RESUMEN
The pull-in voltage
instability associated with a simple MEMS device, consisting of a thin
dielectric elastic membrane supported above a rigid conducting ground plate, is
analyzed. The upper surface of the membrane is coated with a thin conducting
film. In a certain asymptotic limit representing a thin device, the
mathematical model consists of a nonlinear partial differential equation for
the deflection of the thin dielectric membrane. When a voltage V is applied to
the conducting film, the dielectric membrane deflects towards the bottom plate.
For a slab, a circular cylindrical, and a square domain, numerical results are
given for the saddle-node bifurcation value $V_{*}$,
also referred to as the pull-in voltage, for which there is no steady-state
membrane deflection for $V > V_{*}$. For $V > V_{*}$
it is shown numerically that the membrane dynamics are such that the thin
dielectric membrane touches the lower plate in finite time. Results are given
for both spatially uniform and nonuniform dielectric
permittivity profiles in the thin dielectric membrane. By allowing for a
spatially nonuniform permittivity profile, it is
shown that the pull-in voltage instability can be delayed until larger values
of $V$ and that greater pull-in distances can be achieved. Analytical bounds
are given for the pull-in voltage $V_{*}$ for two
classes of spatially variable permittivity profiles. In particular, a rigorous
analytical bound $V_1$, which depends on the class of permittivity profile, is
derived that guarantees for the range $V > V_{1}
> V_{*}$ that there is no steady-state solution for the membrane deflection
and that finite-time touchdown occurs. Numerical results for touchdown
behavior, both for $V > V_1$ and for $V_{*} < V
< V_1$, together with an asymptotic construction of the touchdown profile,
are given for both a spatially uniform and a spatially nonuniform
permittivity profile.