejad et al. [14] presented a pressure load study for an FGM spherical solid by using the plane-elasticity theory (PET) model. In 2021, Dastjerdi et al. [15] presented a mathematical study for thick FGM spherical shells by using the FSDT model. In 2022, Arslan and Mack [16] presented a pressure load stress study for thick FGM spherical shells by using the homogenized-properties model. In 2014, Khaire et al. [17] presented a free vibration numerical study for FGM spherical shells by using the higher-order shear deformation theory (HSDT) model. In 2020, Zeverdejani and Kiani [18] presented a thermal-shock vibration numerical study for FGM spherical shells. In 2019, Shariyat and Ghafourinam [19] presented a pressure load stress study for thick FGM spherical shells by using Norton’s creep model.

Free vibration computational studies with shear correction effect in FGM spherical shells under environment-temperature were investigated for natural frequency. In 2020, Hong [20] used TSDT to present the frequency results for thick-walled FGM spherical shells with a simple homogeneous equation and varied shear correction, which was usually positive, smaller than one, and not affected by the nonlinear TSDT term. It is novel and interesting to investigate the natural frequency for thick-walled FGM spherical shells with advanced shear correction, whose value can be greater than one, can be a negative value, and can be affected by the nonlinear TSDT term. Four parameter effects of advanced shear correction, nonlinear TSDT term, temperature, and FGM power law index on the natural frequencies of thick-walled FGM spherical shells for a given angle with respect to the z axis and radius are investigated.

It is interesting to investigate what is fundamentally new on the effects of natural frequency, e.g., treatment of advanced shear correction presented in Section 2.2, Section 2.3, Section 3 and Section 4; nonlinear TSDT term effects presented in Section 2.1, Section 3 and Section 4; temperature dependency on FGMs presented in Section 3 and Section 4; and compared homogeneous equation in sparse matrix and full matrix presented in Section 2.4 and Section 4.

2. Materials and Methods

A two constituent-material in thick-walled FGM spherical shells has FGM material 1 on the inner layer and FGM material 2 on the outer layer [20]. A position point on FGM spherical shells in two coordinate systems between spherical axes $(r, \theta , \varnothing )$ and Cartesian axes (x, y, z) is displayed in Figure 1, where r denotes the radius, $\theta$ is the circumferential angle, and $\varnothing$ is the angle between the z axis and the r axis. The power law function type material properties of FGM spherical shells are considered. They are functions of the environment-temperature $T$ [21,22].

Figure 1. Coordinate systems of spherical axes and Cartesian axes for FGM spherical shells on $T$ effect.

2.1. TSDT Model of Displacements

The nonlinear TSDT model of displacements $u$, $v$, and $w$ of thick-walled FGM spherical shells in Figure 1 on a given angle $\varnothing$ and x axial length $L$ are expressed with $c_1$ denotes the nonlinear TSDT term of z³ as follows [22],

$$u=u_0(x,\theta ,t)+z{\varphi }_x(x,\theta ,t)-c_1{z}^3({\varphi }_x+{\displaystyle \frac{\partial w}{\partial x}}),$$

(1)

$$v=v_0(x,\theta ,t)+z{\varphi }_{\theta }(x,\theta ,t)-c_1{z}^3({\varphi }_{\theta }+{\displaystyle \frac{\partial w}{R\partial \theta }}),$$

(2)

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

(3)

in which $u_0$ and $v_0$ are tangential-displacement in x and $\theta$ axes, respectively, $w$ denotes the transverse-displacement in the $z$ axis of the middle plane in shells. ${\varphi }_x$ and ${\varphi }_{\theta }$ denote the shear-rotations. $R$ denotes the radius of the middle surface in shells. t denotes the time. $c_1=4/\left(3{h^{\mathrm{*}}}^2\right)$, in which $h^{\ast }$ is the total thickness equal to $h_1$ + $h_2$, where $h_1$ denotes the thickness of constituent-material 1 and $h_2$ denotes the thickness of constituent-material 2.

2.2. Dynamic Equilibrium Equation with TSDT

The dynamic partial differential equation (PDE) of motion represented in TSDT on the given $\varnothing$ of thick-walled FGM spherical shells can be applied [20]. Also, the Von Karman strain-displacement relations with not negligible shear strains are applied to the given $\varnothing$ of thick-walled FGM spherical shells. Thus, the five dynamic equilibrium PDE with TSDT in matrix form can be represented for FGM spherical shells. The coefficients of elements containing stiffness integrals $A_{i^s{j}^s},B_{i^s{j}^s},D_{i^s{j}^s},E_{i^s{j}^s},F_{i^s{j}^s},H_{i^s{j}^s}$,$i^s,j^s=1,2,6$, $A_{i^{\ast }{j}^{\ast }},B_{i^{\ast }{j}^{\ast }},D_{i^{\ast }{j}^{\ast }},E_{i^{\ast }{j}^{\ast }},F_{i^{\ast }{j}^{\ast }},H_{i^{\ast }{j}^{\ast }}$, $i^{\ast },j^{\ast }=4,5$ and $c_1$ terms. The stiffness integrals that are in integrals of stiffness ${\overline{Q}}_{i^s{j}^s}$, ${\overline{Q}}_{i^{\ast }{j}^{\ast }}$ can be applied to the given $\varnothing$ ≠ 0 in the following [22],

$$\left(A_{i^s{j}^s},B_{i^s{j}^s},D_{i^s{j}^s},E_{i^s{j}^s},F_{i^s{j}^s},H_{i^s{j}^s}\right)={\int }_{-{\displaystyle \frac{h^{\ast }}{2}}}^{{\displaystyle \frac{h^{\ast }}{2}}}{\overline{Q}}_{i^s{j}^s}(1,z,z^2,z^3,z^4,z^6)dz,$$

(4)

$$\left(A_{i^{\ast }{j}^{\ast }},B_{i^{\ast }{j}^{\ast }},D_{i^{\ast }{j}^{\ast }},E_{i^{\ast }{j}^{\ast }},F_{i^{\ast }{j}^{\ast }},H_{i^{\ast }{j}^{\ast }}\right)={\int }_{-{\displaystyle \frac{h^{\ast }}{2}}}^{{\displaystyle \frac{h^{\ast }}{2}}}{k}_{\alpha }{\overline{Q}}_{i^{\ast }{j}^{\ast }}(1,z,z^2,z^3,z^4,z^5)dz.$$

(5)

in which $k_{\alpha }$ denotes shear correction, usually considered and used in transverse stresses to make a good adjustment for thick-walled materials.

2.3. Advanced $k_{\alpha }$

The advanced $k_{\alpha }$ expression for the given $\varnothing$ in the thick-walled FGM spherical shells can be applied in the following [22]:

$$k_{\alpha }={\displaystyle \frac{1}{h^{\ast }}}{\displaystyle \frac{FGMZSV}{FGMZIV}},$$

(6)

where $FGMZSV=({FGMZS-c_{1}FGMZSN)}^2$ expressed in $({h^{\ast })}^6$ power and $FGMZIV=FGMZI-2c_{1}FGMZIV1+{c_1}^{2}FGMZIV2$ expressed in $({h^{\ast })}^5$ power, in which $FGMZS$, $FGMZSN$, $FGMZI$, $FGMZIV1$, and $FGMZIV2$ are parameters in the functions of the Young’s modulus on FGM constituent-material 1 and 2 for $E_1$, $E_2$, respectively, and the power law index $R_n$. Thus, the advanced $k_{\alpha }$ values are non-dimensional and nonlinear and functions of $c_1$, $R_n$, and $T$, but the advanced $k_{\alpha }$ values are not functions of parameter $h^{\ast }$.

2.4. Simple Homogeneous Equation

In the free vibration study, there are no thermal loads for temperature-difference $\Delta T$ = 0, no in-plane distributed forces $p_1=p_2=0$ and no external pressure load $q=0$. The free vibration frequency ${\omega }_{mn}$ with subscripts m and n denote that the mode shape number can be used in the following typical four-sided simple supported time sinusoidal displacements $u_0$, $v_0$, $w$ and shear rotations ${\varphi }_x$, ${\varphi }_{\theta }$ expressions with amplitudes $a_{mn}$, $b_{mn}$, $c_{mn}$, $d_{mn}$ and $e_{mn}$.

$${u_0=a_{mn}\mathrm{c}\mathrm{o}\mathrm{s}\left(m\pi x/L\right)\mathrm{s}\mathrm{i}\mathrm{n}\left(n\pi \theta /R\right)\mathrm{s}\mathrm{i}\mathrm{n}(\omega }_{mn}t),$$

(7)

$${v_0=b_{mn}\mathrm{s}\mathrm{i}\mathrm{n}\left(m\pi x/L\right)\mathrm{c}\mathrm{o}\mathrm{s}\left(n\pi \theta /R\right)\mathrm{s}\mathrm{i}\mathrm{n}(\omega }_{mn}t),$$

(8)

$${w=c_{mn}\mathrm{s}\mathrm{i}\mathrm{n}\left(m\pi x/L\right)\mathrm{s}\mathrm{i}\mathrm{n}\left(n\pi \theta /R\right)\mathrm{s}\mathrm{i}\mathrm{n}(\omega }_{mn}t),$$

(9)

$${{\varphi }_x=d_{mn}\mathrm{c}\mathrm{o}\mathrm{s}\left(m\pi x/L\right)\mathrm{s}\mathrm{i}\mathrm{n}\left(n\pi \theta /R\right)\mathrm{s}\mathrm{i}\mathrm{n}(\omega }_{mn}t),$$

(10)

$${{\varphi }_{\theta }=e_{mn}\mathrm{s}\mathrm{i}\mathrm{n}\left(m\pi x/L\right)\mathrm{c}\mathrm{o}\mathrm{s}\left(n\pi \theta /R\right)\mathrm{s}\mathrm{i}\mathrm{n}(\omega }_{mn}t),$$

(11)

where ${\omega }_{mn}$ is the natural frequency with subscripts $m$ is the number of axial half-waves and $n$ is the number of circumferential waves. By substituting Equations (7)–(11) into PDE under free vibration with no external loads $f_1=f_2=\cdots =f_5=0$ and assumed reasonable simplifications of ${FH}_{13}$ = ${FH}_{14}$ = ${FH}_{15}$ = ${FH}_{23}$= ${\mathrm{F}\mathrm{H}}_{24}$ = ${FH}_{25}$ = 0, $B_{ij}=E_{ij}=A_{16}=A_{26}=D_{16}=D_{26}=A_{45}=D_{45}=F_{45}=0$ and $I_1=I_3$ = $J_1$= $I_6$ = $J_4$ = 0 in the homogeneous matrix. Thus, the simple homogeneous equation in a symmetrical sparse matrix can be obtained in the following [22].

$$\left[\begin{array}{ccccc}F{H}_{11}-{\lambda }_{mn}& F{H}_{12}& 0& 0& 0\\ F{H}_{12}& F{H}_{22}-{\lambda }_{mn}& 0& 0& 0\\ 0& 0& F{H}_{33}-{\lambda }_{mn}& F{H}_{34}& F{H}_{35}\\ 0& 0& F{H}_{34}& F{H}_{44}-{\displaystyle \frac{K_2}{I_0}}{\lambda }_{mn}& F{H}_{45}\\ 0& 0& F{H}_{35}& F{H}_{45}& F{H}_{55}-{\displaystyle \frac{K_2}{I_0}}{\lambda }_{mn}\end{array}\right]\left\{\begin{array}{c}{a}_{mn}\hfill \\ b_{mn}\hfill \\ c_{mn}\hfill \\ d_{mn}\hfill \\ e_{mn}\hfill \end{array}\right\}=\left\{\begin{array}{c}0\hfill \\ 0\hfill \\ 0\hfill \\ 0\hfill \\ 0\hfill \end{array}\right\},$$

(12)

where $I_i={\displaystyle \sum _{k=1}^{N^{\ast }}{\displaystyle {\int }_k^{k+1}{\rho }^{(k)}}}{z}^{i}dz$, (i = 0,1,2,…,6), in which $N^{\ast }$ is the total number of layers, ${\rho }^{(k)}$ is the density of the (k)th ply. $J_i=I_i-c_1{I}_{i+2}$, $(i=1,4)$, $K_2=I_2-2c_1{I}_4+{c_1}^2{I}_6$, ${\lambda }_{mn}=I_0{{\omega }_{mn}}^2$, ${FH}_{11},\dots ,{FH}_{55}$ coefficients are listed in the Appendix A.

Thus, the stiffness integrals $A_{i^s{j}^s},B_{i^s{j}^s},D_{i^s{j}^s},E_{i^s{j}^s},F_{i^s{j}^s},H_{i^s{j}^s}$ value, $A_{i^{\ast }{j}^{\ast }},B_{i^{\ast }{j}^{\ast }},D_{i^{\ast }{j}^{\ast }},E_{i^{\ast }{j}^{\ast }},F_{i^{\ast }{j}^{\ast }},H_{i^{\ast }{j}^{\ast }}$ containing $k_{\alpha }$ value, and $c_1$ value of TSDT; they are all included in the homogeneous matrix of (12).

2.5. Numerical Method

The determinant of (12) vanishes for obtaining a non-trivial solution that can be used in the simple fifth-order of ${\lambda }_{mn}$ in a polynomial equation as follows,

$$A(1){{\lambda }_{mn}}^5+A(2){{\lambda }_{mn}}^4+A(3){{\lambda }_{mn}}^3+A(4){{\lambda }_{mn}}^2+A(5){\lambda }_{mn}+A(6)=0,$$

(13)

where $A\left(1\right), \dots ,A\left(6\right)$ coefficients are listed in the Appendix A.

The Lahey-Fujitsu Fortran 7.8 program is used to solve (13) by using the algorithm of Newton’s method [23]. In the numerical calculation for choosing and iterating a ${\lambda }_{mn}$ into (13) until the tolerance is less than or equal to the 1 × 10⁻⁶ value, then the root ${\lambda }_{mn}$ can be solved. Thus, the ${\omega }_{mn}$ can be calculated for the given $\varnothing$ ≠ 0 of the thick-walled FGM spherical shells under free vibration.

3. Numerical Results

The FGM constituent-material 1 on the inner part of the spherical shells is SUS304, the FGM constituent-material 2 on the outer part of the spherical shells is Si₃N₄ on the given $0°<\varnothing \le 90°$. They are used for free vibration frequency computations with a simple homogeneous equation under the effects of $T$, advanced $k_{\alpha }$ and four-sided simple supported boundary conditions. The basic geometric values are $L/R=1$, $h_1=h_2$, $h^{\ast }=1.2$ mm. The calculated values of advanced $k_{\alpha }$, referred to by Hong [21], which are much different from the varied values of $k_{\alpha }$ calculated by Hong [20], e.g., the compared values of advanced $k_{\alpha }$ with varied $k_{\alpha }$ for T = 300 K and 1000 K are listed in Table 1. The values of advanced $k_{\alpha }$ can be greater than one, can be a negative value, and can be affected by $c_1$. The values of varied $k_{\alpha }$ are usually positive, smaller than one, and not affected by $c_1$. Both types of $k_{\alpha }$ values are affected by $R_n$ and T.

Table 1. Values of advanced $k_{\alpha }$ and varied $k_{\alpha }$ vs. $R_n$ under T = 300 K and 1000 K.

${\mathit{c}}_1$ (1/mm2)	T (K)	Advanced kα by Hong [21]
		${\mathit{R}}_{\mathit{n}}=0.1$	${\mathit{R}}_{\mathit{n}}=0.2$	${\mathit{R}}_{\mathit{n}}=0.5$	${\mathit{R}}_{\mathit{n}}=1$	${\mathit{R}}_{\mathit{n}}=2$	${\mathit{R}}_{\mathit{n}}=5$	${\mathit{R}}_{\mathit{n}}=10$
0.925925	300	−0.821565	−0.861923	−1.181503	−4.392341	1.474844	0.583927	0.463617
0	300	0.898426	0.956498	1.087891	1.195721	1.226106	1.121959	1.019034
0.925925	1000	−0.189321	−0.185984	−0.195625	−0.252506	−0.532898	1.590231	0.610227
0	1000	0.932949	1.029293	1.276062	1.516531	1.616820	1.419804	1.206723
		Varied $k_{\alpha }$ by Hong [20]
0.925925	300	-	-	0.102677	0.138573	0.217517	-	0.492255
0	300	-	-	0.102677	0.138573	0.217517	-	0.492255
0.925925	1000	0.057519	0.059382	0.067907	0.088108	0.137812	0.270674	0.355043
0	1000	0.057519	0.059382	0.067907	0.088108	0.137812	0.270674	0.355043

3.1. Non-Dimensional Frequency

The non-dimensional frequency $f^{\ast }=4\pi {\omega }_{11}R\sqrt{I_2/A_{11}}$ defined, in which ${\omega }_{11}$ is the first natural frequency for mode shape $m=n=1$. Values of $f^{\ast }$ under $c_1$ = 0.925925/mm² and $c_1$ = 0/mm² for ${L/h}^{\ast }$ = 5, 8 and 10, respectively, on $\varnothing =10°,45°$ and $90°$ are displayed in Table 2, Table 3 and Table 4 for the present solution of simple homogeneous Equation (12). The $f^{\ast }$ greatest values under environment-temperature 1 K, 100 K, 300 K, 600 K, and 1000 K with advanced $k_{\alpha }$ and $c_1$ = 0.925925/mm² are 208.72235 on $\varnothing =10°$ and 24.658657 on $\varnothing =45°$, 10.076400 on $\varnothing =90°$, respectively, but when $c_1$ = 0/mm² is applied, then $f^{\ast }$ greatest values become 366.27481 on $\varnothing =10°$, 24.460136 on $\varnothing =45°$ and 7.898860 on $\varnothing =90°$. Thus the $f^{\ast }$ values are significantly affected by $c_1$. On $\varnothing =10°$, ${L/h}^{\ast }$ = 10, a non-zero drastically reduces the frequency $f^{\ast }$ value from 366.27 (for $R_n$ = 10, $c_1$ = 0/mm², T = 600 K) to 208.72 (for $R_n$ = 0.5, $c_1$ = 0.925925/mm², T = 1000 K); the $f^{\ast }$ value reduces mainly due to the effects of $R_n$, $c_1$, T, and $k_{\alpha }$. On $\varnothing =90°$ (become a circular cylindrical shell), a non-zero value smoothly increases the frequency $f^{\ast }$ value from 7.90 (for ${L/h}^{\ast }$ = 5, $R_n$ = 10, $c_1$ = 0/mm², T = 600 K) to 10.08 (for ${L/h}^{\ast }$ = 8, $R_n$ = 1, $c_1$ = 0.925925/mm², T = 300 K); the $f^{\ast }$ value increases mainly due to the effects of ${L/h}^{\ast }$, $R_n$, $c_1$, T and $k_{\alpha }$.

Table 2. $f^{\ast }$ for SUS304/Si₃N₄ on $\varnothing =10°$.

${\mathit{L}/\mathit{h}}^{\ast }$	${\mathit{R}}_{\mathit{n}}$	${\mathit{c}}_1$ (1/mm2)	${\mathit{f}}^{\ast }$
			Present Solution, Advanced ${\mathit{k}}_{\mathit{\alpha }}$
			T = 1 K	T = 100 K	T = 300 K	T = 600 K	T = 1000 K
5	0.5	0.925925 0	34.831550 59.372345	38.937732 84.997497	4.556338 35.301105	5.242971 38.132156	78.854614 44.269439
	1	0.925925 0	37.521785 35.943412	31.496433 36.446735	2.192647 69.074363	2.675805 40.791358	82.710907 49.129543
	2	0.925925 0	3.172364 39.290824	10.797158 75.432434	15.345757 40.402904	15.711335 43.702884	56.756023 54.894638
	10	0.925925 0	6.224920 45.137218	6.270420 95.846450	6.432409 44.748584	6.987969 48.461200	8.692406 65.753356
8	0.5	0.925925 0	87.841110 214.01928	6.917023 89.241149	4.426317 156.25784	5.012203 101.19923	204.07486 115.87066
	1	0.925925 0	7.279757 94.427879	5.191968 96.072174	2.306787 99.888313	2.802402 107.94323	129.80630 248.56913
	2	0.925925 0	3.276158 103.03987	19.200660 103.89609	26.983800 106.81732	27.396707 270.33493	11.700458 142.53106
	10	0.925925 0	7.108106 118.49715	7.159274 117.54760	7.341172 118.49424	7.977503 128.28086	9.923837 170.61027
10	0.5	0.925925 0	8.548742 279.42364	6.684215 133.86978	4.432353 141.63438	5.011566 152.74237	208.72235 279.82177
	1	0.925925 0	6.999097 140.05490	5.163668 143.49270	2.336681 150.47529	2.836246 162.35302	194.80217 185.68304
	2	0.925925 0	3.306802 152.07115	24.810934 154.55258	33.525913 160.42941	34.126480 173.18660	11.050667 205.30691
	10	0.925925 0	7.350216 173.77079	7.403072 174.01590	7.590321 177.39599	8.248464 366.27481	10.257929 242.93650

Table 3. $f^{\ast }$ for SUS304/Si₃N₄ on $\varnothing =45°.$

${\mathit{L}/\mathit{h}}^{\ast }$	${\mathit{R}}_{\mathit{n}}$	${\mathit{c}}_1$ (1/mm2)	${\mathit{f}}^{\ast }$
			Present Solution, Advanced ${\mathit{k}}_{\mathit{\alpha }}$
			T = 1 K	T = 100 K	T = 300 K	T = 600 K	T = 1000 K
5	0.5	0.925925 0	8.463405 6.625898	9.136831 7.118206	10.136724 7.947797	10.686658 8.430737	9.802042 7.683148
	1	0.925925 0	8.861703 6.976523	12.407337 7.458821	10.585899 8.277026	11.246615 8.777153	11.014883 8.156356
	2	0.925925 0	9.246535 7.330720	9.915120 7.796439	11.062045 8.596633	11.759777 9.119574	11.014883 8.678068
	10	0.925925 0	10.029774 7.824207	10.607832 8.254026	11.644557 9.016480	12.391345 9.586984	12.175586 9.520373
8	0.5	0.925925 0	5.774473 4.529300	6.245143 4.894484	7.072479 5.521222	7.499098 5.859288	6.646452 15.97286
	1	0.925925 0	6.030556 4.728994	6.499904 5.088683	7.209340 5.709747	7.564799 6.060668	7.008310 22.498374
	2	0.925925 0	12.304396 4.956110	6.783788 5.306417	23.838476 5.917467	24.658657 6.283709	7.438728 24.460136
	10	0.925925 0	6.809462 15.525394	7.224026 17.441642	7.968572 21.958253	8.469730 21.647085	8.213203 16.823829
10	0.5	0.925925 0	4.888590 12.409802	5.286721 13.939558	5.980225 16.848964	6.343058 16.896772	5.625683 12.830549
	1	0.925925 0	5.104107 14.62383	5.499571 16.566768	1.546536 20.627670	9.123397 19.451889	5.931276 14.381142
	2	0.925925 0	5.371245 15.131563	5.784337 16.883903	6.357991 20.445615	6.751663 19.658353	6.294243 15.280694
	10	0.925925 0	5.765657 12.868285	6.117527 13.937146	6.750018 15.918221	7.174003 16.623294	6.951572 14.781123

Table 4. $f^{\ast }$ for SUS304/Si₃N₄ on $\varnothing =90°.$

${\mathit{L}/\mathit{h}}^{\ast }$	${\mathit{R}}_{\mathit{n}}$	${\mathit{c}}_1$ (1/mm2)	${\mathit{f}}^{\ast }$
			Present Solution, Advanced ${\mathit{k}}_{\mathit{\alpha }}$
			T = 1 K	T = 100 K	T = 300 K	T = 600 K	T = 1000 K
5	0.5	0.925925 0	3.180231 5.373269	3.437422 5.900316	3.883499 6.836672	4.120175 7.190872	3.663467 5.986253
	1	0.925925 0	3.320887 5.676170	3.575643 6.208767	3.975845 7.160784	0.922096 7.515075	3.863254 6.343256
	2	0.925925 0	3.492469 5.922058	3.733553 6.440915	4.141279 7.375738	4.398373 7.747935	4.100106 6.706929
	10	0.925925 0	3.756472 6.152549	3.984605 6.615449	4.394351 7.462710	4.670895 7.898860	4.532107 7.257466
8	0.5	0.925925 0	2.239428 4.160760	2.420821 4.516713	2.734613 5.131114	2.901336 5.437644	2.577111 4.752500
	1	0.925925 0	2.337628 4.345048	2.516835 4.697000	10.076400 5.307589	3.009658 5.623309	2.716942 5.000139
	2	0.925925 0	2.452844 4.543049	3.294041 4.886523	2.921348 5.488020	3.102310 5.817079	2.882582 5.291233
	10	0.925925 0	2.643668 4.871469	2.804904 5.191528	3.095393 5.766388	3.289649 6.127082	3.186127 5.840260
10	0.5	0.925925 0	1.926534 3.864588	2.082611 4.189507	2.352230 4.749449	2.495704 5.036603	2.217462 4.427573
	1	0.925925 0	2.010933 4.032133	2.165022 4.352602	2.439111 4.908469	2.586515 5.204863	2.337732 4.656122
	2	0.925925 0	2.109406 4.217228	5.662171 4.530303	2.514311 5.077848	2.670045 5.386767	2.480144 4.928913
	10	0.925925 0	2.274468 4.538136	2.413341 4.830076	2.663402 5.354484	2.830519 5.691110	2.741575 5.452303

Another one non-dimensional frequency $\mathsf{\Omega }=({\omega }_{11}{L}^2/h^{\ast })\sqrt{{\rho }_1/E_1}$ defined, where ${\rho }_1$ is the density of FGM constituent-material 1. Values of $\mathsf{\Omega }$ under $c_1$ = 0.925925/mm² and $c_1$ = 0/mm² for ${L/h}^{\ast }$ = 5, 8, and 10, respectively, on $\varnothing =10°,45°$, and $90°$ are displayed in Table 5, Table 6 and Table 7 for the present solution of simple homogeneous Equation (12). The $\mathsf{\Omega }$ greatest values under environment-temperature 1 K, 100 K, 300 K, 600 K, and 1000 K with advanced $k_{\alpha }$ and $c_1$ = 0.925925/mm² are 811.82086 on $\varnothing =10°$, 63.836673 on $\varnothing =45°$, and 18.566265 on $\varnothing =90°$, respectively, but when $c_1$ = 0/mm² is applied, then $\mathsf{\Omega }$ greatest values become 1119.0484 on $\varnothing =10°$, 68.950126 on $\varnothing =45°$, and 17.473329 on $\varnothing =90°$. Thus, the $\mathsf{\Omega }$ values are significantly affected by $c_1$.

Table 5. $\mathsf{\Omega }$ for SUS304/Si₃N₄ on $\varnothing =10°.$

${\mathit{L}/\mathit{h}}^{\ast }$	${\mathit{R}}_{\mathit{n}}$	${\mathit{c}}_1$ (1/mm2)	$\mathsf{\Omega }$
			Present Solution, Advanced ${\mathit{k}}_{\mathit{\alpha }}$
			T = 1 K	T = 100 K	T = 300 K	T = 600 K	T = 1000 K
5	0.5	0.925925 0	63.351696 107.98654	69.176635 151.00625	7.872272 60.991939	9.094638 66.145347	153.35162 86.092491
	1	0.925925 0	65.387298 62.636749	53.840740 62.302898	3.664575 115.44409	4.488223 68.420784	152.57975 90.631027
	2	0.925925 0	5.275682 65.341133	17.701944 123.67149	24.750640 65.164451	25.421075 70.711647	98.698249 95.461311
	10	0.925925 0	9.585399 69.504226	9.606960 146.84709	9.802168 68.191116	10.674876 74.029701	13.669148 103.39972
8	0.5	0.925925 0	255.62495 622.81396	19.662015 253.67280	12.236204 431.96243	13.910944 280.86987	634.99566 360.54107
	1	0.925925 0	20.297699 263.28744	14.200435 262.76483	6.168540 267.10961	7.520911 289.69122	383.13333 733.67102
	2	0.925925 0	8.717268 274.17056	50.367183 272.54028	69.633987 275.65118	70.924980 699.84686	32.555198 396.57653
	10	0.925925 0	17.512588 291.94723	17.550048 288.15298	17.899211 288.91210	19.498394 313.54055	24.968998 429.26611
10	0.5	0.925925 0	31.096941 1016.4325	23.750307 475.66513	15.316110 489.42126	17.386470 529.90429	811.82086 1088.3603
	1	0.925925 0	24.393941 488.13305	17.653791 490.57959	7.810600 502.97924	9.514674 544.64093	718.71704 685.07226
	2	0.925925 0	10.998508 505.79248	81.355064 506.77801	108.14557 517.50213	110.43387 560.43481	38.434036 714.0540
	10	0.925925 0	22.636358 535.15942	22.684612 533.22229	23.133356 540.65759	25.200836 1119.0484	32.261989 764.05432

Table 6. $\mathsf{\Omega }$ for SUS304/Si₃N₄ on $\varnothing =45°.$

${\mathit{L}/\mathit{h}}^{\ast }$	${\mathit{R}}_{\mathit{n}}$	${\mathit{c}}_1$ (1/mm2)	$\mathsf{\Omega }$
			Present Solution, Advanced ${\mathit{k}}_{\mathit{\alpha }}$
			T = 1 K	T = 100 K	T = 300 K	T = 600 K	T = 1000 K
5	0.5	0.925925 0	15.393258 12.051197	16.232461 12.646179	17.513855 13.731909	18.537445 14.624245	19.062410 14.941716
	1	0.925925 0	15.442839 12.157630	21.209392 12.750283	17.692232 13.833409	18.864345 14.722230	19.154790 15.046321
	2	0.925925 0	15.377103 12.191080	16.255842 12.782263	17.841590 13.865214	19.027420 14.755550	19.154790 15.091087
	10	0.925925 0	15.444276 12.048050	16.252342 12.646059	17.744815 13.739962	18.929115 14.645152	19.146585 14.971159
8	0.5	0.925925 0	16.804197 13.180641	17.752159 13.912838	19.551309 15.262979	20.813110 16.261955	20.680982 49.700874
	1	0.925925 0	16.814628 13.185563	17.777742 13.917944	19.278373 15.268340	20.301931 16.265239	20.685569 66.405693
	2	0.925925 0	32.739784 13.187316	17.795236 13.919798	61.517211 15.270527	63.836673 16.267355	20.697420 68.057556
	10	0.925925 0	16.776804 38.250671	17.708782 42.755966	19.428937 53.538513	20.701482 52.909210	20.664934 42.329807
10	0.5	0.925925 0	17.782756 45.141944	18.784742 49.529937	20.664821 58.222026	22.005773 58.619442	21.880968 49.904129
	1	0.925925 0	17.789335 50.968433	18.802192 56.639240	5.1694598 68.950126	30.605995 65.254684	21.883276 53.058815
	2	0.925925 0	17.864898 50.327957	18.966844 55.362327	20.509168 65.952049	21.848499 63.614772	21.891271 53.145999
	10	0.925925 0	17.756412 39.630275	18.745422 42.706424	20.572330 48.514667	21.918127 50.787746	21.863239 46.487785

Table 7. $\mathsf{\Omega }$ for SUS304/Si₃N₄ on $\varnothing =90°.$

${\mathit{L}/\mathit{h}}^{\ast }$	${\mathit{R}}_{\mathit{n}}$	${\mathit{c}}_1$ (1/mm2)	$\mathsf{\Omega }$
			Present Solution, Advanced ${\mathit{k}}_{\mathit{\alpha }}$
			T = 1 K	T = 100 K	T = 300 K	T = 600 K	T = 1000 K
5	0.5	0.925925 0	5.784212 9.772912	6.106911 10.482480	6.709765 11.812148	7.146997 12.473533	7.124487 11.641697
	1	0.925925 0	5.787141 9.891572	6.112288 10.613412	6.644836 11.967831	1.546665 12.605302	7.126682 11.701632
	2	0.925925 0	5.808020 9.848456	6.121162 10.559882	6.679326 11.896074	7.116606 12.536226	7.130050 11.663293
	10	0.925925 0	5.784378 9.473958	6.104845 10.135582	6.696430 11.372216	7.135295 12.066360	7.126915 11.412649
8	0.5	0.925925 0	6.516922 12.108159	6.881316 12.839003	7.559621 14.184557	8.052412 15.091719	8.018895 14.787794
	1	0.925925 0	6.517866 12.115031	6.883740 12.846661	26.945133 14.192934	8.077132 15.091481	8.019266 14.758296
	2	0.925925 0	6.526577 12.088236	8.640930 12.818329	7.538786 14.162303	8.031305 15.059334	8.020460 14.722258
	10	0.925925 0	6.513334 12.002076	6.875865 12.726371	7.547174 14.059581	8.040471 14.975647	8.016496 14.694459
10	0.5	0.925925 0	7.007969 14.057844	7.399919 14.886129	8.128190 16.411842	8.658269 17.473329	8.624771 17.220947
	1	0.925925 0	7.008700 14.053184	7.401881 14.880881	8.152981 16.407068	8.676906 17.460603	8.624997 17.178630
	2	0.925925 0	7.015937 14.026609	18.566265 14.854867	8.110490 16.379774	8.640313 17.431673	8.625900 17.142679
	10	0.925925 0	7.004647 13.976033	7.394999 14.800395	8.117367 16.319097	8.647848 17.387567	8.622469 17.147920

Comparisons for the present solution of frequency parameters $f^{\ast }$, $\mathsf{\Omega }$ with published available work are displayed in Table 8 and Table 9. The values of $f^{\ast }$ vs. $h^{\ast }$ for SUS304/Si₃N₄, ${L/h}^{\ast }$ = 10 under T = 300 K with advanced $k_{\alpha }$ are displayed in Table 8. The compared value $f^{\ast }$ = 2.336681 is mainly affected by the advanced $k_{\alpha }$ = −4.392341 on $c_1$ = 0.925925/mm². $R_n$ = 1 is much smaller than 11.616583 due to linear variation $k_{\alpha }$ = 0.138573 presented by Hong [20] and is also much smaller than 11.8633 presented by Sayyad and Ghugal [5] for a/h = 10, R/a = 10, in which a is the arc length and h is the thickness, three-layer $0°/90°/0°$ spherical laminates without considering the shear correction. Thus the $f^{\ast }$ values are significantly affected by $k_{\alpha }$. The values of $\mathsf{\Omega }$ vs. $h^{\ast }$ for SUS304/Si₃N₄, ${L/h}^{\ast }$ = 10 under T = 1000 K with advanced $k_{\alpha }$ and varied $k_{\alpha }$ effects are displayed in Table 9. The compared value $\mathsf{\Omega }$ = 38.434036 is mainly affected by the advanced $k_{\alpha }$ = −0.532898, $c_1$ = 0.925925/mm². $R_n$ = 2 is smaller than 59.915550 due to linear variation $k_{\alpha }$ = 0.137812 presented by Hong [20] and is also smaller than 69.520 presented by Li et al. [24] for h/R = 0.02, three-layer $0°/90°/0°$ spherical laminates by using FSDT and constant $k_{\alpha }$ = 5/6. Thus, the $\mathsf{\Omega }$ values are significantly affected by $k_{\alpha }$.

Table 8. Comparison of frequency $f^{\ast }$ for $\varnothing =10°.$

${\mathit{c}}_1$ (1/mm2)	${\mathit{h}}^{\ast }$ (mm)	${\mathit{f}}^{\ast }$
		Present Solution, ${\mathit{L}/\mathit{h}}^{\ast }$ = 10, T = 300 K, Advanced kα, SUS304/Si3N4			Sayyad and Ghugal, 2019, Spherical [5], Without ${\mathit{k}}_{\mathit{\alpha }}$
		${\mathit{R}}_{\mathit{n}}$ = 0.5	${\mathit{R}}_{\mathit{n}}$ = 1	${\mathit{R}}_{\mathit{n}}$ = 2
0.925925	1.2	4.432353	2.336681	33.525913	11.8633
0.333333	2	9.587066	5.052578	67.505981	-
0.000033	200	9649.3388	5082.5991	65,478.824	-
0.000014	300	17,684.128	9230.4921	120,311.75	-
0.000003	600	50,019.789	26,110.931	340,290.03	-
0.000001	900	89,233.835	43,224.820	625,047.06	-

Table 9. Comparison of frequency $\mathsf{\Omega }$ for $\varnothing =10°.$

${\mathit{c}}_1$ (1/mm2)	${\mathit{h}}^{\ast }$ (mm)	$\mathsf{\Omega }$
		Present Solution, ${\mathit{L}/\mathit{h}}^{\ast }$$= 10, \mathit{T}= 1000 \mathbf{K},$ Advanced ${\mathit{k}}_{\mathit{\alpha }}$, SUS304/Si3N4			Li et al., 2019, Spherical [24], Constant ${\mathit{k}}_{\mathit{\alpha }}$ = 5/6
		$R_n=\mathbf{0.5}$	$R_n=\mathbf{1}$	$R_n=\mathbf{2}$
0.925925	1.2	811.82086	718.71704	38.434036	69.520
0.333333	2	1436.6695	1492.0356	49.932869	-
0.000033	200	-	-	503.02319	-
0.000014	300	-	-	616.26330	-
0.000003	600	-	-	866.81646	-
0.000001	900	-	-	1066.6158

3.2. Natural Frequency

The values of dimensional ${\omega }_{mn}$ (1/s) for SUS304/Si₃N₄ FGM thick-walled spherical shells are presented. Values of ${\omega }_{11}$ (1/s) vs. $R_n$ on $\varnothing =10°$ are shown in Table 10 for ${L/h}^{\ast }$ = 5, 8, and 10, advanced $k_{\alpha }$, TSDT with $c_1$ = 0.925925/mm², FSDT with $c_1$ = 0/mm², under environment-temperature 1 K, 100 K, 300 K, 600 K, and 1000 K. Usually, the ${\omega }_{11}$ values on ${L/h}^{\ast }$ = 5 of FSDT are overestimated vs. TSDT except for $R_n$ = 1, T = 1 K, 1000 K and $R_n$ = 2, T = 1000 K, e.g., ${\omega }_{11}$ = 0.030261/s with $c_1$ = 0/mm² is greater than ${\omega }_{11}$ = 0.017753/s with $c_1$ = 0.925925/mm² for $R_n$ = 0.5, T = 1 K. Values of ${\omega }_{mn}$ vs. $m$,$n$ = 1,2,…,9 on $\varnothing =10°$ are shown in Table 11 for ${L/h}^{\ast }$ = 5 and 10, $R_n$ = 0.5, $T$ = 300 K, advanced $k_{\alpha }$, $c_1$ = 0.925925/mm². The ${\omega }_{mn}$ values are smaller than 0.008277/s for ${L/h}^{\ast }$ = 5 and smaller than 0.008056/s for ${L/h}^{\ast }$ = 10.

Table 10. Fundamental natural frequency ${\omega }_{11}$ under advanced $k_{\alpha }$ for $\varnothing =10°.$

${\mathit{L}/\mathit{h}}^{\ast }$	${\mathit{R}}_{\mathit{n}}$	${\mathit{c}}_1$ (1/mm2)	${\mathit{\omega }}_{11}$ (1/s)
			T = 1 K	T = 100 K	T = 300 K	T = 600 K	T = 1000 K
5	0.5	0.925925 0	0.017753 0.030261	0.019616 0.042820	0.002242 0.017373	0.002483 0.018059	0.003476 0.019515
	1	0.925925 0	0.018323 0.017553	0.015267 0.017667	0.001043 0.032885	0.001225 0.018680	0.034586 0.020544
	2	0.925925 0	0.001478 0.018310	0.005019 0.035069	0.007050 0.018562	0.006940 0.019306	0.022372 0.021639
	10	0.925925 0	0.002686 0.019477	0.002724 0.041641	0.002792 0.019424	0.002914 0.020212	0.003098 0.023438
8	0.5	0.925925 0	0.027982 0.068177	0.002177 0.028099	0.001361 0.048065	0.001483 0.029955	0.056226 0.031924
	1	0.925925 0	0.002221 0.028821	0.001572 0.029106	0.000686 0.029721	0.000802 0.030896	0.033925 0.064964
	2	0.925925 0	0.000954 0.030012	0.005579 0.030189	0.007748 0.030672	0.007564 0.074640	0.002882 0.035115
	10	0.925925 0	0.001917 0.031958	0.001944 0.031918	0.001991 0.032147	0.002079 0.033439	0.002210 0.038010
10	0.5	0.925925 0	0.002178 0.071210	0.001683 0.033721	0.001090 0.034853	0.001186 0.036169	0.046005 0.061677
	1	0.925925 0	0.001709 0.034198	0.001251 0.034778	0.000556 0.035819	0.000649 0.037175	0.040729 0.038822
	2	0.925925 0	0.000770 0.035435	0.005767 0.035926	0.007701 0.036853	0.007537 0.038253	0.002178 0.040465
	10	0.925925 0	0.001585 0.037492	0.001608 0.037801	0.001647 0.038502	0.001720 0.076383	0.001828 0.043298

Table 11. ${\omega }_{mn}$ vs. $m$ and $n$ under advanced $k_{\alpha }$, $c_1$, $R_n=0.5$ and T = 300 K for $\varnothing =10°$.

${\mathit{c}}_1$ (1/mm2)	${\mathit{L}/\mathit{h}}^{\ast }$	${\mathit{\omega }}_{1\mathit{n}}$ (1/s)
		$\mathit{n}=1$	$\mathit{n}=2$	$\mathit{n}=3$	$\mathit{n}=4$	$\mathit{n}=5$	$\mathit{n}=6$	$\mathit{n}=7$	$\mathit{n}=8$	$\mathit{n}=9$
0.925925	5 10	0.002242 0.001090	0.002089 0.001088	0.001960 0.001084	0.001851 0.001079	0.001756 0.001072	0.007328 0.001065	0.006254 0.008109	0.005463 0.008041	0.004853 0.001039
	${L/h}^{\ast }$	${\omega }_{2n}$ (1/s)
		$n=1$	$n=2$	$n=3$	$n=4$	$n=5$	$n=6$	$n=7$	$n=8$	$n=9$
	5 10	0.002168 0.001085	0.001994 0.001080	0.007582 0.001074	0.001731 0.001065	0.001627 0.001055	0.006977 0.007626	0.006031 0.007417	0.005312 0.006967	0.004745 0.007840
	${L/h}^{\ast }$	${\omega }_{3n}$ (1/s)
		$n=1$	$n=2$	$n=3$	$n=4$	$n=5$	$n=6$	$n=7$	$n=8$	$n=9$
	5 10	0.008277 0.001076	0.007304 0.001069	0.006635 0.001058	0.016829 0.001045	0.007538 0.001030	0.006521 0.007651	0.005732 0.006915	0.005104 0.006571	0.004595 0.000960
	${L/h}^{\ast }$	${\omega }_{4n}$ (1/s)
		$n=1$	$n=2$	$n=3$	$n=4$	$n=5$	$n=6$	$n=7$	$n=8$	$n=9$
	5 10	0.006493 0.006968	0.006096 0.005591	0.001638 0.001039	0.007733 0.001021	0.006813 0.007836	0.006041 0.007046	0.005398 0.006500	0.004863 0.006990	0.004415 0.004695
	${L/h}^{\ast }$	${\omega }_{5n}$ (1/s)
		$n=1$	$n=2$	$n=3$	$n=4$	$n=5$	$n=6$	$n=7$	$n=8$	$n=9$
	5 10	0.005589 0.001050	0.001724 0.001036	0.007466 0.001017	0.006794 0.007803	0.006152 0.007085	0.005569 0.006505	0.005054 0.006635	0.004606 0.005065	0.004220 0.004736
	${L/h}^{\ast }$	${\omega }_{6n}$ (1/s)
		$n=1$	$n=2$	$n=3$	$n=4$	$n=5$	$n=6$	$n=7$	$n=8$	$n=9$
	5 10	0.007494 0.001033	0.006872 0.008056	0.006462 0.007519	0.006017 0.006956	0.005561 0.006438	0.005121 0.006257	0.004713 0.005249	0.004344 0.004903	0.004015 0.004557
	${L/h}^{\ast }$	${\omega }_{7n}$ (1/s)
		$n=1$	$n=2$	$n=3$	$n=4$	$n=5$	$n=6$	$n=7$	$n=8$	$n=9$
	5 10	0.006218 0.007327	0.005948 0.007031	0.005680 0.006659	0.005372 0.006258	0.005042 0.005948	0.004709 0.005277	0.004387 0.004968	0.004086 0.004640	0.003809 0.004335
	${L/h}^{\ast }$	${\omega }_{8n}$ (1/s)
		$n=1$	$n=2$	$n=3$	$n=4$	$n=5$	$n=6$	$n=7$	$n=8$	$n=9$
	5 10	0.005418 0.006453	0.005243 0.006236	0.005056 0.005970	0.004836 0.005699	0.004592 0.005127	0.004337 0.004938	0.004083 0.004650	0.003838 0.004370	0.003606 0.002626
	${L/h}^{\ast }$	${\omega }_{9n}$ (1/s)
		$n=1$	$n=2$	$n=3$	$n=4$	$n=5$	$n=6$	$n=7$	$n=8$	$n=9$
	5 10	0.004813 0.005772	0.004686 0.005605	0.005550 0.005418	0.004388 0.005482	0.004203 0.004817	0.004006 0.004591	0.003804 0.004349	0.003604 0.002527	0.003411 0.007561

3.3. Compared ${\omega }_{1n}$

The values of dimensional ${\omega }_{mn}$ for $m$ = 1 and $n$ = 1 to 9 vs. $R_n$ and $T$ (K) are presented for SUS304/Si₃N₄. Figure 2 displays the ${\omega }_{1n}$ (1/s) vs. $R_n$ for ${L/h}^{\ast }$ = 5, 10, $\varnothing =10°$, advanced $k_{\alpha }$, $c_1$ = 0.925925/mm² under $T$ = 300 K. Usually for thick-walled ${L/h}^{\ast }$ = 5 in Figure 2a, the ${\omega }_{1n}$ values are oscillating with $n$, versus $R_n$ = 1; the ${\omega }_{1n}$ values are almost constant firstly, increasing then decreasing with $n$, versus $R_n$ = 0.5 and 10; the maximum ${\omega }_{13}$ = 0.014121/s is obtained for $R_n$ = 1. Thus, the ${\omega }_{1n}$ values are affected by $R_n$ or ${L/h}^{\ast }$ = 5. For moderately thick-walled ${L/h}^{\ast }$ = 10 in Figure 2b, the ${\omega }_{1n}$ values are constant with $n$, versus $R_n$ = 10; the ${\omega }_{1n}$ values are a lower constant with $n$ from 1 to 6, increasing then decreasing, versus $R_n$ = 0.5 and 1; the maximum ${\omega }_{17}$ = 0.008109/s is obtained for $R_n$ = 0.5. Thus, the ${\omega }_{1n}$ values are also affected by $R_n$ or ${L/h}^{\ast }$= 10.

Figure 2. ${\omega }_{1n}$ (1/s) vs. $R_n$ under $T$ = 300 K for: (a) ${L/h}^{\ast }$ = 5; (b) ${L/h}^{\ast }$ = 10.

Figure 3 displays the ${\omega }_{1n}$ (1/s) vs. $T$ for ${L/h}^{\ast }$ = 5, 10, ${\varnothing =10}^{°}$, advanced $k_{\alpha }$, $c_1$ = 0.925925/mm² and $R_n$ = 0.5. Usually for thick-walled ${L/h}^{\ast }$ = 5 in Figure 3a, the ${\omega }_{1n}$ values are constant with $n$ from 1 to 5, increasing with $n$ from 5 to 6 versus $T$ = 300 K and 600 K, but decreasing with $n$ from 1 to 6 versus $T$ = 1000 K; the maximum ${\omega }_{11}$ = 0.034761/s is obtained for $T$ = 1000 K. Thus, the ${\omega }_{16}$ and ${\omega }_{19}$ values have the endurance ability on higher environment-temperature 1000 K for ${L/h}^{\ast }$ = 5. For moderately thick-walled ${L/h}^{\ast }$ = 10 in Figure 3b, the ${\omega }_{1n}$ values are constant with $n$ from 1 to 6 versus $T$ = 300 K and 600 K; the maximum ${\omega }_{11}$ = 0.046005/s is obtained for $T$ = 1000 K. Thus, the ${\omega }_{17}$ value has the endurance ability in a higher environment-temperature of 1000 K for ${L/h}^{\ast }$ = 10.

Figure 3. ${\omega }_{1n}$ (1/s) vs. $T$ (K) for: (a) ${L/h}^{\ast }$ = 5; (b) ${L/h}^{\ast }$ = 10.

4. Discussions

The variation of natural frequency compared ${\omega }_{1n}$ (1/s) with respect to $c_1$ and $k_{\alpha }$ requires further clarification on $\varnothing =10°$, ${L/h}^{\ast }$ = 5, $R_n$ = 0.5, T = 1000 K are displayed in Figure 4. The effects of $c_1$ value are displayed in Figure 4a for ${\omega }_{1n}$, values are decreasing for n = 1 to 6, ${\omega }_{1n}$ values on $c_1$ = 0.925925/mm² are greater than on $c_1$ = 0/mm² for n = 1 to 8, ${\omega }_{1n}$ values on $c_1$ = 0.333333/mm² are greater than on $c_1$ = 0.925925/mm² for n = 1 and oscillate close on $c_1$ = 0/mm² for n = 2 to 9. The effects of $k_{\alpha }$ value are displayed in Figure 4b for ${\omega }_{1n}$ values on advanced $k_{\alpha }$ [21] are greater than on varied $k_{\alpha }$ [20] for n = 1 to 5, and the ${\omega }_{1n}$ values are in sinusoidal oscillating on varied $k_{\alpha }$ case. Basically, the values of advanced $k_{\alpha }$ [21] are mainly functions of material properties $E_1$, $E_2$, $R_n$, T, which are primarily from stiffness redistribution, but not a function of $h^{\ast }$ which is from a thickness-wise examination of the strain energy approach, and $c_1$ which is from a correction associated with the assumed kinematic TSDT model. The values of varied $k_{\alpha }$ [20] are mainly functions of material properties $E_1$, $E_2$, $R_n$, T, and $h^{\ast }$, but not a function of $c_1$.

Figure 4. Compared ${\omega }_{1n}$ (1/s) for: (a) $c_1$ = 0/mm², $c_1$ = 0.333333/mm², and $c_1$ = 0.925925/mm²; (b) advanced $k_{\alpha }$ [21] and varied $k_{\alpha }$ [20]; (c) sparse matrix and full matrix [20].

Thus, when $c_1$ = 0/mm² is used, the structure behaves classically for FSDT, whereas when $c_1$ ≠ 0 is used, e.g., $c_1$ = 0.333333/mm² on $h^{\ast }$ = 2 mm and $c_1$ = 0.925925/mm² on $h^{\ast }$ = 1.2 mm dominate the $h^{\ast }$ size effects for TSDT. There is a critical temperature, e.g., greater than 600 K or a thickness ratio, e.g., ${L/h}^{\ast }$ less than 10 for a thick-walled study where local effects become not negligible due to thermal effects for FGMs. The dependence of ${\omega }_{mn}$ on higher mode numbers (m,n ≤ 9) results in very low frequency values. This free vibration behavior is physically expected for thick-walled spherical FGM shells in the given typically sinusoidal displacements and shear rotations. The numerical truncation effects involved in the compared ${\omega }_{1n}$ descriptions are displayed in Figure 4c with a presented simple homogeneous equation in a symmetrical sparse matrix (12) and a fully homogeneous equation in a symmetrical full matrix [20]. The assumed simplification in the sparse matrix causes a numerical truncation error, e.g., a large different ${\omega }_{1n}$ value occurs at n = 1 and 2; the ${\omega }_{1n}$ values of the presented sparse matrix are greater than those in the full matrix. The variation of natural frequency compared ${\omega }_{1n}$, ${\lambda }_{1n}=I_0{{\omega }_{1n}}^2$ with respect to ${L/h}^{\ast }$ value on $\varnothing =10°$, $c_1$ = 0.925925/mm², $R_n$ = 0.5, 1, 10, T = 300 K are displayed in Figure 2. For ${L/h}^{\ast }$ = 5, it is displayed in Figure 2a, and for ${L/h}^{\ast }$ = 10, it is displayed in Figure 2b due to the $R_n$ power function of Young’s modulus in the expression $E_{fgm}=\left(E_2-E_1\right)({\displaystyle \frac{z+h^{\ast }/2}{h^{\ast }}}{)}^{R_n}+E_1$ for FGMs [21]. This $E_{fgm}$ dominates thickness and affects bending-shear coupling for ${\omega }_{1n}$ data with ${L/h}^{\ast }$ = 10 showing slightly lower frequencies than ${L/h}^{\ast }$ = 5. In Figure 3a, the frequency increases with moderate temperatures T = 600 K but decreases at T = 1000 K. The physical phenomenon causes this trend reversal at high temperature due to the individual temperature-natural material property $P_i$, e.g., $E_1$, $E_2$, etc. are expressed in $P_i=P_0(P_{-1}{T}^{-1}+1+P_{1}T+P_2{T}^2+P_3{T}^3)$, in which $P_0$, $P_{-1}$, $P_1$, $P_2$, and $P_3$ denote coefficients of temperature for FGMs [20]. The values of advanced $k_{\alpha }$ expression in (6) are non-dimensional and nonlinear and functions of $c_1$, $R_n$, and $T$, but are not functions of $h^{\ast }$ for thick-walled FGMs [22]. The physical meaning of advanced $k_{\alpha }$ is used to obtain an auto-calculation adjustment value rather than an assumed constant positive value for the transverse stiffness and transverse shear stress. Thus $k_{\alpha }$ behaves unusually, e.g., $k_{\alpha }$ > 1 or $k_{\alpha }$ < 0 due to the given values of $c_1$, $R_n$, and $T$. There are many key parameters studied in the applied mathematical simulation for the FGM spherical shell structure under numerical free vibration. In theory and simulation go first, this is why certain parameters, e.g., advanced shear correction, temperature, and power-law index $R_n$ = 0.1 are closely related to ceramic Si₃N₄; $R_n$ = 10 is closely related to metal SUS304, which leads to the influence on increases or decreases in natural frequencies by solving the homogeneous equation in a sparse matrix or a full matrix.

In particular, it is interesting in the related works on nonlinear vibration in 2025 by Putranto et al. [25] and equivalent single-layer (ESL) modeling in 2024 by Putranto [26] to discuss the broader context of advanced structural dynamics. A brief comparison of modeling philosophy and applicability would be valuable if possible in the future. A numerical flowchart summarizing the modeling steps, assumptions, and solution procedure to improve readability is displayed in Figure 5. The comparisons with selected published results would show more agreement in the exact choice of benchmark studies, e.g., FGM type, $k_{\alpha }$ value, $c_1$ value, sparse matrix, and full matrix. If possible, one further comparison with an ESL-based solution would be more confidence in the future.

Figure 5. A numerical flowchart.

5. Conclusions

The parameters of dimensional natural frequency and non-dimensional frequency, respectively, on $\varnothing =10°, 45°,$ and $90°$ are presented with the simply symmetrical sparse matrix of thick-walled FGM spherical shells subjected to free vibration. Four main effects on frequency values are considered. They are the advanced shear coefficient, the nonlinear TSDT term $c_1$ of displacements, the FGM power law index, and the environment temperature. The novel and important numerical results are obtained in the following: the frequency parameters $f^{\ast }$, $\mathsf{\Omega }$ data are presented and subjected to $c_1$ effect; the natural frequency ${\omega }_{mn}$ data vs. $R_n$ and $T$ subjected to mode-shape number $m$, $n$ are computed; the frequency values are studied for thick-walled FGM SUS304/Si₃N₄ spherical shells by using advanced $k_{\alpha }$. This is a new finding, as the assumed simplification in a sparse matrix causes a numerical truncation error; the ${\omega }_{1n}$ values at n = 1 and 2 of the presented sparse matrix are much greater than those in the full matrix for thick-walled FGM spherical shells.