Analytical treatment on the nonlocal strain gradient vibrational response of postbuckled functionally graded porous micro-/nanoplates reinforced with GPL

Abstract

Through production of porous materials with remarkable complexity in geometry, functionally graded porous materials (FGPMs) have gained considerable attention for use in additive manufacturing in biomedical applications. In the current study, the size-dependent linear and nonlinear vibrational characteristics of axially loaded micro-/nanoplates made of FGPM reinforced with graphene platelets (GPLs) is investigated within both the prebuckling and postbuckling regimes. To this end, the nonlocal strain gradient continuum elasticity in conjunction with geometrical nonlinearity is implemented into the refined exponential shear deformation plate theory. On the basis of the closed-cell Gaussian random field scheme as well as the Halpin–Tsai micromechanical modeling, the mechanical properties of the FGPM reinforced with GPLs are achieved corresponding to the uniform and three different patterns of porosity dispersion. Via the variational approach, the differential equations of motion related to the nonlinear problem are constructed in the presence of nonlocality and strain gradient size dependency. Finally, with the aid of an improved perturbation technique together with the Galerkin method, analytical expressions in explicit form for the size-dependent linear frequency–load and deflection–nonlinear frequency responses of the FGPM micro-/nanoplates within stability and instability domains are obtained. It is displayed that within the prebuckling regime, the nonlocality causes reduction of the linear frequency of the micro-/nanoplate, while the strain gradient size dependency leads to increasing it. But within the postbuckling domain, these patterns are vice versa. Also, it is found that for a specific value of plate deflection, increasing the value of the porosity coefficient leads to increase in the frequency ratio of ω_NL/ω_L within both the prebuckling and postbuckling regimes.

Appendices

Appendix 1

The nonlocal strain gradient stress resultants can be expressed as:

$$\begin{aligned} & \left\{ {\begin{array}{*{20}c} {N_{xx} - \mu^{2} \left( {\frac{{\partial^{2} N_{xx} }}{{\partial x^{2} }} + \frac{{\partial^{2} N_{xx} }}{{\partial y^{2} }}} \right)} \\ {N_{yy} - \mu^{2} \left( {\frac{{\partial^{2} N_{yy} }}{{\partial x^{2} }} + \frac{{\partial^{2} N_{yy} }}{{\partial y^{2} }}} \right)} \\ {N_{xy} - \mu^{2} \left( {\frac{{\partial^{2} N_{xy} }}{{\partial x^{2} }} + \frac{{\partial^{2} N_{xy} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ & \quad = \left[ {\begin{array}{*{20}c} {A_{11}^{*} } & {A_{12}^{*} } & 0 \\ {A_{12}^{*} } & {A_{22}^{*} } & 0 \\ 0 & 0 & {A_{66}^{*} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\varepsilon_{xx}^{0} - l^{2} \left( {\frac{{\partial^{2} \varepsilon_{xx}^{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} \varepsilon_{xx}^{0} }}{{\partial y^{2} }}} \right)} \\ {\varepsilon_{yy}^{0} - l^{2} \left( {\frac{{\partial^{2} \varepsilon_{yy}^{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} \varepsilon_{yy}^{0} }}{{\partial y^{2} }}} \right)} \\ {\gamma_{xy}^{0} - l^{2} \left( {\frac{{\partial^{2} \gamma_{xy}^{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} \gamma_{xy}^{0} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ & \quad + \left[ {\begin{array}{*{20}c} {B_{11}^{*} } & {B_{12}^{*} } & 0 \\ {B_{12}^{*} } & {B_{22}^{*} } & 0 \\ 0 & 0 & {B_{66}^{*} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\kappa_{xx}^{\left( 1 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xx}^{\left( 1 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xx}^{\left( 1 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{yy}^{\left( 1 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{yy}^{\left( 1 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{yy}^{\left( 1 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{xy}^{\left( 1 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xy}^{\left( 1 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xy}^{\left( 1 \right)} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ & \quad + \left[ {\begin{array}{*{20}c} {F_{11}^{*} } & {F_{12}^{*} } & 0 \\ {F_{12}^{*} } & {F_{22}^{*} } & 0 \\ 0 & 0 & {F_{66}^{*} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\kappa_{xx}^{\left( 2 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xx}^{\left( 2 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xx}^{\left( 2 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{yy}^{\left( 2 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{yy}^{\left( 2 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{yy}^{\left( 2 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{xy}^{\left( 2 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xy}^{\left( 2 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xy}^{\left( 2 \right)} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ \end{aligned},$$

$$\begin{aligned} & \left\{ {\begin{array}{*{20}c} {M_{xx} - \mu^{2} \left( {\frac{{\partial^{2} M_{xx} }}{{\partial x^{2} }} + \frac{{\partial^{2} M_{xx} }}{{\partial y^{2} }}} \right)} \\ {M_{yy} - \mu^{2} \left( {\frac{{\partial^{2} M_{yy} }}{{\partial x^{2} }} + \frac{{\partial^{2} M_{yy} }}{{\partial y^{2} }}} \right)} \\ {M_{xy} - \mu^{2} \left( {\frac{{\partial^{2} M_{xy} }}{{\partial x^{2} }} + \frac{{\partial^{2} M_{xy} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ & \quad = \left[ {\begin{array}{*{20}c} {B_{11}^{*} } & {B_{12}^{*} } & 0 \\ {B_{12}^{*} } & {B_{11}^{*} } & 0 \\ 0 & 0 & {B_{66}^{*} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\varepsilon_{xx}^{0} - l^{2} \left( {\frac{{\partial^{2} \varepsilon_{xx}^{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} \varepsilon_{xx}^{0} }}{{\partial y^{2} }}} \right)} \\ {\varepsilon_{yy}^{0} - l^{2} \left( {\frac{{\partial^{2} \varepsilon_{yy}^{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} \varepsilon_{yy}^{0} }}{{\partial y^{2} }}} \right)} \\ {\gamma_{xy}^{0} - l^{2} \left( {\frac{{\partial^{2} \gamma_{xy}^{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} \gamma_{xy}^{0} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ & \quad + \left[ {\begin{array}{*{20}c} {D_{11}^{*} } & {D_{12}^{*} } & 0 \\ {D_{12}^{*} } & {D_{11}^{*} } & 0 \\ 0 & 0 & {D_{66}^{*} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\kappa_{xx}^{\left( 1 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xx}^{\left( 1 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xx}^{\left( 1 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{yy}^{\left( 1 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{yy}^{\left( 1 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{yy}^{\left( 1 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{xy}^{\left( 1 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xy}^{\left( 1 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xy}^{\left( 1 \right)} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ & \quad + \left[ {\begin{array}{*{20}c} {G_{11}^{*} } & {G_{12}^{*} } & 0 \\ {G_{12}^{*} } & {G_{22}^{*} } & 0 \\ 0 & 0 & {G_{66}^{**} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\kappa_{xx}^{\left( 2 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xx}^{\left( 2 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xx}^{\left( 2 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{yy}^{\left( 2 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{yy}^{\left( 2 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{yy}^{\left( 2 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{xy}^{\left( 2 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xy}^{\left( 2 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xy}^{\left( 2 \right)} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ \end{aligned},$$

$$\begin{aligned} & \left\{ {\begin{array}{*{20}c} {R_{xx} - \mu^{2} \left( {\frac{{\partial^{2} R_{xx} }}{{\partial x^{2} }} + \frac{{\partial^{2} R_{xx} }}{{\partial y^{2} }}} \right)} \\ {R_{yy} - \mu^{2} \left( {\frac{{\partial^{2} R_{yy} }}{{\partial x^{2} }} + \frac{{\partial^{2} R_{yy} }}{{\partial y^{2} }}} \right)} \\ {R_{xy} - \mu^{2} \left( {\frac{{\partial^{2} R_{xy} }}{{\partial x^{2} }} + \frac{{\partial^{2} R_{xy} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ & \quad = \left[ {\begin{array}{*{20}c} {F_{11}^{*} } & {F_{12}^{*} } & 0 \\ {F_{12}^{*} } & {F_{22}^{*} } & 0 \\ 0 & 0 & {F_{66}^{*} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\varepsilon_{xx}^{0} - l^{2} \left( {\frac{{\partial^{2} \varepsilon_{xx}^{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} \varepsilon_{xx}^{0} }}{{\partial y^{2} }}} \right)} \\ {\varepsilon_{yy}^{0} - l^{2} \left( {\frac{{\partial^{2} \varepsilon_{yy}^{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} \varepsilon_{yy}^{0} }}{{\partial y^{2} }}} \right)} \\ {\gamma_{xy}^{0} - l^{2} \left( {\frac{{\partial^{2} \gamma_{xy}^{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} \gamma_{xy}^{0} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ & \quad + \left[ {\begin{array}{*{20}c} {G_{11}^{*} } & {G_{12}^{*} } & 0 \\ {G_{12}^{*} } & {G_{22}^{*} } & 0 \\ 0 & 0 & {G_{66}^{**} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\kappa_{xx}^{\left( 1 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xx}^{\left( 1 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xx}^{\left( 1 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{yy}^{\left( 1 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{yy}^{\left( 1 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{yy}^{\left( 1 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{xy}^{\left( 1 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xy}^{\left( 1 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xy}^{\left( 1 \right)} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ & \quad + \left[ {\begin{array}{*{20}c} {H_{11}^{*} } & {H_{12}^{*} } & 0 \\ {H_{12}^{*} } & {H_{22}^{*} } & 0 \\ 0 & 0 & {H_{66}^{*} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\kappa_{xx}^{\left( 2 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xx}^{\left( 2 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xx}^{\left( 2 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{yy}^{\left( 2 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{yy}^{\left( 2 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{yy}^{\left( 2 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{xy}^{\left( 2 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xy}^{\left( 2 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xy}^{\left( 2 \right)} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ \end{aligned},$$

$$\left\{ {\begin{array}{*{20}c} {Q_{x} - \mu^{2} \left( {\frac{{\partial^{2} Q_{x} }}{{\partial x^{2} }} + \frac{{\partial^{2} Q_{x} }}{{\partial y^{2} }}} \right)} \\ {Q_{y} - \mu^{2} \left( {\frac{{\partial^{2} Q_{y} }}{{\partial x^{2} }} + \frac{{\partial^{2} Q_{y} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {A_{44}^{*} } & 0 \\ 0 & {A_{55}^{*} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\psi_{x} - l^{2} \left( {\frac{{\partial^{2} \psi_{x} }}{{\partial x^{2} }} + \frac{{\partial^{2} \psi_{x} }}{{\partial y^{2} }}} \right)} \\ {\psi_{y} - l^{2} \left( {\frac{{\partial^{2} \psi_{y} }}{{\partial x^{2} }} + \frac{{\partial^{2} \psi_{y} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\}.$$

Appendix 2

$$\varphi_{1} = \frac{{A_{11}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{2} = \frac{1}{{A_{66}^{*} }},\quad \varphi_{3} = \frac{{A_{12}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{4} = \frac{{A_{22}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},$$

$$\varphi_{5} = \frac{{A_{11}^{*} B_{12}^{*} - A_{12}^{*} B_{11}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{6} = \frac{{A_{11}^{*} B_{11}^{*} - A_{12}^{*} B_{12}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{7} = \frac{{A_{22}^{*} B_{22}^{*} - A_{12}^{*} B_{12}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{8} = \frac{{B_{66}^{*} }}{{A_{66}^{*} }},$$

$$\varphi_{9} = \frac{{A_{22}^{*} B_{12}^{*} - A_{12}^{*} B_{22}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{10} = \frac{{A_{11}^{*} F_{12}^{*} - A_{12}^{*} F_{11}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{11} = \frac{{A_{11}^{*} F_{11}^{*} - A_{12}^{*} F_{12}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{12} = \frac{{F_{66}^{*} }}{{A_{66}^{*} }},$$

$$\varphi_{13} = \frac{{A_{22}^{*} F_{12}^{*} - A_{12}^{*} F_{22}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{14} = \frac{{A_{22}^{*} F_{22}^{*} - A_{12}^{*} F_{12}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{15} = D_{11}^{*} - \frac{{A_{11}^{*} \left( {\left( {B_{11}^{*} } \right)^{2} + \left( {B_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},$$

$$\varphi_{16} = 2\left( {D_{12}^{*} + 2D_{66}^{*} } \right) - \frac{{A_{11}^{*} \left( {\left( {B_{11}^{*} } \right)^{2} + \left( {B_{12}^{*} } \right)^{2} } \right) + A_{22}^{*} \left( {\left( {B_{22}^{*} } \right)^{2} + \left( {B_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} - 4\frac{{\left( {B_{66}^{*} } \right)^{2} }}{{A_{66}^{*} }},$$

$$\varphi_{17} = D_{22}^{*} - \frac{{A_{22}^{*} \left( {\left( {B_{22}^{*} } \right)^{2} + \left( {B_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} ,\quad \varphi_{18} = \frac{{A_{11}^{*} \left( {\left( {F_{11}^{*} } \right)^{2} + \left( {F_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} - G_{11}^{*},$$

$$\varphi_{19} = \frac{{2A_{11}^{*} F_{11}^{*} F_{12}^{*} - A_{12}^{*} \left( {\left( {F_{11}^{*} } \right)^{2} + \left( {F_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} + 2\frac{{\left( {B_{66}^{**} } \right)^{2} }}{{A_{66}^{*} }} - \left( {G_{12}^{*} + 2G_{66}^{*} } \right),$$

$$\varphi_{20} = \frac{{A_{22}^{*} \left( {\left( {F_{22}^{*} } \right)^{2} + \left( {F_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} - G_{22}^{*},$$

$$\varphi_{21} = \frac{{2A_{22}^{*} F_{22}^{*} F_{12}^{*} - A_{12}^{*} \left( {\left( {F_{22}^{*} } \right)^{2} + \left( {F_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} + 2\frac{{\left( {B_{66}^{**} } \right)^{2} }}{{A_{66}^{*} }} - \left( {G_{12}^{*} + 2G_{66}^{*} } \right),$$

$$\varphi_{22} = \frac{{A_{11}^{*} \left( {B_{11}^{*} F_{11}^{*} + B_{12}^{*} F_{12}^{*} } \right) - A_{12}^{*} \left( {B_{12}^{*} F_{11}^{*} + B_{11}^{*} F_{12}^{*} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} - G_{11}^{*} ,$$

$$\varphi_{23} = \frac{{A_{11}^{*} \left( {B_{11}^{*} F_{12}^{*} + B_{12}^{*} F_{11}^{*} } \right) - A_{12}^{*} \left( {B_{12}^{*} F_{12}^{*} + B_{11}^{*} F_{11}^{*} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} + 2\frac{{B_{66}^{*} F_{66}^{*} }}{{A_{66}^{*} }} - \left( {G_{12}^{*} + 2G_{66}^{*} } \right),$$

$$\varphi_{24} = H_{11}^{*} - \frac{{A_{11}^{*} \left( {\left( {F_{11}^{*} } \right)^{2} + \left( {F_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} ,\quad \varphi_{25} = H_{66}^{*} - \frac{{\left( {F_{66}^{*} } \right)^{2} }}{{A_{66}^{*} }},$$

$$\varphi_{26} = H_{12}^{*} + H_{66}^{*} + \frac{{A_{12}^{*} \left( {\left( {F_{11}^{*} } \right)^{2} + \left( {F_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} - \frac{{\left( {F_{66}^{*} } \right)^{2} }}{{A_{66}^{*} }},$$

$$\varphi_{27} = \frac{{A_{22}^{*} \left( {B_{22}^{*} F_{22}^{*} + B_{12}^{*} F_{12}^{*} } \right) - A_{12}^{*} \left( {B_{12}^{*} F_{22}^{*} + B_{22}^{*} B_{12}^{*} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} - G_{22}^{*},$$

$$\varphi_{28} = \frac{{A_{22}^{*} \left( {B_{22}^{*} F_{12}^{*} + B_{12}^{*} F_{22}^{*} } \right) - A_{12}^{*} \left( {B_{12}^{*} F_{12}^{*} + B_{22}^{*} F_{22}^{*} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} + 2\frac{{B_{66}^{*} F_{66}^{*} }}{{A_{66}^{*} }} - \left( {G_{12}^{*} + 2G_{66}^{*} } \right),$$

$$\varphi_{29} = H_{22}^{*} - \frac{{A_{22}^{*} \left( {\left( {F_{22}^{*} } \right)^{2} + \left( {F_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{30} = H_{12}^{*} + H_{66}^{*} + \frac{{A_{12}^{*} \left( {\left( {F_{22}^{*} } \right)^{2} + \left( {F_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} - \frac{{\left( {F_{66}^{*} } \right)^{2} }}{{A_{66}^{*} }}.$$