Abstract
A MEMS model with an insulating layer is considered and its reinforced limit is derived by means of a Gamma convergence approach when the thickness of the layer tends to zero. The limiting model inherits the dielectric properties of the insulating layer.
Highlights
Idealized microelectromechanical systems (MEMS) consist of two dielectric plates: a rigid ground plate above which an elastic plate is suspended
When the two plates are not prevented from touching each other, a contact of the plates commonly leads to an instability of the device— known in the literature as “pull-in instability”—which is revealed as a singularity in the corresponding mathematical equations, e.g., see [9,16] and the references therein
When the ground plate is coated with an insulating layer preventing a direct contact of the plates, see Fig. 1, a touchdown of the elastic plate on this layer does not result in an instability as the device may continue to operate without interruption
Summary
Idealized microelectromechanical systems (MEMS) consist of two dielectric plates: a rigid ground plate above which an elastic plate is suspended. In the model considered in [10, Section 5], the deflection u of the elastic plate is governed by an evolution equation involving contributions from mechanical and electrostatic forces, the latter depending on the electrostatic potential denoted by ψ in the following. Σδ is the permittivity of the device, which is different in the insulating layer and free space, and hu,δ is a given suitable function describing the boundary values of the electrostatic potential. In the following we shall derive the limiting model obtained from (1.1) as δ → 0 when imposing suitable assumptions on the function hu,δ defining the boundary values of the potential (see (2.2) below) and on the permittivity σδ (see (2.1) below), so that information on the dielectric heterogeneity is inherited. The function u is fixed and assumed to satisfy u ∈ H01(D) ∩ C(D ) with u ≥ −H in D , and (u) satisfies the segment property in the sense of [11, Definition 10.23]
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