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On the vibrations of FG GNPs-RPN annular plates with piezoelectric/metallic coatings on Kerr elastic substrate considering size dependency and surface stress effects

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Abstract

Vibrational behavior of a functionally graded graphene nanoplatelets-reinforced porous nanocomposite (FG GNPs-RPN) annular microplate with piezoelectric/metallic coverings is carried out in the current work in asymmetric condition. The microstructure is sited on Kerr elastic substrate and also is in a thermoelectrical media. The core’s properties are varied over the thickness path based on different assumed forms. To determine the core’s effective properties, Gaussian random field scheme, Halpin–Tsai, and extended rule of mixture models are appointed. Also, Gurtin–Murdoch and modified strain gradient theories are occupied to consider surface stress and small-scale impacts, respectively. The derivation process of the governing equations and associated boundary conditions was conducted via Hamilton’s principle. Generalized differential quadrature scheme is chosen to attain the results. By validating the results’ precision, the effects of different parameters on the frequencies are observed. It is perceived that adding GNPs enhances the frequencies by about 20–28%, and increasing porosity up to seventy percent leads the frequencies to decrease by about 8–15%. The simpler models of the under-evaluation microstructure are used in different productions; thus, the current model can be used to move on the edge of knowledge and future uses in numerous industries based on the obtained results.

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Acknowledgements

The authors would like to thank the reviewers for their valuable comments and suggestions to improve the clarity of this study. Also, they are thankful to the Iranian Nanotechnology Development Committee for their financial support and the University of Kashan for supporting this work.

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  1. Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran

    Ehsan Arshid, Saeed Amir & Abbas Loghman

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  1. Ehsan Arshid
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  2. Saeed Amir
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  3. Abbas Loghman
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Correspondence to Saeed Amir.

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Appendices

Appendix A

The used stress resultants in Eqs. (61)–(79) are defined as:

$$\left\{ {\begin{array}{*{20}c} {N_{rr} ,} & {M_{rr} } \\ \end{array} } \right\} = \int {\sigma_{rr} \left\{ {\begin{array}{*{20}c} {1,} & z \\ \end{array} } \right\}} \,{\text{d}}z + \left\{ {\begin{array}{*{20}c} {\left( {\sigma_{rr}^{t} + \sigma_{rr}^{b} } \right),} & {\frac{h}{2}\left( {\sigma_{rr}^{t} - \sigma_{rr}^{b} } \right)} \\ \end{array} } \right\},$$
$$\left\{ {\begin{array}{*{20}c} {N_{\theta \theta } ,} & {M_{\theta \theta } } \\ \end{array} } \right\} = \int {\sigma_{\theta \theta } \left\{ {\begin{array}{*{20}c} {1,} & z \\ \end{array} } \right\}} \,{\text{d}}z + \left\{ {\begin{array}{*{20}c} {\left( {\sigma_{\theta \theta }^{t} + \sigma_{\theta \theta }^{b} } \right),} & {\frac{h}{2}\left( {\sigma_{\theta \theta }^{t} - \sigma_{\theta \theta }^{b} } \right)} \\ \end{array} } \right\},$$
$$\begin{aligned} & \left\{ {\begin{array}{*{20}c} {N_{r\theta } ,} & {M_{r\theta } } \\ \end{array} } \right\} = \int {\sigma_{r\theta } \left\{ {\begin{array}{*{20}c} {1,} & z \\ \end{array} } \right\}} \,{\text{d}}z \\ & \quad \quad \quad \quad \quad \quad \quad\quad \quad + \left\{ {\begin{array}{*{20}c} {\frac{1}{2}\left( {\sigma_{r\theta }^{t} + \sigma_{r\theta }^{b} + \sigma_{\theta r}^{t} + \sigma_{\theta r}^{b} } \right),} & {\frac{h}{2}\left( {\sigma_{r\theta }^{t} - \sigma_{r\theta }^{b} + \sigma_{\theta r}^{t} - \sigma_{\theta r}^{b} } \right)} \\ \end{array} } \right\}, \\ \end{aligned}$$
$$Q_{r} = \int {\sigma_{rz} } \,{\text{d}}z + \left( {\tau_{s}^{t} + \tau_{s}^{b} } \right)\frac{\partial w}{{\partial r}},\;\;Q_{\theta } = \int {\sigma_{\theta z} } \,{\text{d}}z + \left( {\tau_{s}^{t} + \tau_{s}^{b} } \right)\frac{1}{r}\frac{\partial w}{{\partial \theta }},$$
$$\left\{ {\begin{array}{*{20}l} {Y_{1} ,} \hfill & {Y_{2} ,} \hfill & {Y_{3} ,} \hfill & {Y_{4} ,} \hfill & {Y_{5} } \hfill \\ \end{array} } \right\} = \int {\left\{ {\begin{array}{*{20}l} {m_{rr} ,} \hfill & {m_{\theta \theta } ,} \hfill & {m_{zz} ,} \hfill & {m_{r\theta } ,} \hfill & {m_{\theta z} } \hfill \\ \end{array} } \right\}} \,{\text{d}}z,$$
$$\left\{ {\begin{array}{*{20}c} {Y_{6} ,} & {Y_{8} } \\ \end{array} } \right\} = \int {\left\{ {\begin{array}{*{20}c} {m_{\theta z} ,} & {m_{rz} } \\ \end{array} } \right\}} z\,{\text{d}}z,\;\;Y_{7} = \int {m_{rz} } \,{\text{d}}z,$$
$$\left\{ {\begin{array}{*{20}c} {X_{1} ,} & {X_{3} ,} & {X_{5} } \\ \end{array} } \right\} = \int {\left\{ {\begin{array}{*{20}c} {P_{r} ,} & {P_{\theta } ,} & {P_{z} } \\ \end{array} } \right\}} \,{\text{d}}z,$$
$$\left\{ {\begin{array}{*{20}c} {X_{2} ,} & {X_{4} } \\ \end{array} } \right\} = \int {\left\{ {\begin{array}{*{20}c} {P_{r} ,} & {P_{\theta } } \\ \end{array} } \right\}} z\,{\text{d}}z,\;\;\left\{ {\begin{array}{*{20}c} {T_{1} ,} & {T_{2} } \\ \end{array} } \right\} = \int {\tau_{rrr} \left\{ {\begin{array}{*{20}c} {1,} & z \\ \end{array} } \right\}} \,{\text{d}}z,$$
$$\left\{ {\begin{array}{*{20}c} {T_{3} ,} & {T_{4} } \\ \end{array} } \right\} = \int {\tau_{\theta \theta \theta } \left\{ {\begin{array}{*{20}c} {1,} & z \\ \end{array} } \right\}} \,{\text{d}}z,$$
$$T_{5} = \int {\tau_{zzz} } \,{\text{d}}z,\;\;T_{6} = \int {\tau_{rrz} } \,{\text{d}}z,\;\;T_{6} = \int {\tau_{rrz} } \,{\text{d}}z,$$
$$\left\{ {\begin{array}{*{20}c} {T_{7} ,} & {T_{8} } \\ \end{array} } \right\} = \int {\tau_{r\theta \theta } \left\{ {\begin{array}{*{20}c} {1,} & z \\ \end{array} } \right\}} \,{\text{d}}z,\;\;\left\{ {\begin{array}{*{20}c} {T_{9} ,} & {T_{10} } \\ \end{array} } \right\} = \int {\tau_{zzr} \left\{ {\begin{array}{*{20}c} {1,} & z \\ \end{array} } \right\}} \,{\text{d}}z,$$
$$\left\{ {\begin{array}{*{20}c} {T_{11} ,} & {T_{12} } \\ \end{array} } \right\} = \int {\tau_{rr\theta } \left\{ {\begin{array}{*{20}c} {1,} & z \\ \end{array} } \right\}} \,{\text{d}}z,\;\;\left\{ {\begin{array}{*{20}c} {T_{13} ,} & {T_{14} } \\ \end{array} } \right\} = \int {\tau_{zz\theta } \left\{ {\begin{array}{*{20}c} {1,} & z \\ \end{array} } \right\}} \,{\text{d}}z,$$
$$T_{15} = \int {\tau_{\theta \theta z} } \,{\text{d}}z,\;\;T_{16} = \int {\tau_{r\theta z} } \,{\text{d}}z,$$
$$\overline{D}_{r} = \int_{b} {D_{r} \cos \left( {\frac{{\pi z_{b} }}{{h_{b} }}} \right)} \,{\text{d}}z + \,\int_{t} {D_{r} \cos \left( {\frac{{\pi z_{t} }}{{h_{t} }}} \right)} \,{\text{d}}z,$$
$$\overline{D}_{\theta } = \int_{b} {D_{\theta } \cos \left( {\frac{{\pi z_{b} }}{{h_{b} }}} \right)} \,{\text{d}}z + \,\int_{t} {D_{\theta } \cos \left( {\frac{{\pi z_{t} }}{{h_{t} }}} \right)} \,{\text{d}}z,$$
$$\overline{D}_{z} = \int_{b} {D_{z} \frac{\pi }{{h_{b} }}\sin \left( {\frac{{\pi z_{b} }}{{h_{b} }}} \right)} \,{\text{d}}z + \,\int_{t} {D_{z} \frac{\pi }{{h_{t} }}\sin \left( {\frac{{\pi z_{t} }}{{h_{t} }}} \right)} \,{\text{d}}z,$$
$$\begin{aligned} & I_{0} = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {\rho (z)^{b} } \,{\text{d}}z + \int\limits_{{ - h_{c} /2}}^{{ + h_{c} /2}} {\rho (z)^{c} } \,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {\rho (z)^{t} } \,{\text{d}}z \\ & \quad\quad \quad + \left\{ {\begin{array}{*{20}c} {\rho_{st}^{b} + \rho_{sb}^{b} ,} & {\rho_{st}^{c} + \rho_{sb}^{c} ,} & {\rho_{st}^{t} + \rho_{sb}^{t} } \\ \end{array} } \right\}, \\ \end{aligned}$$
$$\begin{aligned} & I_{1} = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {\rho (z)^{b} \,z} \,{\text{d}}z + \int\limits_{{ - h_{c} /2}}^{{ + h_{c} /2}} {\rho (z)^{c} \,z} \,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {\rho (z)^{t} } \,z\,{\text{d}}z \\ & \quad\quad \quad + \left\{ {\begin{array}{*{20}c} { - \left( {\frac{{h_{c} }}{2}} \right)\rho_{st}^{b} - \left( {\frac{{h_{c} }}{2} + h_{b} } \right)\rho_{sb}^{b} ,} & {\left( {\frac{{h_{c} }}{2}} \right)\rho_{st}^{c} - \left( {\frac{{h_{c} }}{2}} \right)\rho_{sb}^{c} ,} & {\left( {\frac{{h_{c} }}{2} + h_{t} } \right)\rho_{st}^{t} + \left( {\frac{{h_{c} }}{2}} \right)\rho_{sb}^{t} } \\ \end{array} } \right\}, \\ \end{aligned}$$
$$\begin{aligned} & I_{2} = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {\rho (z)^{b} \,z^{2} } \,{\text{d}}z + \int\limits_{{ - h_{c} /2}}^{{ + h_{c} /2}} {\rho (z)^{c} \,z^{2} } \,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {\rho (z)^{t} } \,z^{2} \,{\text{d}}z \\ & \quad\quad \quad + \left\{ {\begin{array}{*{20}c} {\left( {\frac{{h_{c} }}{2}} \right)^{2} \rho_{st}^{b} + \left( {\frac{{h_{c} }}{2} + h_{b} } \right)^{2} \rho_{sb}^{b} ,} & {\left( {\frac{{h_{c} }}{2}} \right)^{2} \rho_{st}^{c} + \left( {\frac{{h_{c} }}{2}} \right)^{2} \rho_{sb}^{c} ,} & {\left( {\frac{{h_{c} }}{2} + h_{t} } \right)^{2} \rho_{st}^{t} + \left( {\frac{{h_{c} }}{2}} \right)^{2} \rho_{sb}^{t} } \\ \end{array} } \right\} \\ \end{aligned}$$

in which

$$z_{b} = z + \frac{{h_{c} }}{2} + \frac{{h_{b} }}{2},\;\;z_{t} = z - \frac{{h_{c} }}{2} - \frac{{h_{t} }}{2}$$

Appendix B

The used parameters in Eqs. (82)–(99) are defined as:

$$F_{1} = 2{\mkern 1mu} {\mkern 1mu} l_{0}^{2} + \frac{4}{5}{\mkern 1mu} {\mkern 1mu} l_{1}^{2} ,\;\;F_{2} = 6{\mkern 1mu} l_{0}^{2} - \frac{{4{\mkern 1mu} {\mkern 1mu} l_{1}^{2} }}{{15{\mkern 1mu} }},$$
$$F_{3} = \frac{{8{\mkern 1mu} l_{1}^{2} }}{{15{\mkern 1mu} }} + \frac{{{\mkern 1mu} l_{2}^{2} }}{4},\;\;F_{4} = - 6{\mkern 1mu} l_{0}^{2} + \frac{4}{5}{\mkern 1mu} l_{1}^{2} + \frac{1}{4}{\mkern 1mu} l_{2}^{2} ,$$
$$F_{5} = 2{\mkern 1mu} l_{0}^{2} + \frac{4}{3}{\mkern 1mu} l_{1}^{2} + \frac{1}{4}{\mkern 1mu} l_{2}^{2} ,\;\;F_{6} = 6{\mkern 1mu} l_{0}^{2} + \frac{2}{3}l_{1}^{2} + \frac{1}{4}{\mkern 1mu} l_{2}^{2} ,$$
$$F_{7} = 2{\mkern 1mu} l_{0}^{2} + \frac{{4{\mkern 1mu} l_{1}^{2} }}{{15{\mkern 1mu} }} - \frac{1}{4}{\mkern 1mu} l_{2}^{2} ,\;\;F_{8} = 4{\mkern 1mu} l_{0}^{2} + \frac{2}{5}{\mkern 1mu} l_{1}^{2} + \frac{1}{2}{\mkern 1mu} l_{2}^{2} ,$$
$$F_{9} = 2{\mkern 1mu} l_{0}^{2} + \frac{2}{15}{\mkern 1mu} l_{1}^{2} + \frac{3}{4}{\mkern 1mu} l_{2}^{2} ,\;\;F_{10} = \frac{{8{\mkern 1mu} l_{1}^{2} }}{{15{\mkern 1mu} }} - \frac{3}{4}{\mkern 1mu} l_{2}^{2} ,$$
$$F_{11} = 2{\mkern 1mu} l_{0}^{2} + \frac{4}{3}{\mkern 1mu} l_{1}^{2} - \frac{3}{4}{\mkern 1mu} l_{2}^{2} ,\;\;F_{12} = \frac{{11{\mkern 1mu} {\mkern 1mu} l_{1}^{2} }}{{15{\mkern 1mu} }} + l_{2}^{2} ,$$
$$F_{13} = \frac{{16{\mkern 1mu} l_{1}^{2} }}{15} - \frac{1}{4}{\mkern 1mu} l_{2}^{2} ,\;\;F_{14} = 2{\mkern 1mu} l_{1}^{2} - \frac{1}{2}{\mkern 1mu} l_{2}^{2} ,$$
$$F_{15} = \frac{2}{15}{\mkern 1mu} l_{1}^{2} + \frac{1}{4}{\mkern 1mu} l_{2}^{2} ,\;\;F_{16} = \frac{1}{3}{\mkern 1mu} l_{1}^{2} - \frac{1}{4}{\mkern 1mu} l_{2}^{2} ,$$
$$F_{17} = \frac{1}{4}{\mkern 1mu} l_{2}^{2} ,\;\;F_{18} = \frac{6}{5}l_{1}^{2} - \frac{1}{4}l_{2}^{2} ,$$
$$F_{19} = \frac{{14{\mkern 1mu} l_{1}^{2} }}{{15{\mkern 1mu} }} - \frac{1}{2}l_{2}^{2} ,\;\;F_{20} = 2{\mkern 1mu} l_{0}^{2} + \frac{{32{\mkern 1mu} {\mkern 1mu} l_{1}^{2} }}{15} + \frac{1}{4}{\mkern 1mu} l_{2}^{2} ,$$
$$F_{21} = 2{\mkern 1mu} l_{0}^{2} + \frac{{8{\mkern 1mu} {\mkern 1mu} l_{1}^{2} }}{{15{\mkern 1mu} }} + \frac{1}{4}l_{2}^{2} ,\;\;F_{22} = \frac{4}{3}l_{1}^{2} + l_{2}^{2} ,$$
$$F_{23} = 2{\mkern 1mu} l_{0}^{2} + \frac{{26{\mkern 1mu} l_{1}^{2} }}{{15{\mkern 1mu} }} + \frac{5}{4}{\mkern 1mu} l_{2}^{2} ,\;\;F_{24} = 2{\mkern 1mu} l_{0}^{2} + \frac{4}{5}{\mkern 1mu} l_{1}^{2} - \frac{3}{4}l_{2}^{2} ,$$
$$F_{25} = \frac{1}{3}{\mkern 1mu} l_{1}^{2} ,\;\;F_{26} = \frac{1}{3}{\mkern 1mu} l_{1}^{2} + l_{2}^{2} ,$$
$$F_{27} = \frac{2}{5}l_{1}^{2} ,\;\;F_{28} = \frac{8}{15}l_{1}^{2} ,$$
$$F_{29} = \frac{16}{{15}}l_{1}^{2} + \frac{1}{4}l_{2}^{2} ,\;\;F_{30} = l_{0}^{2} - \frac{2}{5}l_{1}^{2} ,$$
$$F_{31} = 3l_{0}^{2} - \frac{4}{15}l_{1}^{2} ,\;\;F_{32} = 2l_{0}^{2} + \frac{6}{5}l_{1}^{2} + \frac{1}{2}l_{2}^{2} ,$$
$$F_{33} = 2l_{0}^{2} - \frac{4}{15}l_{1}^{2} - \frac{1}{8}l_{2}^{2} ,\;\;F_{34} = \frac{2}{5}l_{1}^{2} - \frac{1}{4}l_{2}^{2} ,$$
$$F_{35} = \frac{2}{5}l_{1}^{2} + \frac{1}{2}l_{2}^{2} ,\;\;F_{36} = \frac{4}{15}l_{1}^{2} - \frac{1}{4}l_{2}^{2} ,$$
$$F_{37} = 4l_{0}^{2} + \frac{4}{5}l_{1}^{2} ,\;\;F_{38} = 2l_{0}^{2} - \frac{2}{5}l_{1}^{2} ,$$
$$F_{39} = \frac{8}{15}l_{1}^{2} - \frac{1}{8}l_{2}^{2} ,\;\;F_{40} = \frac{2}{5}l_{1}^{2} - \frac{3}{8}l_{2}^{2} ,$$
$$F_{41} = 4l_{0}^{2} + \frac{22}{{15}}l_{1}^{2} + \frac{1}{4}l_{2}^{2} ,\;\;F_{42} = l_{0}^{2} + \frac{2}{3}l_{1}^{2} - \frac{1}{4}l_{2}^{2} ,$$
$$F_{43} = \frac{4}{3}l_{1}^{2} - \frac{1}{2}l_{2}^{2} ,\;\;F_{44} = \frac{2}{3}l_{1}^{2} + l_{2}^{2} ,$$
$$F_{45} = 2l_{0}^{2} - \frac{8}{15}l_{1}^{2} - \frac{1}{4}l_{2}^{2} ,\;\;F_{46} = 2l_{0}^{2} + \frac{16}{{15}}l_{1}^{2} + \frac{1}{4}l_{2}^{2} ,$$
$$F_{47} = 4l_{0}^{2} + \frac{2}{15}l_{1}^{2} ,\;\;F_{48} = l_{0}^{2} - \frac{16}{{15}}l_{1}^{2} - \frac{1}{8}l_{2}^{2} ,$$
$$F_{49} = l_{0}^{2} + \frac{2}{3}l_{1}^{2} + \frac{1}{8}l_{2}^{2} ,\;\;F_{50} = 4l_{0}^{2} + \frac{4}{5}l_{1}^{2} + \frac{1}{8}l_{2}^{2} ,$$
$$F_{51} = \frac{14}{{15}}l_{1}^{2} - \frac{1}{4}l_{2}^{2} ,\;\;F_{52} = \frac{8}{15}l_{1}^{2} - \frac{1}{4}l_{2}^{2} ,$$
$$F_{53} = \frac{2}{15}l_{1}^{2} - \frac{1}{4}l_{2}^{2} ,\;\;F_{54} = 2l_{0}^{2} - \frac{4}{15}l_{1}^{2} - \frac{1}{4}l_{2}^{2} ,$$
$$F_{55} = l_{0}^{2} + \frac{8}{15}l_{1}^{2} + \frac{1}{2}l_{2}^{2} ,\;\;F_{56} = l_{0}^{2} + \frac{2}{3}l_{1}^{2} + \frac{1}{2}l_{2}^{2} ,$$
$$F_{57} = l_{0}^{2} + \frac{16}{{15}}l_{1}^{2} - \frac{1}{8}l_{2}^{2}$$

And also:

$$\begin{aligned} \left\{ {\begin{array}{*{20}c} {A_{1} ,} & {A_{2} ,} & {A_{5} } \\ \end{array} } \right\} & = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {Q_{11}^{b} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}} \,{\text{d}}z \\ & \quad\quad + \int\limits_{{ - h_{c} /2}}^{{ + h_{c} /2}} {Q_{11}^{c} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}} \,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {Q_{11}^{t} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}} \,{\text{d}}z, \\ \end{aligned}$$
$$\begin{aligned} \left\{ {\begin{array}{*{20}c} {A_{3} ,} & {A_{4} ,} & {A_{6} } \\ \end{array} } \right\} & = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {Q_{12}^{b} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}} \,{\text{d}}z \\ & \quad\quad + \int\limits_{{ - h_{c} /2}}^{{ + h_{c} /2}} {Q_{12}^{c} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}} \,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {Q_{12}^{t} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}} \,{\text{d}}z, \\ \end{aligned}$$
$$\begin{aligned} \left\{ {\begin{array}{*{20}c} {A_{7} ,} & {A_{8} ,} & {A_{11} } \\ \end{array} } \right\} & = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {Q_{21}^{b} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}} \,{\text{d}}z \\ & \quad\quad + \int\limits_{{ - h_{c} /2}}^{{ + h_{c} /2}} {Q_{21}^{c} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}} \,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {Q_{21}^{t} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}} \,{\text{d}}z, \\ \end{aligned}$$
$$\begin{aligned} \left\{ {\begin{array}{*{20}c} {A_{9} ,} & {A_{10} ,} & {A_{12} } \\ \end{array} } \right\} & = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {Q_{22}^{b} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}} \,{\text{d}}z \\ & \quad\quad + \int\limits_{{ - h_{c} /2}}^{{ + h_{c} /2}} {Q_{22}^{c} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}} \,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {Q_{22}^{t} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}} \,{\text{d}}z, \\ \end{aligned}$$
$$\begin{aligned} \left\{ {\begin{array}{*{20}c} {A_{13} ,} & {A_{14} ,} & {A_{15} } \\ \end{array} } \right\} & = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {Q_{66}^{b} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}} \,{\text{d}}z \\ & \quad\quad + \int\limits_{{ - h_{c} /2}}^{{ + h_{c} /2}} {Q_{66}^{c} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}} \,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {Q_{66}^{t} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}} \,{\text{d}}z, \\ \end{aligned}$$
$$A_{16} = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {Q_{55}^{b} } \,{\text{d}}z + \int\limits_{{ - h_{c} /2}}^{{ + h_{c} /2}} {Q_{55}^{c} } \,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {Q_{55}^{t} } \,{\text{d}}z,$$
$$A_{17} = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {Q_{44}^{b} } \,{\text{d}}z + \int\limits_{{ - h_{c} /2}}^{{ + h_{c} /2}} {Q_{44}^{c} } \,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {Q_{44}^{t} } \,{\text{d}}z,$$
$$\begin{aligned} \left\{ {\begin{array}{*{20}c} {B_{1} ,} & {B_{2} ,} & {B_{3} } \\ \end{array} } \right\} & = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {\mu (z)^{b} } \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}\,{\text{d}}z \\ & \quad\quad + \int\limits_{{ - h_{c} /2}}^{{ + h_{c} /2}} {\mu (z)^{c} } \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}\,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {\mu (z)^{t} } \,\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned}$$
$$P_{1} = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {\frac{\pi }{{h_{b} }}e_{31} \sin \left( {\frac{{\pi z_{b} }}{{h_{b} }}} \right)} \,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {\frac{\pi }{{h_{t} }}e_{31} \sin \left( {\frac{{\pi z_{t} }}{{h_{t} }}} \right)} \,{\text{d}}z,$$
$$P_{2} = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {\frac{{\pi z_{b} }}{{h_{b} }}e_{31} \sin \left( {\frac{{\pi z_{b} }}{{h_{b} }}} \right)} \,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {\frac{{\pi z_{t} }}{{h_{t} }}e_{31} \sin \left( {\frac{{\pi z_{t} }}{{h_{t} }}} \right)} \,{\text{d}}z,$$
$$P_{3} = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {\frac{\pi }{{h_{b} }}e_{32} \sin \left( {\frac{{\pi z_{b} }}{{h_{b} }}} \right)} \,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {\frac{\pi }{{h_{t} }}e_{32} \sin \left( {\frac{{\pi z_{t} }}{{h_{t} }}} \right)} \,{\text{d}}z,$$
$$P_{4} = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {\frac{{\pi z_{b} }}{{h_{b} }}e_{32} \sin \left( {\frac{{\pi z_{b} }}{{h_{b} }}} \right)} \,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {\frac{{\pi z_{t} }}{{h_{t} }}e_{32} \sin \left( {\frac{{\pi z_{t} }}{{h_{t} }}} \right)} \,{\text{d}}z,$$
$$P_{5} = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {e_{15} \cos \left( {\frac{{\pi z_{b} }}{{h_{b} }}} \right)} \,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {e_{15} \cos \left( {\frac{{\pi z_{t} }}{{h_{t} }}} \right)} \,{\text{d}}z,$$
$$P_{6} = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {e_{24} \cos \left( {\frac{{\pi z_{b} }}{{h_{b} }}} \right)} \,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {e_{24} \cos \left( {\frac{{\pi z_{t} }}{{h_{t} }}} \right)} \,{\text{d}}z,$$
$$P_{7} = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {s_{11} \cos^{2} \left( {\frac{{\pi z_{b} }}{{h_{b} }}} \right)} \,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {s_{11} \cos^{2} \left( {\frac{{\pi z_{t} }}{{h_{t} }}} \right)} \,{\text{d}}z,$$
$$P_{8} = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {s_{22} \cos^{2} \left( {\frac{{\pi z_{b} }}{{h_{b} }}} \right)} \,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {s_{22} \cos^{2} \left( {\frac{{\pi z_{t} }}{{h_{t} }}} \right)} \,{\text{d}}z,$$
$$P_{9} = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {\left( {\frac{{\pi^{2} }}{{h_{b}^{2} }}} \right)s_{33} \sin^{2} \left( {\frac{{\pi z_{b} }}{{h_{b} }}} \right)} \,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {\left( {\frac{{\pi^{2} }}{{h_{t}^{2} }}} \right)s_{33} \sin^{2} \left( {\frac{{\pi z_{t} }}{{h_{t} }}} \right)} \,{\text{d}}z,$$
$$P_{10} = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {\left( {\frac{\pi }{{h_{b}^{2} }}} \right)s_{33} \sin \left( {\frac{{\pi z_{b} }}{{h_{b} }}} \right)} \,{\text{d}}z + \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {\left( {\frac{\pi }{{h_{t}^{2} }}} \right)s_{33} \sin \left( {\frac{{\pi z_{t} }}{{h_{t} }}} \right)} \,{\text{d}}z$$

And for other parameter definition:

$$H_{1} = Q_{11t}^{c} + Q_{11b}^{c} + C_{11t}^{t} + C_{11b}^{t} + C_{11t}^{b} + C_{11b}^{b} ,$$
$$H_{2} = Q_{66t}^{c} + Q_{66b}^{c} + {\mkern 1mu} C_{66t}^{t} + {\mkern 1mu} C_{66b}^{t} + {\mkern 1mu} C_{66t}^{b} + {\mkern 1mu} C_{66b}^{b} ,$$
$$H_{3} = Q_{12t}^{c} + Q_{12b}^{c} + C_{12t}^{t} + C_{12b}^{t} + C_{12t}^{b} + C_{12b}^{b} ,$$
$$H_{4} = \frac{{h_{c} }}{2}\left( {Q_{11b}^{c} - Q_{11t}^{c} } \right){\mkern 1mu} - \frac{{h_{c} }}{2}\left( {C_{11t}^{t} + C_{11b}^{t} } \right) - h_{t} C_{11t}^{t} + \frac{{h_{c} }}{2}\left( {C_{11t}^{b} + C_{11b}^{b} } \right) + h_{b} C_{11b}^{b} ,$$
$$H_{5} = \frac{{h_{c} }}{4}\left( {Q_{66b}^{c} - Q_{66t}^{c} } \right) - \frac{{h_{c} }}{4}\left( {C_{66t}^{t} + C_{66b}^{t} } \right) - \frac{{h_{t} }}{2}C_{66t}^{t} + {\mkern 1mu} \frac{{h_{c} }}{4}\left( {C_{66t}^{b} + C_{66b}^{b} } \right) + \frac{{h_{b} }}{2}C_{66b}^{b} ,$$
$$H_{6} = \frac{{h_{c} }}{2}\left( {Q_{12b}^{c} - Q_{12t}^{c} } \right){\mkern 1mu} - \frac{{h_{c} }}{2}\left( {C_{12t}^{t} + C_{12b}^{t} } \right) - h_{t} C_{12t}^{t} + \frac{{h_{c} }}{2}\left( {C_{12t}^{b} + C_{12b}^{b} } \right) + h_{b} C_{12b}^{b} ,$$
$$H_{7} = \frac{{h_{c}^{2} }}{4}Q_{11t}^{c} + \frac{{h_{c}^{2} }}{4}Q_{11b}^{c} + \left( {\frac{{h_{c} }}{2} + h_{t} {\mkern 1mu} } \right)^{2} C_{11t}^{t} + {\mkern 1mu} \frac{{h_{c}^{2} {\mkern 1mu} }}{4}C_{11b}^{t} + \left( {\frac{{h_{c} }}{2} + h_{b} } \right)^{2} C_{11b}^{b} + {\mkern 1mu} \frac{{h_{c}^{2} {\mkern 1mu} }}{4}C_{11t}^{b} ,$$
$$H_{8} = \frac{{h_{c}^{2} }}{4}Q_{66t}^{c} + \frac{{h_{c}^{2} }}{4}Q_{66b}^{c} + \left( {\frac{{h_{c} }}{2} + h_{t} {\mkern 1mu} } \right)^{2} \frac{{C_{66t}^{t} }}{2} + \frac{{h_{c}^{2} {\mkern 1mu} }}{8}C_{66b}^{t} + \left( {\frac{{h_{c} }}{2} + h_{b} } \right)^{2} \frac{{C_{66b}^{b} }}{2} + {\mkern 1mu} \frac{{h_{c}^{2} {\mkern 1mu} }}{8}C_{66t}^{b} ,$$
$$H_{9} = \frac{{h_{c}^{2} }}{4}Q_{12t}^{c} + \frac{{h_{c}^{2} }}{4}Q_{12b}^{c} + \left( {\frac{{h_{c} }}{2} + h_{t} {\mkern 1mu} } \right)^{2} C_{12t}^{t} + {\mkern 1mu} \frac{{h_{c}^{2} {\mkern 1mu} }}{4}C_{12b}^{t} + \left( {\frac{{h_{c} }}{2} + h_{b} } \right)^{2} C_{12b}^{b} + {\mkern 1mu} \frac{{h_{c}^{2} {\mkern 1mu} }}{4}C_{12t}^{b} ,$$
$$H_{10} = \left( {\frac{{h_{c} }}{2} + h_{b} } \right)\frac{{C_{12b}^{b} }}{2},\;\;H_{11} = \left( {\frac{{h_{c} }}{2} + h_{b} } \right)^{2} \frac{{C_{12b}^{b} }}{2},$$
$$R_{1} = \tau_{st}^{c} + \tau_{sb}^{c} + \tau_{st}^{t} + \tau_{sb}^{t} + \tau_{st}^{b} + \tau_{sb}^{b} ,$$
$$R_{2} = \mu_{st}^{c} + \mu_{sb}^{c} + \mu_{st}^{t} + {\mkern 1mu} \mu_{sb}^{t} + {\mkern 1mu} \mu_{st}^{b} + \mu_{sb}^{b} ,$$
$$\begin{aligned} R_{3} & = S_{1}^{c} \left( {\tau_{sb}^{c} - {\mkern 1mu} \tau_{st}^{c} } \right) + S_{3}^{c} \left( {\tau_{sb}^{c} + {\mkern 1mu} \tau_{st}^{c} } \right) + S_{1}^{t} \left( {{\mkern 1mu} \tau_{sb}^{t} - {\mkern 1mu} \tau_{st}^{t} } \right) \\ & \quad\quad + S_{3}^{t} {\mkern 1mu} \left( {\tau_{sb}^{t} + {\mkern 1mu} \tau_{st}^{t} } \right) + S_{1}^{b} \left( {{\mkern 1mu} \tau_{sb}^{b} - \tau_{st}^{b} } \right) + S_{3}^{b} \left( {{\mkern 1mu} \tau_{sb}^{b} + {\mkern 1mu} \tau_{st}^{b} } \right), \\ \end{aligned}$$
$$R_{4} = \frac{{h_{c} }}{2}\left( {\tau_{st}^{c} - \tau_{sb}^{c} } \right) + \frac{{h_{c} }}{2}\left( {\tau_{st}^{t} + \tau_{sb}^{t} {\mkern 1mu} } \right) + h_{t} {\mkern 1mu} \tau_{st}^{t} - \frac{{h_{c} }}{2}\left( {{\mkern 1mu} \tau_{sb}^{b} + \tau_{st}^{b} } \right) - h_{{b{\mkern 1mu} }} {\mkern 1mu} \tau_{sb}^{b} ,$$
$$R_{5} = \frac{{h_{c} }}{4}\left( {\mu_{st}^{c} - \mu_{sb}^{c} } \right) + \frac{{h_{c} }}{4}\left( {{\mkern 1mu} \mu_{st}^{t} + \mu_{sb}^{t} } \right) + \frac{{h_{t} }}{2}\mu_{st}^{t} - \frac{{h_{c} }}{4}\left( {{\mkern 1mu} \mu_{sb}^{b} + \mu_{st}^{b} } \right) - \frac{{h_{b} }}{2}\mu_{sb}^{b} ,$$
$$\begin{aligned} R_{6} & = S_{1}^{c} {\mkern 1mu} \left( {{\mkern 1mu} \rho_{st}^{c} - \rho_{sb}^{c} } \right) - S_{3}^{c} {\mkern 1mu} \left( {\rho_{sb}^{c} + \rho_{st}^{c} } \right) + S_{1}^{t} {\mkern 1mu} \left( {\rho_{st}^{t} - {\mkern 1mu} \rho_{sb}^{t} } \right) \\ & \quad\quad - S_{3}^{t} \left( {\rho_{st}^{t} {\mkern 1mu} + {\mkern 1mu} \rho_{sb}^{t} } \right){\mkern 1mu} + S_{1}^{b} {\mkern 1mu} \left( {\rho_{st}^{b} {\mkern 1mu} - {\mkern 1mu} \rho_{sb}^{b} } \right){\mkern 1mu} - S_{3}^{b} \left( {{\mkern 1mu} \rho_{st}^{b} + {\mkern 1mu} \rho_{sb}^{b} } \right), \\ \end{aligned}$$
$$\begin{aligned} R_{7} & = S_{2}^{c} {\mkern 1mu} \left( {\tau_{sb}^{c} - \tau_{st}^{c} } \right) + S_{4}^{c} {\mkern 1mu} \left( {\tau_{sb}^{c} + {\mkern 1mu} \tau_{st}^{c} } \right) + S_{2}^{t} {\mkern 1mu} \left( {\tau_{sb}^{t} - {\mkern 1mu} \tau_{st}^{t} } \right) \\ & \quad\quad + S_{4}^{t} \left( {{\mkern 1mu} \tau_{sb}^{t} + {\mkern 1mu} \tau_{st}^{t} } \right) + S_{2}^{b} {\mkern 1mu} \left( {\tau_{sb}^{b} - {\mkern 1mu} \tau_{st}^{b} } \right) + S_{4}^{b} {\mkern 1mu} \left( {\tau_{st}^{b} + {\mkern 1mu} \tau_{sb}^{b} } \right), \\ \end{aligned}$$
$$R_{8} = \frac{{h_{c}^{2} }}{4}\tau_{st}^{c} + \frac{{h_{c}^{2} }}{4}\tau_{sb}^{c} + \left( {\frac{{h_{c} }}{2} + h_{t} {\mkern 1mu} } \right)^{2} \tau_{st}^{t} + {\mkern 1mu} \frac{{h_{c}^{2} }}{4}\tau_{sb}^{t} + \left( {\frac{{h_{c} }}{2} + h_{b} } \right)^{2} \tau_{sb}^{b} + {\mkern 1mu} \frac{{h_{c}^{2} }}{4}\tau_{st}^{b} ,$$
$$R_{9} = \frac{{h_{c}^{2} }}{4}\mu_{st}^{c} + \frac{{h_{c}^{2} }}{4}\mu_{sb}^{c} + \left( {\frac{{h_{c} }}{2} + h_{t} {\mkern 1mu} } \right)^{2} \frac{{\mu_{st}^{t} }}{2} + {\mkern 1mu} \frac{{h_{c}^{2} }}{8}\mu_{sb}^{t} + \left( {\frac{{h_{c} }}{2} + h_{b} } \right)^{2} \frac{{\mu_{sb}^{b} }}{2} + {\mkern 1mu} \frac{{h_{c}^{2} }}{8}\mu_{st}^{b} ,$$
$$\begin{aligned} R_{10} & = S_{2}^{c} {\mkern 1mu} \left( {\rho_{st}^{c} - \rho_{sb}^{c} } \right) - S_{4}^{c} \left( {\rho_{sb}^{c} + {\mkern 1mu} \rho_{st}^{c} } \right) + S_{2}^{t} {\mkern 1mu} \left( {\rho_{st}^{t} {\mkern 1mu} - {\mkern 1mu} \rho_{sb}^{t} } \right) \\ & \quad\quad - S_{4}^{t} {\mkern 1mu} \left( {\rho_{st}^{t} + {\mkern 1mu} \rho_{sb}^{t} } \right) + S_{2}^{b} {\mkern 1mu} \left( {\rho_{st}^{b} - \rho_{sb}^{b} } \right){\mkern 1mu} - S_{4}^{b} {\mkern 1mu} \left( {\rho_{sb}^{b} + {\mkern 1mu} \rho_{st}^{b} } \right), \\ \end{aligned}$$
$$E_{1} = \frac{{{\mkern 1mu} \pi {\mkern 1mu} }}{{h_{t} }}\left( {e_{31b}^{t} - e_{31t}^{t} } \right){\mkern 1mu} + \frac{\pi }{{h_{b} }}\left( {e_{31b}^{b} - e_{31t}^{b} } \right),$$
$$E_{2} = \frac{{{\mkern 1mu} \pi {\mkern 1mu} }}{{h_{t} }}\left( {e_{32t}^{t} - e_{32b}^{t} } \right){\mkern 1mu} {\mkern 1mu} + \frac{\pi }{{h_{b} }}\left( {e_{32t}^{b} - e_{32b}^{b} } \right),$$
$$E_{3} = \frac{{h_{c} }}{2}\left( {\frac{\pi }{{h_{t} }}\left( {{\mkern 1mu} e_{31b}^{t} - {\mkern 1mu} {\mkern 1mu} e_{31t}^{t} {\mkern 1mu} {\mkern 1mu} + {\mkern 1mu} e_{32t}^{t} - e_{32b}^{t} } \right) + \frac{{\pi {\mkern 1mu} }}{{h_{b} }}\left( {e_{31t}^{b} - {\mkern 1mu} e_{31b}^{b} - {\mkern 1mu} e_{32t}^{b} + {\mkern 1mu} e_{32b}^{b} } \right)} \right),$$
$$E_{4} = {\mkern 1mu} \pi {\mkern 1mu} \left( {e_{32t}^{t} + e_{32b}^{b} - e_{31t}^{t} - e_{31b}^{b} } \right),$$
$$E_{5} = \frac{{h_{c} }}{2}\left( {\frac{\pi }{{h_{t} }}\left( {e_{31b}^{t} - {\mkern 1mu} e_{31t}^{t} {\mkern 1mu} } \right){\mkern 1mu} + \frac{\pi }{{h_{b} }}\left( {{\mkern 1mu} e_{31t}^{b} - e_{31b}^{b} } \right)} \right),$$
$$E_{6} = \pi \left( {e_{31t}^{t} {\mkern 1mu} + e_{31b}^{b} {\mkern 1mu} } \right),\;\;E_{7} = \pi \left( {e_{32t}^{t} + e_{32b}^{b} } \right),$$
$$E_{8} = \frac{{h_{c} }}{2}\left( {\frac{\pi }{{h_{t} }}\left( {e_{32b}^{t} {\mkern 1mu} {\mkern 1mu} - e_{32t}^{t} } \right) + \frac{\pi }{{h_{b} }}\left( {e_{32t}^{b} - {\mkern 1mu} {\mkern 1mu} e_{32b}^{b} {\mkern 1mu} {\mkern 1mu} } \right)} \right),$$
$$E_{9} = \frac{{\pi^{2} }}{{h_{t}^{2} }}\left( {S_{33t}^{t} + S_{33b}^{t} } \right) + \frac{{\pi^{2} }}{{h_{b}^{2} }}\left( {S_{33t}^{b} + S_{33b}^{b} } \right)$$

in which

$$Q_{11t}^{c} = \frac{{E_{tc} }}{{1 - \nu_{tc}^{2} }},\;\;Q_{11b}^{c} = \frac{{E_{bc} }}{{1 - \nu_{bc}^{2} }},$$
$$Q_{12t}^{c} = \lambda_{st}^{c} + \tau_{st}^{c} ,\;\;Q_{12b}^{c} = \lambda_{sb}^{c} + \tau_{sb}^{c} ,$$
$$Q_{66t}^{c} = \mu_{st}^{c} - \tau_{st}^{c} ,\;\;Q_{66b}^{c} = \mu_{sb}^{c} - \tau_{sb}^{c}$$

That:

$$\lambda_{t,b}^{c} = \frac{{\nu_{t,b}^{c} E_{t,b}^{c} }}{{1 - \left( {\nu_{t,b}^{c} } \right)^{2} }},\;\;\mu_{t,b}^{c} = \frac{{E_{t,b}^{c} }}{{2(1 + \nu_{t,b}^{c} )}},$$

And:

$$\left\{ {\begin{array}{*{20}c} {S_{1}^{c} ,} & {S_{2}^{c} } \\ \end{array} } \right\} = \int\limits_{{ - h_{c} /2}}^{{ + h_{c} /2}} {\frac{{\nu_{c} }}{{2(1 - \nu_{c} )}}\left\{ {\begin{array}{*{20}c} {1,} & z \\ \end{array} } \right\}} \,{\text{d}}z,$$
$$\left\{ {\begin{array}{*{20}c} {S_{3}^{c} ,} & {S_{4}^{c} } \\ \end{array} } \right\} = \int\limits_{{ - h_{c} /2}}^{{ + h_{c} /2}} {\frac{{\nu_{c} }}{{(1 - \nu_{c} )}}g_{1} (z)\left\{ {\begin{array}{*{20}c} {1,} & z \\ \end{array} } \right\}} \,{\text{d}}z,$$
$$\left\{ {\begin{array}{*{20}c} {S_{1}^{t} ,} & {S_{2}^{t} } \\ \end{array} } \right\} = \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {\frac{{\nu_{t} }}{{2(1 - \nu_{t} )}}\left\{ {\begin{array}{*{20}c} {1,} & z \\ \end{array} } \right\}} \,{\text{d}}z,\;\;\left\{ {\begin{array}{*{20}c} {S_{3}^{t} ,} & {S_{4}^{t} } \\ \end{array} } \right\} = \int\limits_{{ + h_{c} /2}}^{{ + h_{c} /2 + h_{t} }} {\frac{{\nu_{t} }}{{(1 - \nu_{t} )}}g_{2} (z)\left\{ {\begin{array}{*{20}c} {1,} & z \\ \end{array} } \right\}} \,{\text{d}}z,$$
$$\left\{ {\begin{array}{*{20}c} {S_{1}^{b} ,} & {S_{2}^{b} } \\ \end{array} } \right\} = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {\frac{{\nu_{b} }}{{2(1 - \nu_{b} )}}\left\{ {\begin{array}{*{20}c} {1,} & z \\ \end{array} } \right\}} \,{\text{d}}z,$$
$$\left\{ {\begin{array}{*{20}c} {S_{3}^{b} ,} & {S_{4}^{b} } \\ \end{array} } \right\} = \int\limits_{{ - h_{c} /2 - h_{b} }}^{{ - h_{c} /2}} {\frac{{\nu_{b} }}{{(1 - \nu_{b} )}}g_{3} (z)\left\{ {\begin{array}{*{20}c} {1,} & z \\ \end{array} } \right\}} \,{\text{d}}z$$

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Arshid, E., Amir, S. & Loghman, A. On the vibrations of FG GNPs-RPN annular plates with piezoelectric/metallic coatings on Kerr elastic substrate considering size dependency and surface stress effects. Acta Mech 234, 4035–4076 (2023). https://doi.org/10.1007/s00707-023-03593-4

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