## 2 Formulation Procedures
The dynamic equilibrium differential equations with nonlinear TSDT of thick FGM plates could be obtained by [PERSON] [10] with the following integrals for the stiffness \(\overline{Q}_{i^{\prime}j^{\prime}}\) and \(\overline{Q}_{i^{\prime}j^{\prime}}\) in the \(z\) direction, respectively,  
\[(A_{i^{\prime}j^{\prime}},B_{i^{\prime}j^{\prime}},D_{i^{\prime}j^{\prime}},E _{i^{\prime}j^{\prime}},F_{i^{\prime}j^{\prime}},H_{i^{\prime}j^{\prime}})= \int_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}\overline{Q}_{i^{\prime}j^{\prime}}(1,z,z^{2},z^{3},z^{4},z^{6})dz,\ (i^{*},j^{*}=1,2,6), \tag{3a}\] \[(A_{i^{\prime}j^{\prime}},B_{i^{\prime}j^{\prime}},D_{i^{\prime}j^{\prime}},E _{i^{\prime}j^{\prime}},F_{i^{\prime}j^{\prime}},H_{i^{\prime}j^{\prime}})= \int_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}k_{a}\overline{Q}_{i^{\prime}j^{*}}( 1,z,z^{2},z^{3},z^{4},z^{5})dz,\ (i^{*},j^{*}=4,5), \tag{3b}\]  
where \(k_{a}\) is the advanced nonlinear varied \(k_{a}\) expression including the \(c_{1}c_{1}\) terms for the thick FGM plates in terms of [PERSON]'s modulus \(E_{1}\) and \(E_{2}\) in constituent material 1 and 2, respectively, can be obtained as follows,  
\[k_{a}=\frac{1}{h^{*}}\frac{FGMZSV}{FGMZIV} \tag{4}\]  
where  
\[FGMZSV=\left(FGMZS-c_{1}FGMZSN\right)^{2},\] \[FGMZIV=FGMZI-2c_{1}FGMZIV1+{c_{1}}^{2}FGMZIV2,\]  
\[FGMZS=\frac{E_{2}-E_{1}}{h^{*}R_{a}}[\frac{\left(\frac{h^{*}}{2}+\frac{h^{*} }{2}\right)^{R_{a}+3}}{R_{n}+3}-\frac{h^{*}\Big{(}\frac{h^{*}}{2}+\frac{h^{*} }{2}\Big{)}^{R_{n}+2}}{(R_{n}+2)}+\frac{h^{*2}\Big{(}\frac{h^{*}}{2}+\frac{h^{ *}}{2}\Big{)}^{R_{n}+1}}{4(R_{n}+1)}]+E_{1}\Big{(}\frac{h^{*3}}{24}+\frac{h^{* 3}}{24}\Big{)}\]  
\[FGMZSN=\frac{E_{2}-E_{1}}{h^{*}R_{a}}[\frac{\left(\frac{h^{*}}{2}+\frac{h^{*} }{2}\right)^{R_{n}+5}}{R_{n}+5}-\frac{2h^{*}\left(\frac{h^{*}}{2}+\frac{h^{*} }{2}\right)^{R_{n}+4}}{R_{n}+4}+\frac{3h^{*2}\left(\frac{h^{*}}{2}+\frac{h^{ *}}{2}\right)^{R_{n}+3}}{2(R_{n}+3)}\]  
\[FGMZI=(E_{2}-E_{1})^{2}h^{*5}[\frac{1}{(R_{n}+2)^{2}(2R_{n}+5)}-\frac{1}{(R_{n }+1)(R_{n}+2)(2R_{n}+4)}+\frac{1}{4(R_{n}+1)^{2}(2R_{n}+3)}]\]  
\[\Delta T=\Big{[}\frac{z}{h^{*}}\overline{T}_{1}sin\Big{(}\frac{\pi x}{a}\Big{)} sin\Big{(}\frac{\pi y}{b}\Big{)}\Big{]}sin(\gamma t) \tag{2}\]0  
\[\Delta T=\Big{[}\frac{z}{h^{*}}\overline{T}_{1}sin\Big{(}\frac{\pi x}{a}\Big{)} sin\Big{(}\frac{\pi y}{b}\Big{)}\Big{]}sin(\gamma t) \tag{2}\]1  
\[\Delta T=\Big{[}\frac{z}{h^{*}}\overline{T}_{1}sin\Big{(}\frac{\pi x}{a}\Big{)} sin\Big{(}\frac{\pi y}{b}\Big{)}\Big{]}sin(\gamma t) \tag{2}\]2  
\[\Delta T=\Big{[}\frac{z}{h^{*}}\overline{T}_{1}sin\Big{(}\frac{\pi x}{a}\Big{)} sin\Big{(}\frac{\pi y}{b}\Big{)}\Big{]}sin(\gamma t) \tag{2}\]3\[\Delta T=\Big{[}\frac{z}{h^{*}}\overline{T}_{1}sin\Big{(}\frac{\pi x}{a}\Big{)} sin\Big{(}\frac{\pi y}{b}\Big{)}\Big{]}sin(\gamma t) \tag{2}\]4 \[\Delta T=\Big{[}\frac{z}{h^{*}}\overline{T}_{1}sin\Big{(}\frac{\pi x}{a}\Big{)} sin\Big{(}\frac{\pi y}{b}\Big{)}\Big{]}sin(\gamma t) \tag{2}\]5 \[\Delta T=\Big{[}\frac{z}{h^{*}}\overline{T}_{1}sin\Big{(}\frac{\pi x}{a}\Big{)} sin\Big{(}\frac{\pi y}{b}\Big{)}\Big{]}sin(\gamma t) \tag{2}\]6 \[\Delta T=\Big{[}\frac{z}{h^{*}}\overline{T}_{1}sin\Big{(}\frac{\pi x}{a}\Big{)} sin\Big{(}\frac{\pi y}{b}\Big{)}\Big{]}sin(\gamma t) \tag{2}\]7 \[\Delta T=\Big{[}\frac{z}{h^{*}}\overline{T}_{1}sin\Big{(}\frac{\pi x}{a}\Big{)} sin\Big{(}\frac{\pi y}{b}\Big{)}\Big{]}sin(\gamma t) \tag{2}\]8 \[\Delta T=\Big{[}\frac{z}{h^{*}}\overline{T}_{1}sin\Big{(}\frac{\pi x}{a}\Big{)} sin\Big{(}\frac{\pi y}{b}\Big{)}\Big{]}sin(\gamma t) \tag{2}\]9 \[(A_{i^{\prime}j^{\prime}},B_{i^{\prime}j^{\prime}},D_{i^{\prime}j^{\prime}},E _{i^{\prime}j^{\prime}},F_{i^{\prime}j^{\prime}},H_{i^{\prime}j^{\prime}})= \int_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}\overline{Q}_{i^{\prime}j^{\prime}}(1,z,z^{2},z^{3},z^{4},z^{6})dz,\ (i^{*},j^{*}=1,2,6), \tag{3a}\]0 \[(A_{i^{\prime}j^{\prime}},B_{i^{\prime}j^{\prime}},D_{i^{\prime}j^{\prime}},E _{i^{\prime}j^{\prime}},F_{i^{\prime}j^{\prime}},H_{i^{\prime}j^{\prime}})= \int_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}\overline{Q}_{i^{\prime}j^{\prime}}(1,z,z^{2},z^{3},z^{4},z^{6})dz,\ (i^{*},j^{*}=1,2,6), \tag{3a}\]1 \[(A_{i^{\prime}j^{\prime}},B_{i^{\prime}j^{\prime}},D_{i^{\prime}j^{\prime}},E _{i^{\prime}j^{\prime}},F_{i^{\prime}j^{\prime}},H_{i^{\prime}j^{\prime}})= \int_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}\overline{Q}_{i^{\prime}j^{\prime}}(1,z,z^{2},z^{3},z^{4},z^{6})dz,\ (i^{*},j^{*}=1,2,6), \tag{3a}\]2 \[(A_{i^{\prime}j^{\prime}},B_{i^{\prime}j^{\prime}},D_{i^{\prime}j^{\prime}},E _{i^{\prime}j^{\prime}},F_{i^{\prime}j^{\prime}},H_{i^{\prime}j^{\prime}})= \int_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}\overline{Q}_{i^{\prime}j^{\prime}}(1,z,z^{2},z^{3},z^{4},z^{6})dz,\ (i^{*},j^{*}=1,2,6), \tag{3a}\]3 \[(A_{i^{\prime}j^{\prime}},B_{i^{\prime}j^{\prime}},D_{i^{\prime}j^{\prime}},E _{i^{\prime}j^{\prime}},F_{i^{\prime}j^{\prime}},H_{i^{\prime}j^{\prime}})= \int_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}\overline{Q}_{i^{\prime}j^{\prime}}(1,z,z^{2},z^{3},z^{4},z^{6})dz,\ (i^{*},j^{*}=1,2,6), \tag{3a}\]4 \[(A_{i^{\prime}j^{\prime}},B_{i^{\prime}j^{\prime}},D_{i^{\prime}j^{\prime}},E _{i^{\prime}j^{\prime}},F_{i^{\prime}j^{\prime}},H_{i^{\prime}j^{\prime}})= \int_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}\overline{Q}_{i^{\prime}j^{\prime}}(1,z,z^{2},z^{3},z^{4},z^{6})dz,\ (i^{*},j^{*}=1,2,6), \tag{3a}\]5 \[(A_{i^{\prime}j^{\prime}},B_{i^{\prime}j^{\prime}},D_{i^{\prime}j^{\prime}},E _{i^{\prime}j^{\prime}},F_{i^{\prime}j^{\prime}},H_{i^{\prime}j^{\prime}})= \int_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}\overline{Q}_{i^{\prime}j^{\prime}}(1,z,z^{2},z^{3},z^{4},z^{6})dz,\ (i^{*},j^{*}=1,2,6), \tag{3a}\]6 \[(A_{i^{\prime}j^{\prime}},B_{i^{\prime}j^{\prime}},D_{i^{\prime}j^{\prime}},E _{i^{\prime}j^{\prime}},F_{i^{\prime}j^{\prime}},H_{i^{\prime}j^{\prime}})= \int_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}\overline{Q}_{i^{\prime}j^{\prime}}(1,z,z^{2},z^{3},z^{4},z^{6})dz,\ (i^{*},j^{*}=1,2,6), \tag{3a}\]7 \[(A_{i^{\prime}j^{\prime}},B_{i^{\prime}j^{\prime}},D_{i^{\prime}j^{\prime}},E _{i^{\prime}j^{\prime}},F_{i^{\prime}j^{\prime}},H_{i^{\prime}j^{\prime}})= \int_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}\overline{Q}_{i^{\prime}j^{\prime}}(1,z,z^{2},z^{3},z^{4},z^{6})dz,\ (i^{*},j^{*}=1,2,6), \tag{3a}\]8 \[(A_{i^{\prime}j^{\prime}},B_{i^{\prime}j^{\prime}},D_{i^{\prime}j^{\prime}},E _{i^{\prime}j^{\prime}},F_{i^{\prime}j^{\prime}},H_{i^{\prime}j^{\prime}})= \int_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}\overline{Q}_{i^{\prime}j^{\prime}}(1,z,z^{2},z^{3},z^{4},z^{6})dz,\ (i^{*},j^{*}=1,2,6), \tag{3a}\]9 \[(A_{i^{\prime}j^{\prime}},B_{i^{\prime}j^{\prime}},D_{i^{\prime}j^{\prime}},E _{i^{\prime}j^{\prime}},F_{i^{\prime}j^{\prime}},H_{i^{\prime}j^{\prime}})= \int_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}k_{a}\overline{Q}_{i^{\prime}j^{*}}( 1,z,z^{2},z^{3},z^{4},z^{5})dz,\ (i^{*},j^{*}=4,5), \tag{3b}\]0 \[(A_{i^{\prime}j^{\prime}},B_{i^{\prime}j^{\prime}},D_{i^{\prime}j^{\prime}},E _{i^{\prime}j^{\prime}},F_{i^{\prime}j^{\prime}},H_{i^{\prime}j^{\prime}})= \int_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}k_{a}\overline{Q}_{i^{\prime}j^{*}}( 1,z,z^{2},z^{3},z^{4},z^{5})dz,\ (i^{*},j^{*}=4,5), \tag{3b}\]1 \[(A_{i^{\prime}j^{\prime}},B_{i^{\prime}j^{\prime}},D_{i^{\prime}j^{\prime}},E _{i^{\prime}j^{\prime}},F_{i^{\prime}j^{\prime}},H_{i^{\prime}j^{\prime}})= \int_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}k_{a}\overline{Q}_{i^{\prime}j^{*}}( 1,z,z^{2},z^{3},z^{4},z^{5})dz,\ (i^{*},j^{*}=4,5), \tag{3b}\]2 \[(A_{i^{\prime}j^{\prime}},B_{i^{\prime}j^{\prime}},D_{i^{\prime}j^{\prime}},E _{i^{\prime}j^{\prime}},F_{i^{\prime}j^{\prime}},H_{i^{\prime}j^{\prime}})= \int_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}k_{a}\overline{Q}_{i^{\prime}j^{*}}( 1,z,z^{2},z^{3},z^{4},z^{5})dz,\ (i^{*},j^{*}=4,5), \tag{3b}\]3 \[(A_{i^{\prime}j^{\prime}},B_{i^{\prime}j^{\prime}},D_{i^{\prime}j^{\prime}},E _{i^{\prime}j^{\prime}},F_{i^{\prime}j^{\prime}},H_{i^{\prime}j^{\prime}})= \int_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}k_{a}\overline{Q}_{i^{\prime}j^{*}}( 1,z,z^{2},z^{3},z^{4},z^{5})dz,\ (i^{*},j^{*}=4,5), \tag{3b}\]4 \[(A_{i^{\prime}j^{\prime}},B_{i^{\prime}j^{\prime}},D_{i^{\prime}j^{\prime}},E _{i^{\prime}j^{\prime}},F_{i^{\prime}j^{\prime}},H_{i^{\prime}j^{\prime}})= \int_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}k_{a}\overline{Q}_{i^{\prime}j^{*}}( 1,z,z^{2},z^{3},z^{4},z^{5})dz,\ (i^{*},j^{*}=4,5), \tag{3b}\]5 \[(A_{i^{\prime}j^{\prime}},B_{i^{\prime}j^{\prime}},D_{i^{\prime}j^{\prime}},E _{i^{\prime}j^{\prime}},F_{i^{\prime}j^{\prime}},H_{i^{\prime}j^{\prime}})= \int_{\frac{-h^{*}}{2}}^{\frac{h^{*}}{2}}k_{a}\overline{Q}_{i^{\prime}j^{*}}( 1,z,z^{2},z^{3},z^{4},z^{5})dz,\ (i^{*},j^{*}=4,5), \tag{3b}\]6  
The values of advanced nonlinear \(k_{a}\) are usually functions of \(c_{1}\), \(R_{n}\) and \(T\), but is independent to the value of \(h^{*}\).