## 2 Problem statement
Assuming the response of the material to be linear elastic, the constitutive relation under the Kirchhoff assumption is, see for example [42],  
\[\sigma_{ij}=\frac{E}{1+\
u}\left(\varepsilon_{ij}^{\mathrm{e}}+\frac{\
u}{1- \
u}\varepsilon_{kk}^{\mathrm{e}}\delta_{ij}\right)=\frac{E}{1+\
u}\left( \varepsilon_{ij}+\frac{\
u}{1-\
u}\varepsilon_{kk}\delta_{ij}\right)-\frac{E \chi\Theta}{1-\
u}\delta_{ij}. \tag{4}\]  
Substituting for \(\sigma_{ij}\) from equation (4) into equations (1) and using equation (2), we obtain the following partial differential equations for the displacement fields \(u_{i}\) and \(w\)  
\[\varrho h\ddot{w} +D\frac{\partial^{2}}{\partial x_{k}\partial x_{k}}\left(\frac{ \partial^{2}w}{\partial x_{j}\partial x_{j}}\right)-B\frac{\partial}{\partial x _{l}}\left(\left(\frac{1}{2}\left(\frac{\partial u_{l}}{\partial x_{j}}+\frac {\partial u_{j}}{\partial x_{l}}\right)+\delta_{lj}\frac{\partial u_{k}}{ \partial x_{k}}\right)\frac{\partial w}{\partial x_{j}}\right)\] \[-\frac{B}{2}\frac{\partial}{\partial x_{k}}\left(\frac{\partial w }{\partial x_{j}}\frac{\partial w}{\partial x_{j}}\frac{\partial w}{\partial x _{k}}\right)+\frac{E\chi\Theta}{1-\
u}\frac{\partial^{2}w}{\partial x_{j} \partial x_{j}}-F_{\mathrm{e}}-F_{\mathrm{s}}=0, \tag{5a}\] \[\varrho h\ddot{u}_{i} -\frac{B}{2}\left((1-\
u)\frac{\partial^{2}u_{i}}{\partial x_{k} \partial x_{k}}+(1+\
u)\frac{\partial}{\partial x_{i}}\left(\frac{\partial u_ {k}}{\partial x_{k}}\right)\right)\] \[=\frac{B}{2}\left((1-\
u)\frac{\partial}{\partial x_{k}}\left( \frac{\partial w}{\partial x_{i}}\frac{\partial w}{\partial x_{k}}\right)+\
u \frac{\partial}{\partial x_{i}}\left(\frac{\partial w}{\partial x_{k}}\frac{ \partial w}{\partial x_{k}}\right)\right),\quad i=1,2, \tag{5b}\]  
where \(B=Eh/(1-\
u^{2})\). From an electrical point of view, the system depicted in Figure 1 behaves as a variable gap capacitor. By assuming that \(g_{0}/\ell\ll 1\) and by neglecting fringing fields, the magnitude \(F_{\mathrm{e}}\) of the Coulomb or the magnitude of the electrostatic force acting on the deformable electrode along its normal is given by the parallel plate approximation [9]  
\[F_{\mathrm{e}}=-\frac{\epsilon_{0}V^{2}}{2g_{0}^{2}\left(1+\hat{w}\right)^{2}}, \tag{6}\]where \(\hat{w}=w/g_{0}\) is the non-dimensional transverse displacement, \(\epsilon_{0}\) is the dielectric constant in vacuum, and \(V\) is the applied direct current voltage. Therefore, within the parallel plate approximation the expression for the electrostatic force depends only on the gap \(g\). In [43], perturbation theory is used to develop expressions for the electrostatic force with corrections to the parallel plate approximations. It is shown that the parallel plate approximation is adequate when \(g_{0}\ll\ell\), since the first correction multiplies the ratio \(g_{0}^{2}/\ell^{2}\). Consistent with the parallel plate approximation for the electrostatic force, and the small deformations assumption in the mechanical model, the distributed load due to the Casimir force per unit surface area is given by the proximity force approximation [44, 45], which for perfect conductors yields:  
\[F_{\rm c}=-\frac{\hbar c\pi^{2}}{240g_{0}^{4}\left(1+\hat{w}\right)^{4}}, \tag{7}\]  
where \(\hbar\) is the Plank's constant and \(c\) is the speed of light in vacuum. Corrections to equation (7) for geometries with known and fixed departures from the parallel configurations are reported in [44, 45]. For gaps smaller than the retardation length, that is, for gaps smaller than the wavelength of the virtual transitions responsible for the quantum dipole fluctuations [31, 33], the Casimir force reduces to the van der Waals distributed force per unit surface area [46, 47, 48]  
\[F_{\rm vdW}=-\frac{A}{6\pi g_{0}^{3}\left(1+\hat{w}\right)^{3}}. \tag{8}\]  
In equation (8), \(A\) is the Hamaker constant, see for example [31], with values in the range \([0.4,4]\times 10^{-19}\,\mathrm{J}\).