On modeling of wave propagation in a thermally affected GNP-reinforced imperfect nanocomposite shell

Abstract

Due to rapid development of process manufacturing, composite materials with porosity have attracted commercial attention in promoting engineering applications. For this regard, in this research wave propagation-thermal characteristics of a size-dependent graphene nanoplatelet-reinforced composite (GNPRC) porous cylindrical nanoshell in thermal environment are investigated. The effects of small scale are analyzed based on nonlocal strain gradient theory (NSGT). The governing equations of the laminated composite cylindrical nanoshell in thermal environment have been evolved using Hamilton’s principle and solved with the assistance of the analytical method. For the first time, wave propagation-thermal behavior of a GNPRC porous cylindrical nanoshell in thermal environment based on NSGT is examined. The results show that by increasing the thickness, the effect of porosity on the phase velocity decreases. Another important result is that by increasing the value of the radius, the difference between the minimum and maximum values of the phase velocity increases. Finally, influence of temperature change, wave number, angular velocity and different types of porosity distribution on phase velocity are investigated using the mentioned continuum mechanics theory. As a useful suggestion, for designing of a GPLRC nanostructure should be attention to the GNP weight function and radius, simultaneously.

Appendix

The governing equations of the GPLRC cylindrical nanoshell in thermal environment are as follows:

$$\begin{aligned} & \delta u{\text{:}}\,{A_{11}}\left( {\frac{{{\partial ^2}u}}{{\partial {x^2}}}+{l^2}\frac{{{\partial ^4}u}}{{\partial {x^4}}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}u}}{{\partial {x^2}\partial {\theta ^2}}}} \right)+{B_{11}}\left( {\frac{{{\partial ^2}{\psi _x}}}{{\partial {x^2}}}+{l^2}\frac{{{\partial ^4}{\psi _x}}}{{\partial {x^4}}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}{\psi _x}}}{{\partial {x^2}\partial {\theta ^2}}}} \right) \\ & \quad +\,{A_{12}}\left( {\frac{1}{R}\frac{{{\partial ^2}v}}{{\partial x\partial \theta }}+\frac{1}{R}\frac{{\partial w}}{{\partial x}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}v}}{{\partial {x^3}\partial \theta }} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^3}w}}{{\partial {x^3}}} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^4}v}}{{\partial x\partial {\theta ^3}}} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^3}w}}{{\partial x\partial {\theta ^2}}}} \right) \\ & \quad +\,{B_{12}}\left( {\frac{1}{R}\frac{{{\partial ^2}{\psi _\theta }}}{{\partial x\partial \theta }} - \frac{{{l^2}}}{R}\frac{{{\partial ^4}{\psi _\theta }}}{{\partial {x^3}\partial \theta }} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^4}{\psi _\theta }}}{{\partial x\partial {\theta ^3}}}} \right) - {N_{\text{h}}}\left( {\frac{1}{R}\frac{{{\partial ^2}v}}{{\partial x\partial \theta }} - \frac{1}{{{R^2}}}\frac{{{\partial ^2}u}}{{\partial {\theta ^2}}}} \right) \\ & \quad +\,\frac{{{A_{66}}}}{R}\left( {\frac{1}{R}\frac{{{\partial ^2}u}}{{\partial {\theta ^2}}}+\frac{{{\partial ^2}v}}{{\partial x\partial \theta }} - \frac{{{l^2}}}{R}\frac{{{\partial ^4}u}}{{\partial {x^2}\partial {\theta ^2}}} - {l^2}\frac{{{\partial ^4}v}}{{\partial {x^3}\partial \theta }} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^4}u}}{{\partial {\theta ^4}}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}v}}{{\partial x\partial {\theta ^3}}}} \right) \\ & \quad +\,\frac{{{B_{66}}}}{R}\left( {\frac{1}{R}\frac{{{\partial ^2}{\psi _x}}}{{\partial {\theta ^2}}}+\frac{{{\partial ^2}{\psi _\theta }}}{{\partial x\partial \theta }} - \frac{{{l^2}}}{R}\frac{{{\partial ^4}{\psi _x}}}{{\partial {x^2}\partial {\theta ^2}}} - {l^2}\frac{{{\partial ^4}{\psi _\theta }}}{{\partial {x^3}\partial \theta }} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^4}{\psi _x}}}{{\partial {\theta ^4}}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}{\psi _\theta }}}{{\partial x\partial {\theta ^3}}}} \right) \\ & \quad =\,(1 - {\mu ^2}{\nabla ^2})\left( {{I_0}\frac{{{\partial ^2}u}}{{\partial {t^2}}}+{I_1}\frac{{{\partial ^2}{\psi _x}}}{{\partial {t^2}}}} \right), \\ \end{aligned}$$

(29)

$$\begin{aligned} \delta v{\text{: }}\frac{{{A_{12}}}}{R}\left( {\frac{{{\partial ^2}u}}{{\partial x\partial \theta }}+{l^2}\frac{{{\partial ^4}u}}{{\partial {x^3}\partial \theta }} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}u}}{{\partial x\partial {\theta ^3}}}} \right)+\frac{{{B_{12}}}}{R}\left( {\frac{{{\partial ^2}{\psi _x}}}{{\partial x\partial \theta }}+{l^2}\frac{{{\partial ^4}{\psi _x}}}{{\partial {x^3}\partial \theta }} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}{\psi _x}}}{{\partial x\partial {\theta ^3}}}} \right) \\ & \quad +\,\frac{{{A_{22}}}}{R}\left( {\frac{1}{R}\frac{{{\partial ^2}v}}{{\partial {\theta ^2}}}+\frac{1}{R}\frac{{\partial w}}{{\partial \theta }} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}v}}{{\partial {x^2}\partial {\theta ^2}}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^3}w}}{{\partial {x^2}\partial \theta }} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^4}v}}{{\partial {\theta ^4}}} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^3}w}}{{\partial {\theta ^3}}}} \right) \\ & \quad +\,\frac{{{B_{22}}}}{R}\left( {\frac{1}{R}\frac{{{\partial ^2}{\psi _\theta }}}{{\partial {\theta ^2}}}+\frac{{{l^2}}}{R}\frac{{{\partial ^4}{\psi _\theta }}}{{\partial {x^2}\partial {\theta ^2}}} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^4}{\psi _\theta }}}{{\partial {\theta ^4}}}} \right) - N_{2}^{{\text{T}}}\left( {\frac{{{\partial ^2}v}}{{\partial {x^2}}}} \right) \\ & \quad +\,{A_{66}}\left( {\frac{1}{R}\frac{{{\partial ^2}u}}{{\partial x\partial \theta }}+\frac{{{\partial ^2}v}}{{\partial {x^2}}} - \frac{{{l^2}}}{R}\frac{{{\partial ^4}u}}{{\partial {x^3}\partial \theta }} - {l^2}\frac{{{\partial ^4}v}}{{\partial {x^4}}} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^4}u}}{{\partial x\partial {\theta ^3}}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}v}}{{\partial {x^2}\partial {\theta ^2}}}} \right) \\ & \quad +\,{B_{66}}\left( {\frac{1}{R}\frac{{{\partial ^2}{\psi _x}}}{{\partial x\partial \theta }}+\frac{{{\partial ^2}{\psi _\theta }}}{{\partial {x^2}}} - \frac{{{l^2}}}{R}\frac{{{\partial ^4}{\psi _x}}}{{\partial {x^3}\partial \theta }} - {l^2}\frac{{{\partial ^4}{\psi _\theta }}}{{\partial {x^4}}} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^4}{\psi _x}}}{{\partial x\partial {\theta ^3}}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}{\psi _\theta }}}{{\partial {x^2}\partial {\theta ^2}}}} \right) \\ & \quad +\,\frac{{{K_{\text{s}}}{A_{44}}}}{R}\left( {{\psi _\theta }+\frac{1}{R}\frac{{\partial w}}{{\partial \theta }} - \frac{v}{R} - {l^2}\frac{{{\partial ^2}{\psi _\theta }}}{{\partial {x^2}}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^3}w}}{{\partial {x^2}\partial \theta }}+\frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^2}v}}{{\partial {x^2}}}} \right. \\ & \quad - \,\left. {\frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^2}{\psi _\theta }}}{{\partial {\theta ^2}}} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^3}w}}{{\partial {\theta ^3}}}+\frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^2}v}}{{\partial {\theta ^2}}}} \right)=(1 - {\mu ^2}{\nabla ^2})\left( {{I_0}\left[ {\frac{{{\partial ^2}v}}{{\partial {t^2}}}\,\,} \right]+{I_1}\left[ {\frac{{{\partial ^2}{\psi _\theta }}}{{\partial {t^2}}}} \right]} \right), \\ \end{aligned}$$

(30)

$$\begin{aligned} \delta w{\text{: }}\frac{{{A_{12}}}}{R}\left( {\frac{{\partial u}}{{\partial x}}+{l^2}\frac{{{\partial ^3}u}}{{\partial {x^3}}}+\frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^3}u}}{{\partial x\partial {\theta ^2}}}} \right)+\frac{{{B_{12}}}}{R}\left( {\frac{{\partial {\psi _x}}}{{\partial x}}+{l^2}\frac{{{\partial ^3}{\psi _x}}}{{\partial {x^3}}}+\frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^3}{\psi _x}}}{{\partial x\partial {\theta ^2}}}} \right) \\ & \quad +\,\frac{{{A_{22}}}}{R}\left( { - \,\frac{1}{R}\frac{{\partial v}}{{\partial \theta }} - \frac{w}{R}+\frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^3}v}}{{\partial {x^2}\partial \theta }}+\frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^2}w}}{{\partial {x^2}}}+\frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^3}v}}{{\partial {\theta ^3}}}+\frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^2}w}}{{\partial {\theta ^2}}}} \right) \\ & \quad +\,\frac{{{B_{22}}}}{R}\left( { - \,\frac{1}{R}\frac{{\partial {\psi _\theta }}}{{\partial \theta }}+\frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^3}{\psi _\theta }}}{{\partial {x^2}\partial \theta }}+\frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^3}{\psi _\theta }}}{{\partial {\theta ^3}}}} \right) - N_{1}^{{\text{T}}}\left( {\frac{{{\partial ^2}w}}{{\partial {x^2}}}} \right) \\ & \quad +\,{K_{\text{s}}}{A_{55}}\left( {\frac{{\partial {\psi _x}}}{{\partial x}} - \frac{{{\partial ^2}w}}{{\partial {x^2}}} - {l^2}\frac{{{\partial ^3}{\psi _x}}}{{\partial {x^3}}} - {l^2}\frac{{{\partial ^4}w}}{{\partial {x^4}}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^3}{\psi _x}}}{{\partial x\partial {\theta ^2}}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}w}}{{\partial {x^2}\partial {\theta ^2}}}} \right) \\ & \quad +\,\frac{{{K_{\text{s}}}{A_{44}}}}{R}\left( {\frac{{\partial {\psi _\theta }}}{{\partial \theta }} - \frac{1}{R}\frac{{{\partial ^2}w}}{{\partial {\theta ^2}}} - \frac{1}{R}\frac{{\partial v}}{{\partial \theta }} - {l^2}\frac{{{\partial ^3}{\psi _\theta }}}{{\partial {x^2}\partial \theta }} - \frac{{{l^2}}}{R}\frac{{{\partial ^4}w}}{{\partial {x^2}\partial {\theta ^2}}}+\frac{{{l^2}}}{R}\frac{{{\partial ^3}v}}{{\partial {x^2}\partial \theta }}} \right. \\ & \quad - \,\left. {\frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^3}{\psi _\theta }}}{{\partial {\theta ^3}}} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^4}w}}{{\partial {\theta ^4}}}+\frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^3}v}}{{\partial {\theta ^3}}}} \right)=(1 - {\mu ^2}{\nabla ^2})\left( {{I_0}\left( {\frac{{{\partial ^2}w}}{{\partial {t^2}}}} \right)} \right), \\ \end{aligned}$$

(31)

$$\begin{aligned} \delta {\psi _x}{\text{:}}\,{B_{11}}\left( {\frac{{{\partial ^2}u}}{{\partial {x^2}}} - {l^2}\frac{{{\partial ^4}u}}{{\partial {x^4}}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}u}}{{\partial {x^2}\partial {\theta ^2}}}} \right)+{D_{11}}\left( {\frac{{{\partial ^2}{\psi _x}}}{{\partial {x^2}}} - {l^2}\frac{{{\partial ^4}{\psi _x}}}{{\partial {x^4}}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}{\psi _x}}}{{\partial {x^2}\partial {\theta ^2}}}} \right) \\ & \quad +\,{B_{12}}\left( {\frac{1}{R}\frac{{{\partial ^2}v}}{{\partial x\partial \theta }}+\frac{1}{R}\frac{{\partial w}}{{\partial x}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}v}}{{\partial {x^3}\partial \theta }} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^3}w}}{{\partial {x^3}}} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^4}v}}{{\partial x\partial {\theta ^3}}} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^3}w}}{{\partial x\partial {\theta ^2}}}} \right) \\ & \quad +\,{D_{12}}\left( {\frac{1}{R}\frac{{{\partial ^2}{\psi _\theta }}}{{\partial x\partial \theta }}+\frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}{\psi _\theta }}}{{\partial {x^3}\partial \theta }} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^4}{\psi _\theta }}}{{\partial x\partial {\theta ^3}}}} \right) \\ & \quad +\,\frac{{{B_{66}}}}{R}\left( {\frac{1}{R}\frac{{{\partial ^2}u}}{{\partial {\theta ^2}}}+\frac{{{\partial ^2}v}}{{\partial x\partial \theta }} - \frac{{{l^2}}}{R}\frac{{{\partial ^4}u}}{{\partial {x^2}\partial {\theta ^2}}} - {l^2}\frac{{{\partial ^4}v}}{{\partial {x^3}\partial \theta }} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^4}u}}{{\partial {\theta ^4}}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}v}}{{\partial x\partial {\theta ^3}}}} \right) \\ & \quad +\,\frac{{{D_{66}}}}{R}\left( {\frac{1}{R}\frac{{{\partial ^2}{\psi _x}}}{{\partial {\theta ^2}}}+\frac{{{\partial ^2}{\psi _\theta }}}{{\partial x\partial \theta }} - \frac{{{l^2}}}{R}\frac{{{\partial ^4}{\psi _x}}}{{\partial {x^2}\partial {\theta ^2}}} - {l^2}\frac{{{\partial ^4}{\psi _\theta }}}{{\partial {x^3}\partial \theta }} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^4}{\psi _x}}}{{\partial {\theta ^4}}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}{\psi _\theta }}}{{\partial x\partial {\theta ^3}}}} \right) \\ & \quad - \,{K_s}{A_{55}}\left( {{\psi _x}+\frac{{\partial w}}{{\partial x}} - {l^2}\frac{{{\partial ^2}{\psi _x}}}{{\partial {x^2}}} - {l^2}\frac{{{\partial ^3}w}}{{\partial {x^3}}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^2}{\psi _x}}}{{\partial {\theta ^2}}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^3}w}}{{\partial x\partial {\theta ^2}}}} \right) \\ & \quad =\,(1 - {\mu ^2}{\nabla ^2})\left( {{I_1}\frac{{{\partial ^2}u}}{{\partial {t^2}}}+{I_2}\frac{{{\partial ^2}{\psi _x}}}{{\partial {t^2}}}} \right), \\ \end{aligned}$$

(32)

$$\begin{aligned} \delta {\psi _\theta }{\text{:}}\,\frac{{{B_{12}}}}{R}\left( {\frac{{{\partial ^2}u}}{{\partial x\partial \theta }} - {l^2}\frac{{{\partial ^4}u}}{{\partial {x^3}\partial \theta }} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}u}}{{\partial x\partial {\theta ^3}}}} \right)+\frac{{{D_{12}}}}{R}\left( {\frac{{{\partial ^2}{\psi _x}}}{{\partial x\partial \theta }} - {l^2}\frac{{{\partial ^4}{\psi _x}}}{{\partial {x^3}\partial \theta }} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}{\psi _x}}}{{\partial x\partial {\theta ^3}}}} \right) \\ & \quad +\,\frac{{{B_{22}}}}{R}\left( {\frac{1}{R}\frac{{{\partial ^2}v}}{{\partial {\theta ^2}}} - \frac{1}{R}\frac{{\partial w}}{{\partial \theta }} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}v}}{{\partial {x^2}\partial {\theta ^2}}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^3}w}}{{\partial {x^2}\partial \theta }} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^4}v}}{{\partial {\theta ^4}}} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^3}w}}{{\partial {\theta ^3}}}} \right) \\ & \quad +\,\frac{{{D_{22}}}}{R}\left( {\frac{1}{R}\frac{{{\partial ^2}{\psi _\theta }}}{{\partial {\theta ^2}}} - \frac{{{l^2}}}{R}\frac{{{\partial ^4}{\psi _\theta }}}{{\partial {x^2}\partial {\theta ^2}}} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^4}{\psi _\theta }}}{{\partial {\theta ^4}}}} \right) \\ & \quad +\,{B_{66}}\left( {\frac{1}{R}\frac{{{\partial ^2}u}}{{\partial x\partial \theta }} - \frac{{{\partial ^2}v}}{{\partial {x^2}}} - \frac{{{l^2}}}{R}\frac{{{\partial ^4}u}}{{\partial {x^3}\partial \theta }} - {l^2}\frac{{{\partial ^4}v}}{{\partial {x^4}}} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^4}u}}{{\partial x\partial {\theta ^3}}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}v}}{{\partial {x^2}\partial {\theta ^2}}}} \right) \\ & \quad +\,{D_{66}}\left( {\frac{1}{R}\frac{{{\partial ^2}{\psi _x}}}{{\partial x\partial \theta }} - \frac{{{\partial ^2}{\psi _\theta }}}{{\partial {x^2}}} - \frac{{{l^2}}}{R}\frac{{{\partial ^4}{\psi _x}}}{{\partial {x^3}\partial \theta }} - {l^2}\frac{{{\partial ^4}{\psi _\theta }}}{{\partial {x^4}}} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^4}{\psi _x}}}{{\partial x\partial {\theta ^3}}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^4}{\psi _\theta }}}{{\partial {x^2}\partial {\theta ^2}}}} \right) \\ & \quad - \,{K_{\text{s}}}{A_{44}}\left( {{\psi _\theta }+\frac{1}{R}\frac{{\partial w}}{{\partial \theta }} - \frac{v}{R} - {l^2}\frac{{{\partial ^2}{\psi _\theta }}}{{\partial {x^2}}} - \frac{{{l^2}}}{R}\frac{{{\partial ^3}w}}{{\partial {x^2}\partial \theta }}+\frac{{{l^2}}}{R}\frac{{{\partial ^2}v}}{{\partial {x^2}}} - \frac{{{l^2}}}{{{R^2}}}\frac{{{\partial ^2}{\psi _\theta }}}{{\partial {\theta ^2}}} - \frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^3}w}}{{\partial {\theta ^3}}}} \right. \\ & \quad +\,\left. {\frac{{{l^2}}}{{{R^3}}}\frac{{{\partial ^2}v}}{{\partial {\theta ^2}}}} \right)=(1 - {\mu ^2}{\nabla ^2})\left[ {{I_1}\left( {\frac{{{\partial ^2}v}}{{\partial {t^2}}}} \right)+{I_2}\left( {\frac{{{\partial ^2}{\psi _\theta }}}{{\partial {t^2}}}} \right)} \right], \\ \end{aligned}$$

(33)

where the defined parameter in Eqs. (29)–(33) are described as:

$$\begin{aligned} & \left\{ {\begin{array}{*{20}{c}} {{A_{11}}}&{{A_{12}}}&{{A_{22}}}&{{A_{66}}}&{{A_{44}}}&{{A_{55}}} \end{array}} \right\}=\int\limits_{{ - h/2}}^{{h/2}} {\left\{ {\begin{array}{*{20}{c}} {{C_{11}}}&{{C_{12}}}&{{C_{22}}}&{{C_{66}}}&{{C_{44}}}&{{C_{55}}} \end{array}} \right\}} {\text{d}}z \\ & \left\{ {\begin{array}{*{20}{c}} {{B_{11}}}&{{B_{12}}}&{{B_{22}}}&{{B_{66}}} \end{array}} \right\}=\int\limits_{{ - h/2}}^{{h/2}} {\left\{ {\begin{array}{*{20}{c}} {{C_{11}}}&{{C_{12}}}&{{C_{22}}}&{{C_{66}}} \end{array}} \right\}} z{\text{d}}z \\ & \left\{ {\begin{array}{*{20}{c}} {{D_{11}}}&{{D_{12}}}&{{D_{22}}}&{{D_{66}}} \end{array}} \right\}=\int\limits_{{ - h/2}}^{{h/2}} {\left\{ {\begin{array}{*{20}{c}} {{C_{11}}}&{{C_{12}}}&{{C_{22}}}&{{C_{66}}} \end{array}} \right\}} {z^2}{\text{d}}z \\ & \left\{ {\begin{array}{*{20}{c}} {{I_0}}&{{I_1}}&{{I_2}} \end{array}} \right\}=\int\limits_{{ - h/2}}^{{h/2}} {\rho (z,T)\left\{ {\begin{array}{*{20}{c}} 1&z&{{z^2}} \end{array}} \right\}} z{\text{d}}z. \\ \end{aligned}$$

(34)