## 2 Problem statement  
We consider a rectangular plate-like body of longer side \(\ell\) and shorter side \(\varphi\ell,\,\varphi\in]0,1]\), occupying the three-dimensional region \(\Omega\times(-h/2,h/2)\) as depicted in Figure 1. The mid-surface \(\Omega\) is described by rectangular Cartesian coordinates \((x_{1},x_{2})\) aligned with the longer and with the shorter sides, respectively. We assume that the initial gap \(g_{0}\) between the two conductors and the thickness \(h\) of the deformable plate are much smaller than the characteristic length \(\ell\). In the proposed model, we assume that \(g_{0}\) and \(h\) can be of the same order of magnitude. Therefore, the maximum displacement that the device can undergo is of the order of the plate thickness \(h\), but it is much smaller than the characteristic length \(\ell\), since \(h/\ell\ll 1\). This implies that strains in the deformable electrode are small. Under these assumptions, we use the von Karman plate theory to account for large deflections and small strains, see for example [41]. Neglecting the effect of the rotatory inertia, the von Karman plate equations in Cartesian coordinates are [41]  
\[\varrho h\ddot{w}+D\frac{\partial^{2}}{\partial x_{k}\partial x_ {k}}\left(\frac{\partial^{2}w}{\partial x_{j}\partial x_{j}}\right)-h\frac{ \partial}{\partial x_{j}}\left(\sigma_{jk}\frac{\partial w}{\partial x_{k}} \right)-F_{\rm e}-F_{\rm s}=0, \tag{1a}\] \[\varrho\ddot{u}_{i}-\frac{\partial\sigma_{ij}}{\partial x_{j}}=0, \qquad i=1,2. \tag{1b}\]  
We use the [PERSON] summation convention, meaning that when an index variable appears twice in a single term we are summing over the range \(\{1,2\}\); free index also span the range \(\{1,2\}\). Furthermore, \(D=Eh^{3}/(12(1-\
u^{2}))\) is the bending stiffness of the plate, \(E\), \(\
u\), and \(\varrho\) are [PERSON]'s modulus, [PERSON]'s  
Figure 1: Sketch of an electrostatically actuated rectangular microplate. ratio and mass density of the plate material, that is assumed to be homogeneous and isotropic, \(u_{i}\) and \(w\) are the in-plane and the out-of-plane displacements of a point on the mid-surface, \(F_{\mathrm{e}}\) is the Coulomb force, and \(F_{\mathrm{s}}\) is the nanoscale surface force. From equation (1a), we note that Coulomb and nanoscale forces act along the normal to the microplate mid-surface. This is due to the assumption of perfectly conducting material. Expressions for the Coulomb and the nanoscale surface forces are discussed below. We note that in the linear theory of a plate the transverse displacement and in-plane displacements are decoupled. Therefore, in the linear theory, the Coulomb force and nanoscale forces do not induce in-plane deformations of the microplate. In the [PERSON] plate theory, the components \(\varepsilon_{ij}\) of the in-plane strain tensor are given by  
\[\varepsilon_{ij}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac {\partial u_{j}}{\partial x_{i}}+\frac{\partial w}{\partial x_{i}}\frac{ \partial w}{\partial x_{j}}\right). \tag{2}\]  
We assume that the strain tensor admits the additive decomposition \(\varepsilon_{ij}=\varepsilon_{ij}^{\mathrm{e}}+\varepsilon_{ij}^{\Theta}\), where \(\varepsilon_{ij}^{\mathrm{e}}\) is the elastic strain and \(\varepsilon_{ij}^{\Theta}\) the thermal one. We set \(\varepsilon_{ij}^{\Theta}=\chi\Theta\delta_{ij}\), where \(\Theta\) is the uniform temperature rise, \(\chi\) is the coefficient of thermal expansion assumed to be constant, and  
\[\delta_{ij}=\left\{ \begin{array}{ccc}1&\mathrm{if}&i=j\\ 0&\mathrm{if}&i\
eq j\end{array} \right., \tag{3}\]  
is the Kronecker delta.