## 2 Modeling Assumptions  
### Problem Definition  
The analysis considers a corrugated plate with length \(L_{x}\), width \(L_{y}\) and thickness \(h\), as shown in Figure 1a. More details about the corrugations are shown in Figure 1b, where the parameter defines the length of half wave of a sinusoidal wave, and refers to the height of the sinusoidal wave. To derive the constitutive equations, we adopt a new One-Variable First-order Shear Deformation Theory (OVFSDT) which is compared to some further plate theories from literature. According to the proposed theory, the displacement field \(U(x,y,z,t),V(x,y,z,t),W(x,y,z,t)\) is defined as  
\[\left\{ \begin{array}{l}U(x,y,z,t)\\ V(x,y,z,t)\\ W(x,y,z,t)\end{array} \right\}=\left\{ \begin{array}{l}u(x,y,t)-z\frac{\partial w (x,y,t)}{\partial x}\\ v(x,y,t)-z\frac{\partial w(x,y,t)}{\partial y}\\ w(x,y,t)+A\frac{\partial^{2}w(x,y,t)}{\partial x^{2}}+B\frac{\partial^{2}w(x, y,t)}{\partial y^{2}}\end{array} \right\} \tag{1}\]  
where \(\varphi\) and \(\psi\) are the rotations around the \(y\) and \(x\) axis, respectively, while \(u\), \(v\) and \(w\) denote the variations of the kinematic quantities along the \(x\), \(y\) and \(z\) axis, respectively. In addition, we define \(A=Eh^{2}/\left(12G\left(1-v^{2}\right)\right),B=h^{2}\left(E+G\left(1-v^{2} \right)\right)/\left(12G\left(1-v^{2}\right)\right)\). By applying the Hamilton's principle, we define the potential energy \(V\) of the domain as [2; 20]  
\[\delta V=\delta\int\limits_{0}^{t}(S+\Omega-T\ )dt=0 \tag{2}\]  
where \(\delta\) refers to the variation of the energy quantities, \(S\) is the strain energy, \(T\) is the kinetic energy, and \(\Omega\) is the work related to external forces or foundations, here neglected in the following. The strain energy \(S\) in Equation (2) can be computed as  
\[\delta S=\iint\limits_{v}\sigma_{ij}\ \delta\varepsilon_{ij}dV=0 \tag{3}\]  
\(\sigma_{ij}\) and \(\varepsilon_{ij}\) being the stress and strain tensors, respectively [6]. Based on the OVFSDT, the strain tensor is defined as follows  
Figure 1: Corrugated nanoplate: (**a**) 3D view, and (**b**) in-plane geometry. \[\left\{ \begin{array}{l}\varepsilon_{xx}\\ \varepsilon_{yy}\\ \gamma_{xx}\\ \gamma_{yz}\\ \gamma_{xy}\\ \gamma_{xy}\\ \gamma_{xy}\\ \end{array} \right\}=\left\{ \begin{array}{l}\frac{\frac{\partial u}{\partial x }-z\frac{\partial^{2}w}{\partial x^{2}}+\frac{1}{2}\Big{(}A\frac{\partial^{3} w}{\partial x^{3}}+B\frac{\partial^{3}w}{\partial x\partial y^{2}}+\frac{ \partial w}{\partial x}\Big{)}^{2}}{\frac{\partial w}{\partial y}-z\frac{ \partial^{2}w}{\partial x^{2}}+\frac{\partial^{3}w}{\partial x^{2}}+B\frac{ \partial^{3}w}{\partial y^{3}}+\frac{\partial w}{\partial y}\Big{)}^{2}}\\ A\frac{\partial^{3}w}{\partial x^{3}}+B\frac{\partial^{3}w}{\partial x \partial y^{2}}\\ A\frac{\partial^{3}w}{\partial x^{3}w}+B\frac{\partial^{3}w}{\partial y^{3}} \\ \left(\frac{\partial u}{\partial y}+\frac{\partial w}{\partial x}\right)-2z \frac{\partial^{2}w}{\partial x\partial y}+\Big{(}A\frac{\partial^{3}w}{ \partial x^{3}}+B\frac{\partial^{3}w}{\partial x\partial y^{2}}+\frac{ \partial w}{\partial x}\Big{)}\Big{(}A\frac{\partial^{3}w}{\partial x^{2} \partial y}+B\frac{\partial^{3}w}{\partial y^{3}}+\frac{\partial w}{\partial y }\Big{)}\end{array} \right\} \tag{4}\]  
In addition, the kinetic energy \(T\) can be determined as [2,20]  
\[T=\frac{1}{2}\int_{A}\int_{-h/2}^{h/2}\rho(z,T)\Bigg{(}\left(\frac{\partial U }{\partial t}\right)^{2}+\left(\frac{\partial V}{\partial t}\right)^{2}+\left( \frac{\partial W}{\partial t}\right)^{2}\Bigg{)}dzdA=0 \tag{5}\]  
whose variation reads  
\[