# Dynamic Stability of Orthotropic Viscoelastic Rectangular Plate of an Arbitrarily Varying Thickness
## 3 Results and Discussion
### Analytical Solution for an Orthotropic Viscoelastic Rectangular Plate of Variable Thickness
\right) \tag{39}\]  
In this case of plate's orthotropic properties, the coefficients \(C_{ij}\) and \(D_{ij}\) are expressed in terms of the elastic constants \(E_{1}\), \(E_{2}\), \(G_{12}\), \(\mu_{1}\), and \(\mu_{2}\) as follows:  
\[C_{11}=B_{11}(1-\Gamma_{11}^{*})h=\frac{E_{1}\big{(}1-\Gamma_{11}^{*})h}{1-\mu_ {1}\mu_{2}} \tag{40}\]  
\[C_{22}=B_{22}(1-\Gamma_{22}^{*})h=\frac{E_{2}(1-\Gamma_{22}^{*})h}{1-\mu_{1}\mu_ {2}} \tag{41}\]  
\[C_{12}=B_{12  
If the plate has isotropic properties (\(E_{1}=E_{2},\mu_{1}=\mu_{2}\)), then the elements of the stiffness matrix take on a more straightforward form with two elastic constants \(E\) (shear modulus) and \(\mu\) (Poisson's ratio):  
\[C_{11}=C_{22}=B(1-\Gamma^{*})h=\frac{E(1-\Gamma^{*})h}{1-\mu^{2}} \tag{48}\]  
\[C_{12}=\mu B(1-\Gamma^{*})h=\frac{\mu E(1-\Gamma^{*})h}{1-\mu^{2}} \tag{49}\]  
\[C_{66}=\frac{1-\mu}{2}B(1-\Gamma^{*})h=\frac{E(1-\Gamma^{*})h}{2(1+\mu)} \tag{50}\]  
\[D_{11}=D_{22}=B(1-\Gamma^{*})\frac{h^{3}}{12}=\frac{E(1-\Gamma^{*})h^{3}}{12(1- \mu^{2})} \tag{51}\]  
\[D_{12}=\mu B(1-\Gamma^{*})\frac{h^{3}}{12}=\frac{\mu E(1-\Gamma^{*})h^{3}}{12( 1-\mu^{2})} \tag{52}\]  
\[D_{66}=\frac{1-\mu}{2}B(1-\Gamma^{*})\frac{h^{3}}{12}=\frac{E(1-\Gamma^{*})h^ {3}}{24(1+\mu)} \tag{53}\]  
The Koltunov-Rzhanitsyn singular kernel [49] is used in the calculations as relaxation kernel \(\Gamma(t),\Gamma_{ij}(t),i,j=1,2\):  
\[\Gamma(t)=Ae^{-\beta t}t^{a-1},\ (0<\alpha<1) \tag{54}\]  
\[\Gamma_{ij}(t)=A_{ij}e^{-\beta_{ij}t}t^{a_{ij}-1},\ (0<\alpha_{ij}<1) \tag{55}\]  
If solving problems of the dynamics of viscoelastic plates based on the Kirchhoff-Love hypothesis in an isotropic formulation, then only one relaxation kernel \(\Gamma(t)\) with three different rheological viscosity parameters \(A\), \(\alpha\), and \(\beta\) is involved in the system of integrodifferential equations. Then, in the orthotropic formulation, already 5 different kernels \(\Gamma(t),\Gamma_{ij}(t),i,j=1,2\) are involved with 15 rheological parameters \(A_{ij},\alpha_{ij}\) and \(\beta_{ij}\). The system of Equations (28)-(30) is the most general. The motion equations of isotropic and orthotropic viscoelastic rectangular plates of variable thickness can be obtained from this system. The corresponding initial and boundary conditions are assumed to be given. For example, consider a rectangular orthotropic viscoelastic plate of variable thickness \(h(x,y)\) with sides \(a\) and \(b\) that is dynamically loaded in-plane along the side \(a\) by a periodic load. The equation of motion, based on the Kirchhoff-Love hypothesis, in a geometrically nonlinear setting has the form (28)-(30), taking into account (38)-(47). If the following dimensionless variables are introduced into Equations (28)-(30):  
\[C_{22}=B_{22}(1-\Gamma_{22}^{*})h=\frac{E_{2}(1-\Gamma_{22}^{*})h}{1-\mu_{1}\mu_ {2}} \tag{41}\]0  
\[C_{22}=B_{22}(1-\Gamma_{22}^{*})h=\frac{E_{2}(1-\Gamma_{22}^{*})h}{1-\mu_{1}\mu_ {2}} \tag{41}\]1  
\[C_{22}=B_{22}(1-\Gamma_{22}^{*})h=\frac{E_{2}(1-\Gamma_{22}^{*})h}{1-\mu_{1}\mu_ {2}} \tag{41}\]2  
and the previous notation is retained, then a dimensionless system of nonlinear integrodifferential equations for the problems of parametric vibrations of an orthotropic viscoelastic rectangular plate of variable thickness is obtained. In this case, the operators \(L_{ij}\) from the expressions (31)-(37) take the following form:\[ \begin{array}{c}L_{11}(u)=h\Delta\big{(}1-\Gamma_{11}^{*}\big{)}\frac{\partial^{2 }u}{\partial x^{2}}+\lambda^{2}h(1-\mu_{1}\mu_{2})g(1-\Gamma^{*})\frac{\partial ^{2}u}{\partial y^{2}}+\\ \qquad+\frac{\partial u}{\partial x}\Delta\big{(}1-\Gamma_{11}^{*}\big{)}\frac{ \partial u}{\partial x}+\lambda^{2}\frac{\partial u}{\partial y}(1-\mu_{1}\mu_ {2})g(1-\Gamma^{*})\frac{\partial u}{\partial y}\end{array}  \tag{56}\]  
\[