2.1 Geometry and material property
Consider an FGM rectangular plate excited by a moving load, as shown in Fig. 1. The structure is constructed of a ceramic–metal mixture, and the mechanical properties are determined to change smoothly along the z direction. FSDT theory is applied according to the thickness of plate. Hence, the displacement fields are assumed as [34]:
$$\begin{aligned} u_{x} \left( {x,y,z,t} \right) & = u_{0} \left( {x,y,t} \right) + z\psi_{x} \left( {x,y,t} \right) \\ u_{y} \left( {x,y,z,t} \right) & = v_{0} \left( {x,y,t} \right) + z\psi_{y} \left( {x,y,t} \right) \\ u_{z} \left( {x,y,z,t} \right) & = w_{0} \left( {x,y,t} \right) \\ \end{aligned}$$
(1)
The plate is made of a metal-ceramic FGM. The volume fraction of the metal material is supposed to change along the z direction as follows:
$$V_{m} = \left( {\frac{z}{h} + \frac{1}{2}} \right)^{n}$$
(2)
The upper and lower faces of the structure are made of metal and ceramic, respectively. Also, n is the positive-definite power law exponent. Therefore, the mechanical properties of the structure such as the modulus of elasticity and mass density are defined as:
$$E = E_{c} + \left( {E_{m} - E_{c} } \right)\left( {\frac{z}{h} + \frac{1}{2}} \right)^{n}$$
(3)
$$\rho = \rho_{c} + \left( {\rho_{m} - \rho_{c} } \right)\left( {\frac{z}{h} + \frac{1}{2}} \right)^{n}$$
E and \(\rho\) are the elasticity modules and density of FGM structure. Also, indices c and m denote the material properties of ceramic and metal. The Poisson's ratio is supposed to be constant.
In Eq. (1), u, v, w, are displacement fields across the x, y and z axes, respectively. Besides, each axis with 0 subscript denotes the mid-plane displacement field. Also,\({\psi }_{x}\) and \({\psi }_{y}\) are normal transverse rotations around y and x axes, respectively. Von-Karman strain–displacement relationships are defined as:
$$\begin{aligned} \varepsilon_{xx} & = \frac{1}{2}\left( {\frac{{\partial u_{x} }}{\partial x} + \frac{{\partial u_{x} }}{\partial x} + \frac{{\partial u_{z} }}{\partial x}\frac{{\partial u_{z} }}{\partial x}} \right) = \frac{{\partial u_{0} }}{\partial x} + z\frac{{\partial \psi_{x} }}{\partial x} + \frac{1}{2}\left( {\frac{{\partial w_{0} }}{\partial x}} \right)^{2} \\ \varepsilon_{yy} & = \frac{1}{2}\left( {\frac{{\partial u_{y} }}{\partial y} + \frac{{\partial u_{y} }}{\partial y} + \frac{{\partial u_{z} }}{\partial y}\frac{{\partial u_{z} }}{\partial y}} \right) = \frac{{\partial v_{0} }}{\partial y} + z\frac{{\partial \psi_{y} }}{\partial y} + \frac{1}{2}\left( {\frac{{\partial w_{0} }}{\partial y}} \right)^{2} \\ \gamma_{xy} & = \left( {\frac{{\partial u_{x} }}{\partial y} + \frac{{\partial u_{y} }}{\partial x} + \frac{{\partial u_{z} }}{\partial x}\frac{{\partial u_{z} }}{\partial y}} \right) = \frac{{\partial u_{0} }}{\partial y} + \frac{{\partial v_{0} }}{\partial x} + z\left( {\frac{{\partial \psi_{x} }}{\partial y} + \frac{{\partial \psi_{y} }}{\partial x}} \right) + \frac{{\partial w_{0} }}{\partial x}\frac{{\partial w_{0} }}{\partial y} \\ \gamma_{xz} & = \left( {\frac{{\partial u_{x} }}{\partial z} + \frac{{\partial u_{z} }}{\partial x} + \frac{{\partial u_{z} }}{\partial x}\frac{{\partial u_{z} }}{\partial z}} \right) = \psi_{x} + \frac{{\partial w_{0} }}{\partial x} \\ \gamma_{yz} & = \left( {\frac{{\partial u_{y} }}{\partial z} + \frac{{\partial u_{z} }}{\partial y} + \frac{{\partial u_{z} }}{\partial y}\frac{{\partial u_{z} }}{\partial z}} \right) = \psi_{y} + \frac{{\partial w_{0} }}{\partial y} \\ \end{aligned}$$
(4)
2.2 Governing equation of motion
The Relationship between stress and strain of FGM rectangular plate based on FSDT is as the following:
$$\begin{aligned} \sigma_{xx} & = \frac{E}{{1 - \upsilon^{2} }}\left( {\varepsilon_{xx} + \upsilon \varepsilon_{yy} } \right) = \frac{E}{{1 - \upsilon^{2} }}\left[ {\frac{{\partial u_{0} }}{\partial x} + \upsilon \frac{{\partial v_{0} }}{\partial y} + \frac{1}{2}\left[ {\left( {\frac{{\partial w_{0} }}{\partial x}} \right)^{2} + \upsilon \left( {\frac{{\partial w_{0} }}{\partial y}} \right)^{2} } \right] + z\left( {\frac{{\partial \psi_{x} }}{\partial x} + \upsilon \frac{{\partial \psi_{y} }}{\partial y}} \right)} \right] \\ \sigma_{yy} & = \frac{E}{{1 - \upsilon^{2} }}\left( {\upsilon \varepsilon_{xx} + \varepsilon_{yy} } \right) = \frac{E}{{1 - \upsilon^{2} }}\left[ {\upsilon \frac{{\partial u_{0} }}{\partial x} + \frac{{\partial v_{0} }}{\partial y} + \frac{1}{2}\left[ {\upsilon \left( {\frac{{\partial w_{0} }}{\partial x}} \right)^{2} + \left( {\frac{{\partial w_{0} }}{\partial y}} \right)^{2} } \right] + z\left( {\upsilon \frac{{\partial \psi_{x} }}{\partial x} + \frac{{\partial \psi_{y} }}{\partial y}} \right)} \right] \\ \tau_{xy} & = \frac{E}{{2\left( {1 + \upsilon } \right)}}\gamma_{xy} = \frac{E}{{2\left( {1 + \upsilon } \right)}}\left[ {\frac{{\partial u_{0} }}{\partial y} + \frac{{\partial v_{0} }}{\partial x} + z\left( {\frac{{\partial \psi_{x} }}{\partial y} + \frac{{\partial \psi_{y} }}{\partial x}} \right) + \frac{{\partial w_{0} }}{\partial x}\frac{{\partial w_{0} }}{\partial y}} \right] \\ \tau_{xz} & = \kappa \frac{E}{{2\left( {1 + \upsilon } \right)}}\gamma_{xz} = \kappa \frac{E}{{2\left( {1 + \upsilon } \right)}}\left( {\psi_{x} + \frac{{\partial w_{0} }}{\partial x}} \right) \\ \tau_{yz} & = \kappa \frac{E}{{2\left( {1 + \upsilon } \right)}}\gamma_{yz} = \kappa \frac{E}{{2\left( {1 + \upsilon } \right)}}\left( {\psi_{y} + \frac{{\partial w_{0} }}{\partial y}} \right) \\ \end{aligned}$$
(5)
where E and ν are the elasticity modules and Poisson's ratio, respectively. In addition, \(\kappa\) = \(\frac{{\uppi }^{2}}{12}\) [35] is the shear correction factor. The strain energy of structure is as the following:
$$\delta U = \iiint_{V} {\left( {\sigma_{xx} \delta \varepsilon_{xx} + \sigma_{yy} \delta \varepsilon_{yy} + \tau_{xy} \delta \gamma_{xy} + \tau_{xz} \delta \gamma_{xz} + \tau_{yz} \delta \gamma_{yz} } \right)dV}$$
(6)
By replacing strain component, we have:
$$\delta U = \iiint_{V} {\left( \begin{gathered} \sigma_{xx} \delta \left( {\frac{{\partial u_{0} }}{\partial x} + z\frac{{\partial \psi_{x} }}{\partial x} + \frac{1}{2}\left( {\frac{{\partial w_{0} }}{\partial x}} \right)^{2} } \right) + \sigma_{yy} \delta \left( {\frac{{\partial v_{0} }}{\partial y} + z\frac{{\partial \psi_{y} }}{\partial y} + \frac{1}{2}\left( {\frac{{\partial w_{0} }}{\partial y}} \right)^{2} } \right) \hfill \\ + \tau_{xy} \delta \left( {\frac{{\partial u_{0} }}{\partial y} + \frac{{\partial v_{0} }}{\partial x} + z\left( {\frac{{\partial \psi_{x} }}{\partial y} + \frac{{\partial \psi_{y} }}{\partial x}} \right) + \frac{{\partial w_{0} }}{\partial x}\frac{{\partial w_{0} }}{\partial y}} \right) + \tau_{xz} \delta \left( {\psi_{x} + \frac{{\partial w_{0} }}{\partial x}} \right) + \tau_{yz} \delta \left( {\psi_{y} + \frac{{\partial w_{0} }}{\partial y}} \right) \hfill \\ \end{gathered} \right)dV}$$
(7)
The resultants of normal and shear forces and moments are obtained by integrating the stress field along the thickness direction:
$$\left\{ {\left. {\begin{array}{*{20}c} {N_{xx} } \\ {N_{yy} } \\ {N_{xy} } \\ \end{array} } \right\}} \right. = \mathop \smallint \nolimits_{{ - {\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}^{{ + {\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} \left\{ {\left. {\begin{array}{*{20}c} {\sigma_{xx} } \\ {\sigma_{yy} } \\ {\tau_{xy} } \\ \end{array} } \right\}dz} \right.\;\left\{ {\left. {\begin{array}{*{20}c} {M_{xx} } \\ {M_{yy} } \\ {M_{xy} } \\ \end{array} } \right\}} \right. = \mathop \smallint \nolimits_{{ - {\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}^{{ + {\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} \left\{ {\left. {\begin{array}{*{20}c} {\sigma_{xx} } \\ {\sigma_{yy} } \\ {\tau_{xy} } \\ \end{array} } \right\}zdz} \right.\;\left\{ {\left. {\begin{array}{*{20}c} {Q_{x} } \\ {Q_{y} } \\ \end{array} } \right\}} \right. = \mathop \smallint \nolimits_{{ - {\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}^{{ + {\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} \left\{ {\left. {\begin{array}{*{20}c} {\tau_{xz} } \\ {\tau_{yz} } \\ \end{array} } \right\}dz} \right.$$
(8)
We can rewrite Eq. (6) by using Eq. (7) as following:
$$\delta U = \iint_{A} {\left( \begin{gathered} N_{xx} \delta \left( {\frac{{\partial u_{0} }}{\partial x}} \right) + M_{xx} \delta \left( {\frac{{\partial \psi_{x} }}{\partial x}} \right) + \frac{{N_{xx} }}{2}\delta \left( {\frac{{\partial w_{0} }}{\partial x}} \right)^{2} + N_{yy} \delta \left( {\frac{{\partial v_{0} }}{\partial y}} \right) + M_{yy} \delta \left( {\frac{{\partial \psi_{y} }}{\partial y}} \right) \hfill \\ + \frac{{N_{yy} }}{2}\delta \left( {\frac{{\partial w_{0} }}{\partial y}} \right)^{2} + N_{xy} \delta \left( {\frac{{\partial u_{0} }}{\partial y}} \right) + N_{xy} \delta \left( {\frac{{\partial v_{0} }}{\partial x}} \right) + M_{xy} \delta \left( {\frac{{\partial \psi_{x} }}{\partial y} + \frac{{\partial \psi_{y} }}{\partial x}} \right) \hfill \\ + N_{xy} \delta \left( {\frac{{\partial w_{0} }}{\partial x}\frac{{\partial w_{0} }}{\partial y}} \right) + Q_{x} \delta \left( {\psi_{x} + \frac{{\partial w_{0} }}{\partial x}} \right) + Q_{y} \delta \left( {\psi_{y} + \frac{{\partial w_{0} }}{\partial y}} \right) \hfill \\ \end{gathered} \right)dA}$$
(9)
The kinetic energy rectangular of plate is expressed as:
$$K = \frac{1}{2}\mathop \smallint \nolimits_{{ - \frac{h}{2}}}^{\frac{h}{2}} \mathop \smallint \nolimits_{A}^{{}} \rho \left( z \right)\left( {V^{P} } \right)^{2} dAdz$$
(10)
\(\rho \left( z \right)\) is the density of FGM plate and \(V^{P}\) is the velocity of force which calculate as the following:
$$\overline{{V^{P} }} = \sqrt {\left( {V_{x}^{P} } \right)^{2} + \left( {V_{y}^{P} } \right)^{2} + \left( {V_{z}^{P} } \right)^{2} }$$
where \(V_{x}^{P}\),\(V_{y}^{P}\),\(V_{z}^{P}\) are the component of moving force through the x,y,z axes and can be calculated from Eq. (1) as the following
$$\left\{ \begin{gathered} V_{x}^{P} = \dot{u}_{x} = \dot{u}_{0} + z\dot{\psi }_{x} \hfill \\ V_{y}^{P} = \dot{u}_{y} = \dot{v}_{0} + z\dot{\psi }_{y} \hfill \\ V_{z}^{P} = \dot{u}_{z} = \dot{w}_{0} \hfill \\ \end{gathered} \right.$$
(11)
By replacing Eq. (10) in Eq. (9), the kinetic energy of structure is simplified as:
$$\delta K = \int_{{ - \frac{h}{2}}}^{{ + \frac{h}{2}}} {\int_{A} {\rho \left( z \right)\left[ \begin{gathered} \left( {\dot{u}_{0} \delta \dot{u}_{0} + \dot{u}_{0} z\delta \dot{\psi }_{x} + \delta \dot{u}_{0} z\dot{\psi }_{x} + z^{2} \dot{\psi }_{x} \delta \dot{\psi }_{x} } \right) \hfill \\ + \left( {\dot{v}_{0} \delta \dot{v}_{0} + \dot{v}_{0} z\delta \dot{\psi }_{y} + \delta \dot{v}_{0} z\dot{\psi }_{y} + z^{2} \dot{\psi }_{y} \delta \dot{\psi }_{y} } \right) + \dot{w}_{0} {\updelta }\dot{w}_{0} \hfill \\ \end{gathered} \right]dAdz} }$$
(12)
The external work that related to moving load (P0) is considered as following:
$$\overline{W} = \iint_{A} {P_{0} \delta \left( {x - vt} \right)\delta \left( {x - e} \right)w\left( {x,y,t} \right)dA}$$
(13)
The Hamilton's principle as the following:
$$\mathop \smallint \nolimits_{{t_{1} }}^{{t_{2} }} \left( {\delta K - \delta U + \delta \overline{W}} \right)dt = 0$$
(14)
$$\left\{ {\begin{array}{*{20}c} {U_{0} = \frac{{u_{0} }}{a}} \\ {V_{0} = \frac{{v_{0} }}{b}} \\ {W_{0} = \frac{{w_{0} }}{a}} \\ {X = \frac{x}{a}} \\ {Y = \frac{y}{b}} \\ {\overline{P} = \frac{{P_{0} h}}{{C_{xx} }}\mathop \Rightarrow \limits^{{}} P_{0} = \frac{{\overline{P}C_{xx} }}{h}} \\ {\frac{{C_{xx} }}{{I_{A} a^{4} }}\frac{1}{{T^{2} }} = \frac{1}{{t^{2} }}} \\ \end{array} } \right.$$
(15)
$$\left\{ {\begin{array}{*{20}c} {I_{A} = \int_{{ - {\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}^{{ + {\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} {\rho \left( z \right)dz} } \\ {I_{B} = \int_{{ - {\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}^{{ + {\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} {\rho \left( z \right)zdz} } \\ {I_{C} = \int_{{ - {\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}^{{ + {\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} {\rho \left( z \right)z^{2} dz} } \\ \end{array} } \right.$$
(16)
In Eq. (15), U0, V0, W0 are the dimensionless mid-plane displacement fields through the x, y, z axes. a, b and h represent the length, width and thickness of the rectangular plate. X and Y are the dimensionless variable. Also, t and T are the non dimensionless and dimensionless time. Besides, \(\overline{P}\) is the dimensionless force.
By substituting Eqs. (9), (12), (13) in Eq. (14) and dimensionless form of variables that is mentioned in Eqs. (15) and (16). Also, calculus of variations, the governing equations of motion are extracted as following (Eqs. 17–21):
$$\left\{ {\begin{array}{*{20}l} {\lambda_{2}^{2} E_{1}^{*} \left[ \begin{gathered} \frac{{\partial^{2} U_{0} }}{{\partial X^{2} }} + \frac{{\partial^{2} W_{0} }}{{\partial X^{2} }}\frac{{\partial W_{0} }}{\partial X} + \left( {\frac{1 + \upsilon }{2}} \right)\left( {\frac{{\partial^{2} V_{0} }}{\partial Y\partial X} + \lambda_{1}^{2} \frac{{\partial^{2} W_{0} }}{\partial Y\partial X}\frac{{\partial W_{0} }}{\partial Y}} \right) \hfill \\ + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\lambda_{1}^{2} \frac{{\partial^{2} U_{0} }}{{\partial Y^{2} }} + \lambda_{1}^{2} \frac{{\partial W_{0} }}{\partial X}\frac{{\partial^{2} W_{0} }}{{\partial Y^{2} }}} \right) \hfill \\ \end{gathered} \right]} \hfill \\ { + \lambda_{2} E_{2}^{*} \left[ {\frac{{\partial^{2} \psi_{x} }}{{\partial X^{2} }} + \left( {\frac{1 + \upsilon }{2}} \right)\lambda_{1} \frac{{\partial^{2} \psi_{y} }}{\partial Y\partial X} + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\lambda_{1}^{2} \frac{{\partial^{2} \psi_{x} }}{{\partial Y^{2} }}} \right)} \right] = \frac{{\partial^{2} U_{0} }}{{\partial T^{2} }} + \frac{{I_{1}^{*} }}{{\lambda_{2} }}\frac{{\partial^{2} \psi_{x} }}{{\partial T^{2} }}} \hfill \\ \end{array} } \right.$$
(17)
$$\left\{ {\begin{array}{*{20}l} {\lambda_{2}^{2} E_{1}^{*} \left[ \begin{gathered} \lambda_{1} \frac{{\partial^{2} V_{0} }}{{\partial Y^{2} }} + \lambda_{1}^{3} \frac{{\partial W_{0} }}{\partial Y}\frac{{\partial^{2} W_{0} }}{{\partial Y^{2} }} + \left( {\frac{1 + \upsilon }{2}} \right)\left( {\lambda_{1} \frac{{\partial^{2} U_{0} }}{\partial X\partial Y} + \lambda_{1} \frac{{\partial W_{0} }}{\partial X}\frac{{\partial^{2} W_{0} }}{\partial Y\partial X}} \right) \hfill \\ + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\frac{1}{{\lambda_{1} }}\frac{{\partial^{2} V_{0} }}{{\partial X^{2} }} + \lambda_{1} \frac{{\partial^{2} W_{0} }}{{\partial X^{2} }}\frac{{\partial W_{0} }}{\partial Y}} \right) \hfill \\ \end{gathered} \right]} \hfill \\ { + \lambda_{2} E_{2}^{*} \left[ {\lambda_{1}^{2} \frac{{\partial^{2} \psi_{y} }}{{\partial Y^{2} }} + \left( {\frac{1 + \upsilon }{2}} \right)\lambda_{1} \frac{{\partial^{2} \psi_{x} }}{\partial Y\partial X} + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\frac{{\partial^{2} \psi_{y} }}{{\partial X^{2} }}} \right)} \right] = \frac{1}{{\lambda_{1} }}\frac{{\partial^{2} V_{0} }}{{\partial T^{2} }} + \frac{{I_{1}^{*} }}{{\lambda_{2} }}\frac{{\partial^{2} \psi_{y} }}{{\partial T^{2} }}} \hfill \\ \end{array} } \right.$$
(18)
$$\left\{ {\begin{array}{*{20}l} {\lambda_{2}^{2} E_{1}^{*} \left[ {\kappa \left( {\frac{1 - \upsilon }{2}} \right)\left( {\frac{{\partial^{2} W_{0} }}{{\partial X^{2} }} + \frac{{\partial \psi_{x} }}{\partial X} + \lambda_{1}^{2} \frac{{\partial^{2} W_{0} }}{{\partial Y^{2} }} + \lambda_{1} \frac{{\partial \psi_{y} }}{\partial Y}} \right)} \right]} \hfill \\ { + \lambda_{2}^{2} E_{1}^{*} \left[ \begin{gathered} \frac{{\partial^{2} U_{0} }}{{\partial X^{2} }}\frac{{\partial W_{0} }}{\partial X} + \frac{{\partial U_{0} }}{\partial X}\frac{{\partial^{2} W_{0} }}{{\partial X^{2} }} + \upsilon \frac{{\partial V_{0} }}{\partial Y}\frac{{\partial^{2} W_{0} }}{{\partial X^{2} }} + \frac{3}{2}\left( {\frac{{\partial W_{0} }}{\partial X}} \right)^{2} \frac{{\partial^{2} W_{0} }}{{\partial X^{2} }} \hfill \\ + \lambda_{1}^{2} \left( \frac{1}{2} \right)\left( {\frac{{\partial W_{0} }}{\partial Y}} \right)^{2} \frac{{\partial^{2} W_{0} }}{{\partial X^{2} }} + \lambda_{1}^{2} \frac{{\partial^{2} V_{0} }}{{\partial Y^{2} }}\frac{{\partial W_{0} }}{\partial Y} + \lambda_{1}^{2} \frac{{\partial V_{0} }}{\partial Y}\frac{{\partial^{2} W_{0} }}{{\partial Y^{2} }} \hfill \\ + \lambda_{1}^{2} \upsilon \frac{{\partial U_{0} }}{\partial X}\frac{{\partial^{2} W_{0} }}{{\partial Y^{2} }} + \lambda_{1}^{4} \frac{3}{2}\left( {\frac{{\partial W_{0} }}{\partial Y}} \right)^{2} \frac{{\partial^{2} W_{0} }}{{\partial Y^{2} }} + \lambda_{1}^{2} \frac{1}{2}\left( {\frac{{\partial W_{0} }}{\partial X}} \right)^{2} \frac{{\partial^{2} W_{0} }}{{\partial Y^{2} }} \hfill \\ + \lambda_{1}^{2} 2\frac{{\partial W_{0} }}{\partial X}\frac{{\partial W_{0} }}{\partial Y}\frac{{\partial^{2} W_{0} }}{\partial X\partial Y} + \left( {\frac{1 + \upsilon }{2}} \right)\left( {\frac{{\partial^{2} V_{0} }}{\partial Y\partial X}\frac{{\partial W_{0} }}{\partial X} + \lambda_{1}^{2} \frac{{\partial^{2} U_{0} }}{\partial X\partial Y}\frac{{\partial W_{0} }}{\partial Y}} \right) \hfill \\ + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\frac{{\partial^{2} V_{0} }}{{\partial X^{2} }}\frac{{\partial W_{0} }}{\partial Y} + \lambda_{1}^{2} \frac{{\partial^{2} U_{0} }}{{\partial Y^{2} }}\frac{{\partial W_{0} }}{\partial X} + 2\lambda_{1}^{2} \frac{{\partial U_{0} }}{\partial Y}\frac{{\partial^{2} W_{0} }}{\partial X\partial Y} + 2\frac{{\partial V_{0} }}{\partial X}\frac{{\partial^{2} W_{0} }}{\partial X\partial Y}} \right) \hfill \\ \end{gathered} \right]} \hfill \\ { + \lambda_{2} E_{2}^{*} \left[ \begin{gathered} \frac{{\partial^{2} \psi_{x} }}{{\partial X^{2} }}\frac{{\partial W_{0} }}{\partial X} + \frac{{\partial \psi_{x} }}{\partial X}\frac{{\partial^{2} W_{0} }}{{\partial X^{2} }} + \lambda_{1} \upsilon \frac{{\partial \psi_{y} }}{\partial Y}\frac{{\partial^{2} W_{0} }}{{\partial X^{2} }} + \lambda_{1}^{3} \frac{{\partial^{2} \psi_{y} }}{{\partial Y^{2} }}\frac{{\partial W_{0} }}{\partial Y} \hfill \\ + \lambda_{1}^{3} \frac{{\partial \psi_{y} }}{\partial Y}\frac{{\partial^{2} W_{0} }}{{\partial Y^{2} }} + \upsilon \lambda_{1}^{2} \frac{{\partial \psi_{x} }}{\partial X}\frac{{\partial^{2} W_{0} }}{{\partial Y^{2} }} + \left( {\frac{1 + \upsilon }{2}} \right)\left( {\lambda_{1} \frac{{\partial^{2} \psi_{y} }}{\partial Y\partial X}\frac{{\partial W_{0} }}{\partial X} + \lambda_{1}^{2} \frac{{\partial^{2} \psi_{x} }}{\partial Y\partial X}\frac{{\partial W_{0} }}{\partial Y}} \right) \hfill \\ + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\lambda_{1} \frac{{\partial^{2} \psi_{y} }}{{\partial X^{2} }}\frac{{\partial W_{0} }}{\partial Y} + \lambda_{1}^{2} \frac{{\partial^{2} \psi_{x} }}{{\partial Y^{2} }}\frac{{\partial W_{0} }}{\partial X} + 2\lambda_{1}^{2} \frac{{\partial \psi_{x} }}{\partial Y}\frac{{\partial^{2} W_{0} }}{\partial X\partial Y} + 2\lambda_{1} \frac{{\partial \psi_{y} }}{\partial X}\frac{{\partial^{2} W_{0} }}{\partial X\partial Y}} \right) \hfill \\ \end{gathered} \right]} \hfill \\ { + \lambda_{1} \lambda_{2} \overline{{P_{0} }} \delta \left( {x - vt} \right)\delta \left( {x - e} \right) = \frac{{\partial^{2} W_{0} }}{{\partial T^{2} }}} \hfill \\ \end{array} } \right.$$
(19)
$$\left\{ {\begin{array}{*{20}l} {\lambda_{2} E_{2}^{*} \left[ \begin{gathered} \frac{{\partial^{2} U_{0} }}{{\partial X^{2} }} + \frac{{\partial^{2} W_{0} }}{{\partial X^{2} }}\frac{{\partial W_{0} }}{\partial X} + \left( {\frac{1 + \upsilon }{2}} \right)\left( {\frac{{\partial^{2} V_{0} }}{\partial Y\partial X} + \lambda_{1}^{2} \frac{{\partial^{2} W_{0} }}{\partial Y\partial X}\frac{{\partial W_{0} }}{\partial Y}} \right) \hfill \\ + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\lambda_{1}^{2} \frac{{\partial^{2} U_{0} }}{{\partial Y^{2} }} + \lambda_{1}^{2} \frac{{\partial W_{0} }}{\partial X}\frac{{\partial^{2} W_{0} }}{{\partial Y^{2} }}} \right) \hfill \\ \end{gathered} \right]} \hfill \\ { + \left[ {\frac{{\partial^{2} \psi_{x} }}{{\partial X^{2} }} + \left( {\frac{1 + \upsilon }{2}} \right)\lambda_{1} \frac{{\partial^{2} \psi_{y} }}{\partial Y\partial X} + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\lambda_{1}^{2} \frac{{\partial^{2} \psi_{x} }}{{\partial Y^{2} }}} \right)} \right]} \hfill \\ { - \lambda_{2}^{2} E_{1}^{*} \left[ {\left( {\frac{1 - \upsilon }{2}} \right)\kappa \left( {\frac{{\partial W_{0} }}{\partial X} + \psi_{x} } \right)} \right] = \frac{{I_{1}^{*} }}{{\lambda_{2} }}\frac{{\partial^{2} U_{0} }}{{\partial T^{2} }} + \frac{{I_{2}^{*} }}{{\lambda_{2}^{2} }}\frac{{\partial^{2} \psi_{x} }}{{\partial T^{2} }}} \hfill \\ \end{array} } \right.$$
(20)
$$\left\{ {\begin{array}{*{20}l} {\lambda_{2} E_{2}^{*} \left[ \begin{gathered} \lambda_{1} \frac{{\partial^{2} V_{0} }}{{\partial Y^{2} }} + \lambda_{1}^{3} \frac{{\partial W_{0} }}{\partial Y}\frac{{\partial^{2} W_{0} }}{{\partial Y^{2} }} + \left( {\frac{1 + \upsilon }{2}} \right)\left( {\lambda_{1} \frac{{\partial^{2} U_{0} }}{\partial Y\partial X} + \lambda_{1} \frac{{\partial^{2} W_{0} }}{\partial Y\partial X}\frac{{\partial W_{0} }}{\partial X}} \right) \hfill \\ + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\frac{1}{{\lambda_{1} }}\frac{{\partial^{2} V_{0} }}{{\partial X^{2} }} + \lambda_{1} \frac{{\partial^{2} W_{0} }}{{\partial X^{2} }}\frac{{\partial W_{0} }}{\partial Y}} \right) \hfill \\ \end{gathered} \right]} \hfill \\ { + \left[ {\lambda_{1}^{2} \frac{{\partial^{2} \psi_{y} }}{{\partial Y^{2} }} + \left( {\frac{1 + \upsilon }{2}} \right)\lambda_{1} \frac{{\partial^{2} \psi_{x} }}{\partial Y\partial X} + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\frac{{\partial^{2} \psi_{y} }}{{\partial X^{2} }}} \right)} \right]} \hfill \\ { - \lambda_{2}^{2} E_{1}^{*} \left[ {\left( {\frac{1 - \upsilon }{2}} \right)\kappa \left( {\lambda_{1} \frac{{\partial W_{0} }}{\partial Y} + \psi_{y} } \right)} \right] = \frac{{I_{1}^{*} }}{{\lambda_{1} \lambda_{2} }}\frac{{\partial^{2} V_{0} }}{{\partial T^{2} }} + \frac{{I_{2}^{*} }}{{\lambda_{2}^{2} }}\frac{{\partial^{2} \psi_{y} }}{{\partial T^{2} }}} \hfill \\ \end{array} } \right.$$
(21)
where:
$$\left\{ {\begin{array}{*{20}l} {\lambda_{2} E_{2}^{*} \left[ \begin{gathered} \lambda_{1} \frac{{\partial^{2} V_{0} }}{{\partial Y^{2} }} + \lambda_{1}^{3} \frac{{\partial W_{0} }}{\partial Y}\frac{{\partial^{2} W_{0} }}{{\partial Y^{2} }} + \left( {\frac{1 + \upsilon }{2}} \right)\left( {\lambda_{1} \frac{{\partial^{2} U_{0} }}{\partial Y\partial X} + \lambda_{1} \frac{{\partial^{2} W_{0} }}{\partial Y\partial X}\frac{{\partial W_{0} }}{\partial X}} \right) \hfill \\ + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\frac{1}{{\lambda_{1} }}\frac{{\partial^{2} V_{0} }}{{\partial X^{2} }} + \lambda_{1} \frac{{\partial^{2} W_{0} }}{{\partial X^{2} }}\frac{{\partial W_{0} }}{\partial Y}} \right) \hfill \\ \end{gathered} \right]} \hfill \\ { + \left[ {\lambda_{1}^{2} \frac{{\partial^{2} \psi_{y} }}{{\partial Y^{2} }} + \left( {\frac{1 + \upsilon }{2}} \right)\lambda_{1} \frac{{\partial^{2} \psi_{x} }}{\partial Y\partial X} + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\frac{{\partial^{2} \psi_{y} }}{{\partial X^{2} }}} \right)} \right]} \hfill \\ { - \lambda_{2}^{2} E_{1}^{*} \left[ {\left( {\frac{1 - \upsilon }{2}} \right)\kappa \left( {\lambda_{1} \frac{{\partial W_{0} }}{\partial Y} + \psi_{y} } \right)} \right] = \frac{{I_{1}^{*} }}{{\lambda_{1} \lambda_{2} }}\frac{{\partial^{2} V_{0} }}{{\partial T^{2} }} + \frac{{I_{2}^{*} }}{{\lambda_{2}^{2} }}\frac{{\partial^{2} \psi_{y} }}{{\partial T^{2} }}} \hfill \\ \end{array} } \right.$$
$$E_{1}^{*} = \frac{{\left( {E_{m} + \frac{{\left( {E_{c} - E_{m} } \right)}}{n + 1}} \right)}}{{\left[ {\frac{1}{12}E_{m} + \left( {E_{c} - E_{m} } \right)\left( {\frac{1}{n + 3} - \frac{1}{n + 2} + \frac{1}{{4\left( {n + 1} \right)}}} \right)} \right]}}$$
(22)
$$E_{2}^{*} = \frac{{\left( {E_{c} - E_{m} } \right)\left( {\frac{1}{n + 2} - \frac{1}{{2\left( {n + 1} \right)}}} \right)}}{{\left[ {\frac{1}{12}E_{m} + \left( {E_{c} - E_{m} } \right)\left( {\frac{1}{n + 3} - \frac{1}{n + 2} + \frac{1}{{4\left( {n + 1} \right)}}} \right)} \right]}}$$
$${{\lambda }_{1}=\frac{a}{b},\lambda }_{2}=\frac{a}{h}$$
$$I_{1}^{*} = \frac{{\left( {\rho_{c} - \rho_{m} } \right)\left( {\frac{1}{n + 2} - \frac{1}{{2\left( {n + 1} \right)}}} \right)}}{{\left( {\rho_{m} + \frac{{\rho_{m} }}{n + 1}} \right)}}$$
$$I_{2}^{*} = \frac{{\left[ {\frac{1}{12}\rho_{m} + \left( {\rho_{c} - \rho_{m} } \right)\left( {\frac{1}{n + 3} - \frac{1}{n + 2} + \frac{1}{{4\left( {n + 1} \right)}}} \right)} \right]}}{{\left( {\rho_{m} + \frac{{\rho_{m} }}{n + 1}} \right)}}$$
The boundary conditions are as the following (Eqs. 23 to 31):
$$N_{x} = \left\{ \begin{gathered} A_{xx} \left( {\frac{{\partial u_{0} }}{\partial x} + \upsilon \frac{{\partial v_{0} }}{\partial y} + \frac{1}{2}\left( {\frac{{\partial w_{0} }}{\partial x}} \right)^{2} + \left( {\frac{\upsilon }{2}} \right)\left( {\frac{{\partial w_{0} }}{\partial y}} \right)^{2} } \right) + B_{xx} \left( {\frac{{\partial \psi_{x} }}{\partial x} + \upsilon \frac{{\partial \psi_{y} }}{\partial y}} \right) = 0 \hfill \\ \to \lambda_{2}^{2} E_{1}^{*} \left( {\frac{{\partial U_{0} }}{\partial X} + \upsilon \frac{{\partial V_{0} }}{\partial Y} + \frac{1}{2}\left( {\frac{{\partial W_{0} }}{\partial X}} \right)^{2} + \lambda_{1}^{2} \left( {\frac{\upsilon }{2}} \right)\left( {\frac{{\partial W_{0} }}{\partial Y}} \right)^{2} } \right) + \lambda_{2} E_{2}^{*} \left( {\frac{{\partial \psi_{x} }}{\partial X} + \lambda_{1} \upsilon \frac{{\partial \psi_{y} }}{\partial Y}} \right) = 0 \hfill \\ \end{gathered} \right.$$
(23)
$$N_{y} = \left\{ \begin{gathered} \frac{1}{2}A_{xx} \left( {2\frac{{\partial v_{0} }}{\partial y} + 2\upsilon \frac{\partial u}{{\partial x}} + \left( {\frac{{\partial w_{0} }}{\partial y}} \right)^{2} + \upsilon \left( {\frac{{\partial w_{0} }}{\partial x}} \right)^{2} } \right) + B_{xx} \left( {\frac{{\partial \psi_{y} }}{\partial y} + \upsilon \frac{{\partial \psi_{x} }}{\partial x}} \right) = 0 \hfill \\ \to \frac{1}{2}\lambda_{2}^{2} E_{1}^{*} \left( {2\frac{{\partial V_{0} }}{\partial Y} + 2\upsilon \frac{{\partial U_{0} }}{\partial X} + \lambda_{1}^{2} \left( {\frac{{\partial W_{0} }}{\partial Y}} \right)^{2} + \upsilon \left( {\frac{{\partial W_{0} }}{\partial X}} \right)^{2} } \right) + \lambda_{2} E_{2}^{*} \left( {\lambda_{1} \frac{{\partial \psi_{y} }}{\partial Y} + \upsilon \frac{{\partial \psi_{x} }}{\partial X}} \right) = 0 \hfill \\ \end{gathered} \right.$$
(24)
$$N_{xy} = \left\{ \begin{gathered} A_{xx} \left( {\frac{1 - \upsilon }{2}} \right)\left( {\frac{{\partial u_{0} }}{\partial y} + \frac{{\partial v_{0} }}{\partial x} + \frac{{\partial w_{0} }}{\partial x}\frac{{\partial w_{0} }}{\partial y}} \right) + B_{xx} \left( {\frac{1 - \upsilon }{2}} \right)\left( {\frac{{\partial \psi_{x} }}{\partial y} + \frac{{\partial \psi_{y} }}{\partial x}} \right) = 0 \hfill \\ \to \lambda_{2}^{2} E_{1}^{*} \left( {\lambda_{1} \frac{{\partial U_{0} }}{\partial Y} + \frac{1}{{\lambda_{1} }}\frac{{\partial V_{0} }}{\partial X} + \lambda_{1} \frac{{\partial W_{0} }}{\partial X}\frac{{\partial W_{0} }}{\partial Y}} \right) + \lambda_{2} E_{2}^{*} \left( {\lambda_{1} \frac{{\partial \psi_{x} }}{\partial Y} + \frac{{\partial \psi_{y} }}{\partial X}} \right) = 0 \hfill \\ \end{gathered} \right.$$
(25)
$${\varvec{M}}_{{\varvec{x}}} = \left\{ {\begin{array}{*{20}c} {\frac{1}{2}B_{xx} \left( {2\frac{{\partial u_{0} }}{\partial x} + 2\upsilon \frac{{\partial v_{0} }}{\partial y} + \upsilon \left( {\frac{{\partial w_{0} }}{\partial y}} \right)^{2} + \left( {\frac{\partial w}{{\partial x}}} \right)^{2} } \right) + C_{xx} \left( {\frac{{\partial \psi_{x} }}{\partial x} + \upsilon \frac{{\partial \psi_{y} }}{\partial y}} \right) = 0} \\ { \to \frac{1}{2}\lambda_{2} E_{2}^{*} \left( {2\frac{{\partial U_{0} }}{\partial X} + 2\upsilon \frac{{\partial V_{0} }}{\partial Y} + \lambda_{1}^{2} \upsilon \left( {\frac{{\partial W_{0} }}{\partial Y}} \right)^{2} + \left( {\frac{{\partial W_{0} }}{\partial X}} \right)^{2} } \right) + \left( {\frac{{\partial \psi_{x} }}{\partial X} + \lambda_{1} \upsilon \frac{{\partial \psi_{y} }}{\partial Y}} \right) = 0} \\ \end{array} } \right.$$
(26)
$${\varvec{M}}_{{\varvec{y}}} = \left\{ \begin{gathered} \frac{1}{2}B_{xx} \left( {2\upsilon \frac{{\partial u_{0} }}{\partial x} + 2\frac{{\partial v_{0} }}{\partial y} + \upsilon \left( {\frac{{\partial w_{0} }}{\partial x}} \right)^{2} + \left( {\frac{{\partial w_{0} }}{\partial y}} \right)^{2} } \right) + C_{xx} \left( {\upsilon \frac{{\partial \psi_{x} }}{\partial x} + \frac{{\partial \psi_{y} }}{\partial y}} \right) = 0 \hfill \\ \to \frac{1}{2}\lambda_{2} E_{2}^{*} \left( {2\upsilon \frac{{\partial U_{0} }}{\partial X} + 2\frac{{\partial V_{0} }}{\partial Y} + \upsilon \left( {\frac{{\partial W_{0} }}{\partial X}} \right)^{2} + \lambda_{1}^{2} \left( {\frac{{\partial W_{0} }}{\partial Y}} \right)^{2} } \right) + \left( {\upsilon \frac{{\partial \psi_{x} }}{\partial X} + \lambda_{1} \frac{{\partial \psi_{y} }}{\partial Y}} \right) = 0 \hfill \\ \end{gathered} \right.$$
(27)
$${\varvec{M}}_{{{\varvec{xy}}}} = \left\{ \begin{gathered} B_{xx} \left( {\frac{1 - \upsilon }{2}} \right)\left( {\frac{{\partial u_{0} }}{\partial y} + \frac{{\partial v_{0} }}{\partial x} + \frac{{\partial w_{0} }}{\partial x}\frac{{\partial w_{0} }}{\partial y}} \right) + C_{xx} \left( {\frac{1 - \upsilon }{2}} \right)\left( {\frac{{\partial \psi_{x} }}{\partial y} + \frac{{\partial \psi_{y} }}{\partial x}} \right) = 0 \hfill \\ \to \lambda_{2} E_{2}^{*} \left( {\lambda_{1} \frac{{\partial U_{0} }}{\partial Y} + \frac{1}{{\lambda_{1} }}\frac{{\partial V_{0} }}{\partial X} + \lambda_{1} \frac{{\partial W_{0} }}{\partial X}\frac{{\partial W_{0} }}{\partial Y}} \right) + \left( {\lambda_{1} \frac{{\partial \psi_{x} }}{\partial Y} + \frac{{\partial \psi_{y} }}{\partial X}} \right) = 0 \hfill \\ \end{gathered} \right.$$
(28)
$$\left\{ {\begin{array}{*{20}l} {{\varvec{\kappa}}\left( {\frac{1 - \upsilon }{2}} \right)A_{xx} \left( {\frac{{\partial w_{0} }}{\partial x} + \psi_{x} } \right) + A_{xx} \left( {\frac{{\partial u_{0} }}{\partial x} + \upsilon \frac{{\partial v_{0} }}{\partial y} + \frac{1}{2}\left( {\frac{{\partial w_{0} }}{\partial x}} \right)^{2} + \left( {\frac{\upsilon }{2}} \right)\left( {\frac{{\partial w_{0} }}{\partial y}} \right)^{2} } \right)\frac{{\partial w_{0} }}{\partial x}} \hfill \\ { + B_{xx} \left( {\frac{{\partial \psi_{x} }}{\partial x} + \upsilon \frac{{\partial \psi_{y} }}{\partial y}} \right)\frac{{\partial w_{0} }}{\partial x} + \frac{1 - \upsilon }{2}A_{xx} \left( {\frac{{\partial u_{0} }}{\partial y} + \frac{{\partial v_{0} }}{\partial x} + \frac{{\partial w_{0} }}{\partial x}\frac{{\partial w_{0} }}{\partial y}} \right)\frac{{\partial w_{0} }}{\partial y}} \hfill \\ { + B_{xx} \left( {\frac{1 - \upsilon }{2}} \right)\left( {\frac{{\partial \psi_{x} }}{\partial y} + \frac{{\partial \psi_{y} }}{\partial x}} \right)\frac{{\partial w_{0} }}{\partial y} = 0} \hfill \\ { \to {\varvec{\kappa}}\left( {\frac{1 - \upsilon }{2}} \right)\lambda_{2}^{2} E_{1}^{*} \left( {\frac{{\partial W_{0} }}{\partial X} + \psi_{x} } \right)} \hfill \\ { + \lambda_{2}^{2} E_{1}^{*} \left( \begin{gathered} \frac{{\partial U_{0} }}{\partial X}\frac{{\partial W_{0} }}{\partial X} + \upsilon \frac{{\partial V_{0} }}{\partial Y}\frac{{\partial W_{0} }}{\partial X} + \frac{1}{2}\left( {\frac{{\partial W_{0} }}{\partial X}} \right)^{2} \frac{{\partial W_{0} }}{\partial X} + \lambda_{1}^{2} \left( \frac{1}{2} \right)\left( {\frac{{\partial W_{0} }}{\partial Y}} \right)^{2} \frac{{\partial W_{0} }}{\partial X} \hfill \\ + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\lambda_{1}^{2} \frac{{\partial U_{0} }}{\partial Y}\frac{{\partial W_{0} }}{\partial Y} + \frac{{\partial V_{0} }}{\partial X}\frac{{\partial W_{0} }}{\partial Y}} \right) \hfill \\ \end{gathered} \right)} \hfill \\ { + \lambda_{2} E_{2}^{*} \left( {\frac{{\partial \psi_{x} }}{\partial X}\frac{{\partial W_{0} }}{\partial X} + \upsilon \lambda_{1} \frac{{\partial \psi_{y} }}{\partial Y}\frac{{\partial W_{0} }}{\partial X} + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\lambda_{1}^{2} \frac{{\partial \psi_{x} }}{\partial Y}\frac{{\partial W_{0} }}{\partial Y} + \lambda_{1} \frac{{\partial \psi_{y} }}{\partial X}\frac{{\partial W_{0} }}{\partial Y}} \right)} \right) = 0} \hfill \\ \end{array} } \right.$$
(29)
$$\left\{ {\begin{array}{*{20}l} {{\varvec{\kappa}}\left( {\frac{1 - \upsilon }{2}} \right)A_{xx} \left( {\frac{{\partial w_{0} }}{\partial y} + \psi_{y} } \right) + \frac{1}{2}A_{xx} \left( {2\frac{{\partial v_{0} }}{\partial y}\frac{{\partial w_{0} }}{\partial y} + 2\upsilon \frac{\partial u}{{\partial x}}\frac{{\partial w_{0} }}{\partial y} + \left( {\frac{{\partial w_{0} }}{\partial y}} \right)^{2} \frac{{\partial w_{0} }}{\partial y} + \upsilon \left( {\frac{{\partial w_{0} }}{\partial x}} \right)^{2} \frac{{\partial w_{0} }}{\partial y}} \right)} \hfill \\ { + B_{xx} \left( {\frac{{\partial \psi_{y} }}{\partial y}\frac{{\partial w_{0} }}{\partial y} + \upsilon \frac{{\partial \psi_{x} }}{\partial x}\frac{{\partial w_{0} }}{\partial y}} \right) + \frac{1 - \upsilon }{2}A_{xx} \left( {\frac{{\partial u_{0} }}{\partial y}\frac{{\partial w_{0} }}{\partial x} + \frac{{\partial v_{0} }}{\partial x}\frac{{\partial w_{0} }}{\partial x} + \frac{{\partial w_{0} }}{\partial x}\frac{{\partial w_{0} }}{\partial y}\frac{{\partial w_{0} }}{\partial x}} \right)} \hfill \\ { + B_{xx} \left( {\frac{1 - \upsilon }{2}} \right)\left( {\frac{{\partial \psi_{x} }}{\partial y}\frac{{\partial w_{0} }}{\partial x} + \frac{{\partial \psi_{y} }}{\partial x}\frac{{\partial w_{0} }}{\partial x}} \right) = 0} \hfill \\ { \to {\varvec{\kappa}}\left( {\frac{1 - \upsilon }{2}} \right)\lambda_{2}^{2} E_{1}^{*} \left( {\lambda_{1} \frac{{\partial W_{0} }}{\partial Y} + \psi_{y} } \right)} \hfill \\ { + \lambda_{2}^{2} E_{1}^{*} \left( \begin{gathered} \frac{{\lambda_{1} }}{1}\frac{{\partial V_{0} }}{\partial Y}\frac{{\partial W_{0} }}{\partial Y} + \upsilon \lambda_{1} \frac{\partial U}{{\partial X}}\frac{{\partial W_{0} }}{\partial Y} + \frac{1}{2}\lambda_{1}^{3} \left( {\frac{{\partial W_{0} }}{\partial Y}} \right)^{2} \frac{{\partial W_{0} }}{\partial Y} + \frac{1}{2}\lambda_{1} \left( {\frac{{\partial W_{0} }}{\partial X}} \right)^{2} \frac{{\partial W_{0} }}{\partial Y} \hfill \\ + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\lambda_{1} \frac{{\partial U_{0} }}{\partial Y}\frac{{\partial W_{0} }}{\partial X} + \frac{1}{{\lambda_{1} }}\frac{{\partial V_{0} }}{\partial X}\frac{{\partial W_{0} }}{\partial X}} \right) \hfill \\ \end{gathered} \right)} \hfill \\ { + \lambda_{2} E_{2}^{*} \left( {\lambda_{1}^{2} \frac{{\partial \psi_{y} }}{\partial Y}\frac{{\partial W_{0} }}{\partial Y} + \lambda_{1} \upsilon \frac{{\partial \psi_{x} }}{\partial X}\frac{{\partial W_{0} }}{\partial Y} + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\lambda_{1} \frac{{\partial \psi_{x} }}{\partial Y}\frac{{\partial W_{0} }}{\partial X} + \frac{{\partial \psi_{y} }}{\partial X}\frac{{\partial W_{0} }}{\partial X}} \right)} \right) = 0} \hfill \\ \end{array} } \right.$$
(30)
where
$$\left\{ \begin{gathered} A_{xx} = \frac{h}{{1 - \upsilon^{2} }}\left( {E_{m} + \frac{{E_{cm} }}{n + 1}} \right) \hfill \\ B_{xx} = \frac{{h^{2} }}{{1 - \upsilon^{2} }}E_{cm} \left( {\frac{1}{n + 2} - \frac{1}{{2\left( {n + 1} \right)}}} \right) \hfill \\ C_{xx} = \frac{{h^{3} }}{{1 - \upsilon^{2} }}\left[ {\frac{1}{12}E_{m} + E_{cm} \left( {\frac{1}{n + 3} - \frac{1}{n + 2} + \frac{1}{{4\left( {n + 1} \right)}}} \right)} \right] \hfill \\ \end{gathered} \right.$$
2.3 Employing GDQ method:
Since the GDQ method is concerned, this approach discretizes the spatial derivatives of a function \(f(z,t)\) as a weighted linear sum of the functional values at all nodes in the solution domain, by means of some fixed weighting coefficients. Thus, the first- and second-order derivatives of a one-dimensional function, read as follows
$$\begin{aligned} \left. {\frac{{\partial f\left( {z,t} \right)}}{{\partial z}}} \right|_{{z = z_{i} }} & = \mathop \sum \limits_{{j = 1}}^{{N_{z} }} A_{{ij}}^{z} f\left( {z_{j} ,t} \right) = \mathop \sum \limits_{{j = 1}}^{{N_{z} }} A_{{ij}}^{z} f_{j} \left( t \right) \\ \left. {\frac{{\partial ^{2} f\left( {z,t} \right)}}{{\partial z^{2} }}} \right|_{{z = z_{i} }} & = \mathop \sum \limits_{{j = 1}}^{{N_{z} }} B_{{ij}}^{z} f\left( {z_{j} ,t} \right) = \mathop \sum \limits_{{j = 1}}^{{N_{z} }} B_{{ij}}^{z} f_{j} \left( t \right) \\ \end{aligned}$$
(32)
where \(A_{ij}^{z}\), \(B_{ij}^{z}\), are the weighted coefficients at the grid nodes of solution domain. To derive the weighting coefficients, the following relations are employed
$$A_{ij}^{z} = \left\{ {\begin{array}{*{20}c} {\frac{{M\left( {z_{i} } \right)}}{{\left( {z_{i} - z_{j} } \right)M\left( {z_{i} } \right)}}\quad {\text{for}}\quad i \ne j} \\ { - \mathop \sum \limits_{k = 1,k \ne i}^{{N_{z} }} A_{ik}^{z} \quad {\text{for}}\quad i = j} \\ \end{array} } \right.\quad i,j = 1,2, \ldots ,N_{z} ,$$
(33)
$$B_{ij}^{z} = \left\{ {\begin{array}{*{20}c} {2\left[ {A_{ii}^{z} A_{ij}^{z} - \frac{{A_{ij}^{z} }}{{z_{i} - z_{j} }}} \right]\quad {\text{for}}\quad i \ne j,} \\ { - \mathop \sum \limits_{k = 1,k \ne i}^{{N_{z} }} B_{ik}^{z} \quad {\text{for}}\quad i = j} \\ \end{array} } \right.\quad i,j = 1,2, \ldots ,N_{z} ,$$
(34)
being \(M^{\left( 1 \right)} \left( {z_{i} } \right) = \mathop \prod \limits_{j = 1,j \ne i}^{N} \left( {z_{i} - z_{j} } \right) {\text{for}}\,i = 1,2, \ldots ,N.\)
To obtain more accurate results, a Chebyshev–Gauss–Lobatto quadrature-mesh size is here assumed, in line with findings by Malik and Bert [36] (see Fig. 2)
$$z_{i} = \frac{h}{2}\left[ {1 - \cos \left[ {\frac{{\left( {i - 1} \right)\pi }}{{\left( {N_{z} - 1} \right)}}} \right]} \right].$$
(35)
By applying GDQ method [36,37,38] to the boundary conditions and governing equations, we have:
$$\left\{ \begin{gathered} \lambda_{2}^{2} E_{1}^{*} \left[ \begin{gathered} \mathop \sum \limits_{n = 1}^{N} A_{in}^{\left( 2 \right)} U_{0 nj} + \mathop \sum \limits_{n = 1}^{N} A_{in}^{\left( 2 \right)} W_{0 nj} \mathop \sum \limits_{n = 1}^{N} A_{in}^{\left( 1 \right)} W_{0 nj} \hfill \\ + \left( {\frac{1 + \upsilon }{2}} \right)\left( {\mathop \sum \limits_{n = 1}^{N} \mathop \sum \limits_{m = 1}^{M} A_{in}^{\left( 1 \right)} B_{jm}^{\left( 1 \right)} V_{0 nm} + \lambda_{1}^{2} \mathop \sum \limits_{n = 1}^{N} \mathop \sum \limits_{m = 1}^{M} A_{in}^{\left( 1 \right)} B_{jm}^{\left( 1 \right)} W_{0 nm} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{\left( 1 \right)} W_{0 im} } \right) \hfill \\ + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\lambda_{1}^{2} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{\left( 2 \right)} U_{0 im} + \lambda_{1}^{2} \mathop \sum \limits_{n = 1}^{N} A_{in}^{\left( 1 \right)} W_{0 nj} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{\left( 2 \right)} W_{0 im} } \right) \hfill \\ \end{gathered} \right] \hfill \\ + \lambda_{2} E_{2}^{*} \left[ \begin{gathered} \mathop \sum \limits_{n = 1}^{N} A_{in}^{\left( 2 \right)} \psi_{x nj} + \left( {\frac{1 + \upsilon }{2}} \right)\lambda_{1} \mathop \sum \limits_{n = 1}^{N} \mathop \sum \limits_{m = 1}^{M} A_{in}^{\left( 1 \right)} B_{jm}^{\left( 1 \right)} \psi_{y nm} \hfill \\ + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\lambda_{1}^{2} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{\left( 2 \right)} \psi_{x im} } \right) \hfill \\ \end{gathered} \right] = \frac{{\partial^{2} U_{0} }}{{\partial T^{2} }} + \frac{{I_{1}^{*} }}{{\lambda_{2} }}\frac{{\partial^{2} \psi_{x} }}{{\partial T^{2} }} \hfill \\ \end{gathered} \right.$$
(36)
$$\left\{ \begin{gathered} \lambda_{2}^{2} E_{1}^{*} \left[ \begin{gathered} \lambda_{1} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{\left( 2 \right)} V_{0 im} + \lambda_{1}^{3} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{\left( 1 \right)} W_{0 im} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{\left( 2 \right)} W_{0 im} \hfill \\ + \left( {\frac{1 + \upsilon }{2}} \right)\left( {\lambda_{1} \mathop \sum \limits_{n = 1}^{N} \mathop \sum \limits_{m = 1}^{M} A_{in}^{(1)} B_{jm}^{\left( 1 \right)} U_{0 nm} + \lambda_{1} \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} \mathop \sum \limits_{n = 1}^{N} \mathop \sum \limits_{m = 1}^{M} A_{in}^{(1)} B_{jm}^{\left( 1 \right)} W_{0 nm} } \right) \hfill \\ + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\frac{1}{{\lambda_{1} }}\mathop \sum \limits_{n = 1}^{N} A_{in}^{(2)} V_{0 nj} + \lambda_{1} \mathop \sum \limits_{n = 1}^{N} A_{in}^{(2)} W_{0 nj} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{\left( 1 \right)} W_{0 im} } \right) \hfill \\ \end{gathered} \right] \hfill \\ + \lambda_{2} E_{2}^{*} \left[ \begin{gathered} \lambda_{1}^{2} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{\left( 2 \right)} \psi_{y im} + \left( {\frac{1 + \upsilon }{2}} \right)\lambda_{1} \mathop \sum \limits_{n = 1}^{N} \mathop \sum \limits_{m = 1}^{M} A_{in}^{(1)} B_{jm}^{\left( 1 \right)} \psi_{x nm} \hfill \\ + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\mathop \sum \limits_{n = 1}^{N} A_{in}^{(2)} \psi_{y nj} } \right) \hfill \\ \end{gathered} \right] = \frac{1}{{\lambda_{1} }}\frac{{\partial^{2} V_{0} }}{{\partial T^{2} }} + \frac{{I_{1}^{*} }}{{\lambda_{2} }}\frac{{\partial^{2} \psi_{y} }}{{\partial T^{2} }} \hfill \\ \end{gathered} \right.$$
(37)
$$\left\{ \begin{gathered} \lambda _{2}^{2} E_{1}^{*} \left[ {\kappa \left( {\frac{{1 - \upsilon }}{2}} \right)\left( {\mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(2)}} W_{{0~nj}} + \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(1)}} \psi _{{x~nj}} + \lambda _{1}^{2} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(2)}} W_{{0~im}} + \lambda _{1} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(1)}} \psi _{{y~im}} } \right)} \right] \hfill \\ + \lambda _{2}^{2} E_{1}^{*} \left[ \begin{gathered} \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(2)}} U_{{0~nj}} \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(1)}} W_{{0~nj}} + \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(1)}} U_{{0~nj}} \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(2)}} W_{{0~nj}} + \upsilon \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(1)}} V_{{0~im}} \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(2)}} W_{{0~nj}} \hfill \\ + \frac{3}{2}\left( {\mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(1)}} W_{{0~nj}} } \right)^{2} \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(2)}} W_{{0~nj}} + \lambda _{1}^{2} \left( {\frac{1}{2}} \right)\left( {\mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(1)}} W_{{0~im}} } \right)^{2} \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(2)}} W_{{0~nj}} \hfill \\ + \lambda _{1}^{2} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(2)}} V_{{0~im}} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(1)}} W_{{0~im}} + \lambda _{1}^{2} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(1)}} V_{{0~im}} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(2)}} W_{{0~im}} \hfill \\ + \lambda _{1}^{2} \upsilon \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(1)}} U_{{0~nj}} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(2)}} W_{{0~im}} + \lambda _{1}^{4} \frac{3}{2}\left( {\mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(1)}} W_{{0~im}} } \right)^{2} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(2)}} W_{{0~im}} \hfill \\ + \lambda _{1}^{2} \frac{1}{2}\left( {\mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(1)}} W_{{0~nj}} } \right)^{2} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(2)}} W_{{0~im}} + \lambda _{1}^{2} 2\mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(1)}} W_{{0~nj}} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(1)}} W_{{0~im}} \mathop \sum \limits_{{n = 1}}^{N} \mathop \sum \limits_{{m = 1}}^{M} A_{{in}}^{{(1)}} B_{{jm}}^{{(1)}} W_{{0~nm}} \hfill \\ + \left( {\frac{{1 + \upsilon }}{2}} \right)\left( {\mathop \sum \limits_{{n = 1}}^{N} \mathop \sum \limits_{{m = 1}}^{M} A_{{in}}^{{(1)}} B_{{jm}}^{{(1)}} V_{{0~nm}} \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(1)}} W_{{0~nj}} + \lambda _{1}^{2} \mathop \sum \limits_{{n = 1}}^{N} \mathop \sum \limits_{{m = 1}}^{M} A_{{in}}^{{(1)}} B_{{jm}}^{{(1)}} U_{{0~nm}} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(1)}} W_{{0~im}} } \right) \hfill \\ + \left( {\frac{{1 - \upsilon }}{2}} \right)\left( \begin{gathered} \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(2)}} V_{{0~nj}} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(1)}} W_{{0~im}} + \lambda _{1}^{2} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(2)}} U_{{0~im}} \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(1)}} W_{{0~nj}} \hfill \\ + 2\lambda _{1}^{2} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(1)}} U_{{0~im}} \mathop \sum \limits_{{n = 1}}^{N} \mathop \sum \limits_{{m = 1}}^{M} A_{{in}}^{{(1)}} B_{{jm}}^{{(1)}} W_{{0~nm}} + 2\mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(1)}} V_{{0~nj}} \mathop \sum \limits_{{n = 1}}^{N} \mathop \sum \limits_{{m = 1}}^{M} A_{{in}}^{{(1)}} B_{{jm}}^{{(1)}} W_{{0~nm}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} \right] \hfill \\ + \lambda _{2} E_{2}^{*} \left[ \begin{gathered} \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(2)}} \psi _{{x~nj}} \mathop \sum \limits_{{n = 1}}^{N} A_{{in}} ^{{\left( 1 \right)}} W_{{0~nj}} + \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(1)}} \psi _{{x~nj}} \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(2)}} W_{{0~nj}} + \hfill \\ \lambda _{1} \upsilon \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(1)}} \psi _{{y~im}} \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(2)}} W_{{0~nj}} \hfill \\ + \lambda _{1} ^{3} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(2)}} \psi _{{y~im}} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(1)}} W_{{0~im}} + \lambda _{1} ^{3} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(1)}} \psi _{{y~im}} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(2)}} W_{{0~im}} \hfill \\ + \upsilon \lambda _{1} ^{2} \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(1)}} \psi _{{x~nj}} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(2)}} W_{{0~im}} \hfill \\ + \left( {\frac{{1 + \upsilon }}{2}} \right)\left( \begin{gathered} \lambda _{1} \mathop \sum \limits_{{n = 1}}^{N} \mathop \sum \limits_{{m = 1}}^{M} A_{{in}}^{{(1)}} B_{{jm}}^{{(1)}} \psi _{{y~nm}} \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(1)}} W_{{0~nj}} \hfill \\ + \lambda _{1} ^{2} \mathop \sum \limits_{{n = 1}}^{N} \mathop \sum \limits_{{m = 1}}^{M} A_{{in}}^{{(1)}} B_{{jm}}^{{(1)}} \psi _{{x~nm}} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(1)}} W_{{0~im}} \hfill \\ \end{gathered} \right) \hfill \\ + \left( {\frac{{1 - \upsilon }}{2}} \right)\left( \begin{gathered} \lambda _{1} \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(2)}} \psi _{{y~nj}} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(1)}} W_{{0~im}} + \lambda _{1}^{2} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(2)}} \psi _{{x~im}} \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(1)}} W_{{0~nj}} \hfill \\ + 2\lambda _{1}^{2} \mathop \sum \limits_{{m = 1}}^{M} B_{{jm}}^{{(1)}} \psi _{{x~im}} \mathop \sum \limits_{{n = 1}}^{N} \mathop \sum \limits_{{m = 1}}^{M} A_{{in}}^{{(1)}} B_{{jm}}^{{(1)}} W_{{0~nm}} + \hfill \\ 2\lambda _{1} \mathop \sum \limits_{{n = 1}}^{N} A_{{in}}^{{(1)}} \psi _{{y~nj}} \mathop \sum \limits_{{n = 1}}^{N} \mathop \sum \limits_{{m = 1}}^{M} A_{{in}}^{{(1)}} B_{{jm}}^{{(1)}} W_{{0~nm}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} \right] \hfill \\ + \lambda _{1} \lambda _{2}^{2} \overline{{P_{{ij}} }} \Delta x_{i} \Delta y_{j} = \frac{{\partial ^{2} W_{0} }}{{\partial T^{2} }} \hfill \\ \end{gathered} \right.$$
(38)
$$\left\{ {\begin{array}{*{20}l} {\lambda_{2} E_{2}^{*} \left[ \begin{gathered} \mathop \sum \limits_{n = 1}^{N} A_{in}^{\left( 2 \right)} U_{0 nj} + \mathop \sum \limits_{n = 1}^{N} A_{in}^{\left( 2 \right)} W_{0 nj} \mathop \sum \limits_{n = 1}^{N} A_{in}^{\left( 1 \right)} W_{0 nj} \hfill \\ + \left( {\frac{1 + \upsilon }{2}} \right)\left( {\mathop \sum \limits_{n = 1}^{N} \mathop \sum \limits_{m = 1}^{M} A_{in}^{\left( 1 \right)} B_{jm}^{(1)} V_{0 nm} + \lambda_{1}^{2} \mathop \sum \limits_{n = 1}^{N} \mathop \sum \limits_{m = 1}^{M} A_{in}^{\left( 1 \right)} B_{jm}^{(1)} W_{0 nm} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} } \right) \hfill \\ + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\lambda_{1}^{2} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(2)} U_{0 im} + \lambda_{2}^{2} \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(2)} W_{0 im} } \right) \hfill \\ \end{gathered} \right]} \hfill \\ { + \left[ {\mathop \sum \limits_{n = 1}^{N} A_{in}^{\left( 2 \right)} \psi_{x nj} + \left( {\frac{1 + \upsilon }{2}} \right)\lambda_{1} \mathop \sum \limits_{n = 1}^{N} \mathop \sum \limits_{m = 1}^{M} A_{in}^{(1)} B_{jm}^{(1)} \psi_{y nm} + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\lambda_{1}^{2} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(2)} \psi_{x im} } \right)} \right]} \hfill \\ { - \left[ {\left( {\frac{1 - \upsilon }{2}} \right)\lambda_{2}^{2} E_{1}^{*} \kappa \left( {\mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} + \psi_{x} } \right)} \right] = \frac{{I_{1}^{*} }}{{\lambda_{2} }}\frac{{\partial^{2} U_{0} }}{{\partial T^{2} }} + \frac{{I_{2}^{*} }}{{\lambda_{2}^{2} }}\frac{{\partial^{2} \psi_{x} }}{{\partial T^{2} }}} \hfill \\ \end{array} } \right.$$
(39)
$$\left\{ \begin{gathered} \lambda_{2} E_{2}^{*} \left[ \begin{gathered} \lambda_{1} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(2)} V_{0 im} + \lambda_{1}^{3} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(2)} W_{0 im} \hfill \\ + \left( {\frac{1 + \upsilon }{2}} \right)\left( {\lambda_{1} \mathop \sum \limits_{n = 1}^{N} \mathop \sum \limits_{m = 1}^{M} A_{in}^{(1)} B_{jm}^{(1)} U_{0 nm} + \lambda_{1} \mathop \sum \limits_{n = 1}^{N} \mathop \sum \limits_{m = 1}^{M} A_{in}^{(1)} B_{jm}^{(1)} W_{0 nm} \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} } \right) \hfill \\ + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\frac{1}{{\lambda_{1} }}\mathop \sum \limits_{n = 1}^{N} A_{in}^{(2)} V_{0 nj} + \lambda_{1} \mathop \sum \limits_{n = 1}^{N} A_{in}^{(2)} W_{0 nj} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} } \right) \hfill \\ \end{gathered} \right] \hfill \\ + \left[ {\lambda_{1}^{2} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(2)} \psi_{y im} + \left( {\frac{1 + \upsilon }{2}} \right)\lambda_{1} \mathop \sum \limits_{n = 1}^{N} \mathop \sum \limits_{m = 1}^{M} A_{in}^{(1)} B_{jm}^{(1)} \psi_{x nm} + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\mathop \sum \limits_{n = 1}^{N} A_{in}^{(2)} \psi_{y nj} } \right)} \right] \hfill \\ - \lambda_{2}^{2} E_{1}^{*} \left[ {\left( {\frac{1 - \upsilon }{2}} \right)\kappa \left( {\lambda_{1} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} + \psi_{y} } \right)} \right] = \frac{{I_{1}^{*} }}{{\lambda_{1} \lambda_{2} }}\frac{{\partial^{2} V_{0} }}{{\partial T^{2} }} + \frac{{I_{2}^{*} }}{{\lambda_{2}^{2} }}\frac{{\partial^{2} \psi_{y} }}{{\partial T^{2} }} \hfill \\ \end{gathered} \right.$$
(40)
Also, the boundary condition can be discretized by using GDQM as following:
$$N_{x} = \left\{ \begin{gathered} \lambda_{2}^{2} E_{1}^{*} \left( {\mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} U_{0 nj} + \upsilon \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} V_{0 im} + \frac{1}{2}\left( {\mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} } \right)^{2} + \lambda_{1}^{2} \left( {\frac{\upsilon }{2}} \right)\left( {\mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} } \right)^{2} } \right) \hfill \\ + \lambda_{2} E_{2}^{*} \left( {\mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} \psi_{x nj} + \lambda_{1} \upsilon \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} \psi_{y im} } \right) = 0 \hfill \\ \end{gathered} \right.$$
(41)
$${\varvec{N}}_{{\varvec{y}}} = \left\{ \begin{gathered} \frac{1}{2}\lambda_{2}^{2} E_{1}^{*} \left( {2\mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} V_{0 im} + 2\upsilon \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} U_{0 nj} + \lambda_{1}^{2} \left( {\mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} } \right)^{2} + \upsilon \left( {\mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} } \right)^{2} } \right) \hfill \\ + \lambda_{2} E_{2}^{*} \left( {\lambda_{1} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} \psi_{y im} + \upsilon \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} \psi_{x nj} } \right) = 0 \hfill \\ \end{gathered} \right.$$
(42)
$${\varvec{N}}_{{{\varvec{xy}}}} = \left\{ \begin{gathered} \lambda_{2}^{2} E_{1}^{*} \left( {\lambda_{1} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{\left( 1 \right)} U_{0 im} + \frac{1}{{\lambda_{1} }}\mathop \sum \limits_{n = 1}^{N} A_{in}^{\left( 1 \right)} V_{0 nj} + \lambda_{1} \mathop \sum \limits_{n = 1}^{N} A_{in}^{\left( 1 \right)} W_{0 nj} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{\left( 1 \right)} W_{0 im} } \right) \hfill \\ + \lambda_{2} E_{2}^{*} \left( {\lambda_{1} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{\left( 1 \right)} \psi_{x im} + \mathop \sum \limits_{n = 1}^{N} A_{in}^{\left( 1 \right)} \psi_{y nj} } \right) = 0 \hfill \\ \end{gathered} \right.$$
(43)
$${\varvec{M}}_{{\varvec{x}}} = \left\{ \begin{gathered} \frac{1}{2}\lambda_{2} E_{2}^{*} \left( {2\mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} U_{0 nj} + 2\upsilon \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} V_{0 im} + \lambda_{1}^{2} \upsilon \left( {\mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} } \right)^{2} + \left( {\mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} } \right)^{2} } \right) \hfill \\ + \left( {\mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} \psi_{x nj} + \lambda_{1} \upsilon \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} \psi_{y im} } \right) = 0 \hfill \\ \end{gathered} \right.$$
(44)
$${\varvec{M}}_{{\varvec{y}}} = \left\{ \begin{gathered} \frac{1}{2}\lambda_{2} E_{2}^{*} \left( {2\upsilon \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} U_{0 nj} + 2\mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} V_{0 im} + \upsilon \left( {\mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} } \right)^{2} + \lambda_{1}^{2} \left( {\mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} } \right)^{2} } \right) \hfill \\ + \left( {\upsilon \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} \psi_{x nj} + \lambda_{1} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} \psi_{y im} } \right) = 0 \hfill \\ \end{gathered} \right.$$
(45)
$${\varvec{M}}_{{{\varvec{xy}}}} = \left\{ \begin{gathered} \lambda_{2} E_{2}^{*} \left( {\lambda_{1} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} U_{0 im} + \frac{1}{{\lambda_{1} }}\mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} V_{0 nj} + \lambda_{1} \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} } \right) \hfill \\ + \left( {\lambda_{1} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} \psi_{x im} + \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} \psi_{y nj} } \right) = 0 \hfill \\ \end{gathered} \right.$$
(46)
$$\left\{ \begin{gathered} \kappa \left( {\frac{1 - \upsilon }{2}} \right)\lambda_{2}^{2} E_{1}^{*} \left( {\mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} + \psi_{x} } \right) \hfill \\ + \lambda_{2}^{2} E_{1}^{*} \left( \begin{gathered} \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} U_{0 nj} \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} + \upsilon \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} V_{0 im} \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} \hfill \\ + \frac{1}{2}\left( {\mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} } \right)^{2} \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} + \lambda_{1}^{2} \left( \frac{1}{2} \right)\left( {\mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} } \right)^{2} \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} \hfill \\ + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\lambda_{1}^{2} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} U_{0 im} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} + \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} V_{0 nj} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} } \right) \hfill \\ \end{gathered} \right) \hfill \\ + \lambda_{2} E_{2}^{*} \left( \begin{gathered} \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} \psi_{x nj} \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} + \upsilon \lambda_{1} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} \psi_{y im} \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} \hfill \\ + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\lambda_{1}^{2} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} \psi_{x im} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} + \lambda_{1} \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} \psi_{y nj} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} } \right) \hfill \\ \end{gathered} \right) = 0 \hfill \\ \end{gathered} \right.$$
(47)
$$\left\{ \begin{gathered} \kappa \left( {\frac{1 - \upsilon }{2}} \right)\lambda_{2}^{2} E_{1}^{*} \left( {\lambda_{1} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} + \psi_{y} } \right) \hfill \\ + \lambda_{2}^{2} E_{1}^{*} \left( \begin{gathered} \lambda_{1} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} V_{0 im} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} + \upsilon \lambda_{1} \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} U_{0 nj} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} \hfill \\ + \frac{1}{2}\lambda_{1}^{3} \left( {\mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} } \right)^{3} + \frac{1}{2}\lambda_{1} \left( {\mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} } \right)^{2} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} \hfill \\ + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\lambda_{1} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} U_{0 im} \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} + \frac{1}{{\lambda_{1} }}\mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} V_{0 nj} \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} } \right) \hfill \\ \end{gathered} \right) \hfill \\ + \lambda_{2} E_{2}^{*} \left( \begin{gathered} \lambda_{1}^{2} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} \psi_{y im} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} + \lambda_{1} \upsilon \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} \psi_{x nj} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} W_{0 im} \hfill \\ + \left( {\frac{1 - \upsilon }{2}} \right)\left( {\lambda_{1} \mathop \sum \limits_{m = 1}^{M} B_{jm}^{(1)} \psi_{x im} \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} + \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} \psi_{y nj} \mathop \sum \limits_{n = 1}^{N} A_{in}^{(1)} W_{0 nj} } \right) \hfill \\ \end{gathered} \right) = 0 \hfill \\ \end{gathered} \right.$$
(48)
The different boundary conditions that are considered in this article can be expressed as:
a) For simply supported condition:
$$\left\{ {\begin{array}{*{20}c} {X = 0,1 \;\psi_{y} = U_{0} = V_{0} = W_{0} = M_{x} = 0} \\ {Y = 0,1 \;\psi_{x} = U_{0} = V_{0} = W_{0} = M_{y} = 0} \\ \end{array} } \right.$$
(49)
b) For clamed boundary condition:
$$\left\{ {\begin{array}{*{20}c} {X = 0,1\begin{array}{*{20}c} {} & {} \\ \end{array} \psi_{x} = \psi_{y} = U_{0} = V_{0} = W_{0} = 0} \\ {Y = 0,1\begin{array}{*{20}c} {} & {} \\ \end{array} \psi_{x} = \psi_{y} = U_{0} = V_{0} = W_{0} = 0} \\ \end{array} } \right.$$
(50)
c) Edges for free boundary condition
$$\left\{ {\begin{array}{*{20}c} {X = 0,1\begin{array}{*{20}c} {} & {} \\ \end{array} Q_{x} = N_{x} = N_{y} = M_{x} = M_{xy} = 0} \\ {Y = 0,1\begin{array}{*{20}c} {} & {} \\ \end{array} Q_{y} = N_{x} = N_{y} = M_{y} = M_{xy} = 0} \\ \end{array} } \right.$$
(51)
The governing equations for free vibration analysis can be obtained by ignoring the force term in Eqs. ((36) to (40)) as the following form:
$$[k_{bb} \,][d_{b} ] + [k_{bd} ][d_{d} ] = 0$$
(52)
$$[k_{db} \,][d_{b} ] + [k_{dd} ][d_{b} ] = \omega [I][d_{b} ]$$
(53)
where the stiffness quantities \({\mathbf{k}}_{bb} ,\)\({\mathbf{k}}_{db} ,\)\({\mathbf{k}}_{bd} ,\)\({\mathbf{k}}_{dd}\) refer to the boundary \(b\) and domain \(d\) weighting coefficients of the plate, respectively, \(\left[ {{\mathbf{d}}_{b} \,\,{\mathbf{d}}_{d} } \right]^{T}\) is the displacement vector, [I] refers to the identify matrix, and \(\omega\) is the natural frequencies.
At the same time, from Eq. (52), the boundary weight coefficients can be replaced by the domain weight coefficients as follows
$${\mathbf{d}}_{b} = {\mathbf{k}}_{bb}^{ - 1} {\mathbf{k}}_{bd} {\mathbf{d}}_{d}$$
(54)
Finally, by replacing Eqs. (53), (54), the dimensionless natural frequencies can be calculated as the following:
$$\omega_{i} = eigenvalue( - [k_{db} ]([k_{bb} ]^{ - 1} [k_{bd} ]) + [k_{dd} ])$$
For nonlinear bending, the governing equation may be obtained in GDQ matrix form:
$$\left[ {K_{L} + K_{NL} } \right]\left[ {d_{0} } \right] + [M]\left[ {\ddot{d}_{0} } \right] = [F]$$
(52)
For solving free vibration, it is sufficient that Eq. (52) be equal zero. Also for transient vibration, the Newmark procedure should be applied [39,40,41].