## 1 Introduction
The microscopic approach to the modeling of both van derWaals and Casimir forces can be formulated in a unified way using Quantum Field Theory, see for example [30, 31, 32, 33]. It is found that the Casimir force is generally effective at larger separation distances between the bodies than the van der Waals force. Whereas the Casimir force between semi-infinite parallel plates is inversely proportional to the fourth power of the gap, van der Waals force is inversely proportional to the third power of the gap. The dependence of these forces on the dielectric properties of the plates and the filling medium is studied in details in [31], Section 4.1.1. It is important to note that van der Waals and Casimir forces cannot in general be considered to simultaneously act in MEMS, since they describe the same physical phenomenon at two different length scales. MEMS can be subjected to considerable temperature variations during sensing operation, such as in monitoring aircraft condition and distributed satellites communication, as well as during device packaging [35]. Modeling of the effect of temperature changes on pull-in instability of electrostatically actuated microelectromechanical switches is addressed in [36, 37, 38]. In these studies, it is observed that even a moderate temperature increase may cause premature failure of the device due to a significant decrease in the pull-in voltage. On the other hand, temperature drops may lead to considerable increase in the pull-in voltage even outside the operational range of the switch [39, 40]. We consider rectangular microplates undergoing large displacements under the combined effect of electrostatic and nanoscale forces. We use the Galerkin method to develop a tractable reduced-order model for electrostatically actuated microplates. The reduced-order model is derived by taking a family of linearly independent kinematically admissible functions as basis functions for the transverse displacement and by decomposing the in-plane displacement vector as the sum of displacements for irrotational and isochoric waves in a 2-D medium. Basis functions for the transverse and the in-plane displacements are related through the nonlinear equation governing the plate in-plane motion. The governing equations of the reduced-order model are derived from the equation governing the transverse motion of the microplate. The model is specialized to the case where a single basis function for the transverse displacement is used to yield manageable solutions. In the static analysis, pull-in parameters are found by solving a system of two nonlinear algebraic equations for the transverse displacement amplitude and the load parameter. The eigenvalue problem corresponding to linear vibrations of the system about its statically deflected position is solved for the fundamental frequency. We show that the fundamental frequency goes to zero as pull-in conditions are approached. The pull-in parameters found from the eigenvalue analysis agree well with those derived from the static analysis. We investigate effects of nanoscale forces and thermal stresses on pull-in parameters and small vibrations of electrostatically actuated microplates. The rest of the paper is organized as follows. In section 2, we present the governing nonlinear equations of motion for a von Karman microplate under the simultaneous effects of thermal loading, electrostatic force, and nanoscale forces. We present expressions of the distributed loads due to either the Casimir or the van der Waals forces. In section 3, we introduce a reduced-order model for the considered device that is capable of accurately predicting its dynamics. The derivation of the reduced-order model follows a procedure typically used for studying deformations of thin two-dimensional structures. That is, in-plane inertial effects are neglected, and the resulting equation is solved for in-plane displacements in terms of transverse deflections which are then substituted in the equation governing the evolution of transverse deflection.