## 3 The Plate Theories (RPT and IRPT)  
Consider the rectangular plate shown in Figure 1, using the assumptions shown below:  
1. In-plane displacement and transverse displacement consists of bending and shear components. 2. The bending component of the in-plane displacement is assumed to be analogous to the displacement that is given by classical plate theory. 3. The shear part of the in-plane displacement causes a parabolic change in the shear strain, thereby generating shear stress through the thickness of the plate so that the shear stress on the top and bottom surfaces disappears. Based on the above three assumptions, the displacement field of RPT is defined as follows [32,41]:  
\[ \begin{array}{ll}u_{1}(x,y,z,t)=u(x,y,t)-z\frac{\partial w_{\text{G}}}{ \partial x}-f(z)\frac{\partial w_{z}}{\partial x}\\ u_{2}(x,y,z,t)=v(x,y,t)-z\frac{\partial w_{z}}{\partial y}-f(z)\frac{\partial w _{z}}{\partial y}\\ u_{3}(x,y,z,t)=w_{b}(x,y,t)+w_{s}(x,y,t)\end{array}  \tag{7}\]  
where \(f(z)=-z/4+5z^{3}/3h^{2}\); \((u_{1},u_{2},u_{3})\) are the displacements along the \((x,y,z)\) axis direction, respectively. \(u\) and \(v\) represent the displacements of a point on the middle surface of the plate along the \(x\)-axis and \(y\)-axis; \(w_{b}\) is the bending part of the transverse displacement; and \(w_{s}\) is the shear part of the transverse displacement. \(h\) indicates the thickness of the plate. Considering the effect of thickness stretching based on RPT, we can obtain the displacement field of IRPT as follows [41]:  
\[ \begin{array}{ll}u_{1}(x,y,z,t)=u(x,y,t)-z\frac{\partial w_{\text{G}}}{ \partial x}-f(z)\frac{\partial w_{z}}{\partial x}\\ u_{2}(x,y,z,t)=v(x,y,t)-z\frac{\partial w_{\text{G}}}{\partial y}-f(z)\frac{ \partial w_{z}}{\partial y}\\ u_{3}(x,y,z,t)=w_{b}(x,y,t)+w_{s}(x,y,t)+g(z)w_{z}(x,y,t)\end{array}  \tag{8}\]  
where \(g(z)=1-f^{\prime}(z)\). \(w_{z}\) is the unknown displacement function with respect to the effect of thickness stretching. The linear relationship between strain and displacement is as follows:  
\[\varepsilon_{x}=\frac{\partial u}{\partial x}-z\frac{\partial^{2}w_{b}}{ \partial x^{2}}-f(z)\frac{\partial^{2}w_{s}}{\partial x^{2}} \tag{9}\]  
\[\varepsilon_{y}=\frac{\partial v}{\partial y}-z\frac{\partial^{2}w_{b}}{ \partial y^{2}}-f(z)\frac{\partial^{2}w_{s}}{\partial y^{2}} \tag{10}\]  
\[\varepsilon_{z}=g^{\prime}(z)w_{z} \tag{11}\]  
\[\gamma_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}-2z \frac{\partial^{2}w_{b}}{\partial x\partial y}-2f(z)\frac{\partial^{2}w_{s}}{ \partial x\partial y} \tag{12}\]  
\[\gamma_{xz}=g(z)\bigg{(}\frac{\partial w_{s}}{\partial x}+\frac{\partial w_{z}} {\partial x}\bigg{)} \tag{13}\]\[\gamma_{yz}=g(z)\bigg{(}\frac{\partial w_{s}}{\partial y}+\frac{\partial w_{z}}{ \partial y}\bigg{)} \tag{14}\]  
If the effect of thickness stretching is neglected, Equation (11) becomes \(\varepsilon_{z}=0\). The relationship between strain and stress is as follows:  
\[\left\{ \begin{array}{c}\sigma_{xx}\\ \sigma_{yy}\\ \sigma_{zz}\\ \tau_{xy}\\ \tau_{xz}\\ \tau_{yz}\end{array} \right\}=\left[ \begin{array}{cccccc}C_{1}^{(k)}&C_{1 2}^{(k)}&C_{13}^{(k)}&0&0&0\\ C_{21}^{(k)}&C_{22}^{(k)}&C_{23}^{(k)}&0&0&0\\ C_{31}^{(k)}&C_{32}^{(k)}&C_{33}^{(k)}&0&0&0\\ 0&0&0&C_{66}^{(k)}&0&0\\ 0&0&0&C_{55}^{(k)}&0\\ 0&0&0&0&C_{44}^{(k)}\end{array} \right]\left\{ \begin{array}{c}\varepsilon_{xx }\\ \varepsilon_{yy}\\ \varepsilon_{zz}\\ \gamma_{xy}\\ \gamma_{xz}\\ \gamma_{yz}\end{array}