  Pull-in (or electro-mechanical) instability occurs when a drastic decrease in
the thickness of a dielectric elastomer results in electrical breakdown, which
limits the applications of dielectric devices. Here we derive the criteria for
determining the pull-in instability of dielectrics actuated by different
loading methods: voltage-control, charge-control, fixed pre-stress and fixed
pre-stretch, by analyzing the free energy of the actuated systems. The Hessian
criterion identifies a maximum in the loading curve beyond which the elastomer
will stretch rapidly and lose stability, and can be seen as a path to failure.
We present numerical calculations for neo-Hookean ideal dielectrics, and obtain
the maximum allowable actuation stretch of a dielectric before failure by
electrical breakdown. We find that applying a fixed pre-stress or a fixed
pre-stretch to a charge-driven dielectric may decrease the stretchability of
the elastomer, a scenario which is the opposite of what happens in the case of
a voltage-driven dielectric. Results show that a reversible large actuation of
a dielectric elastomer, free of the pull-in instability, can be achieved by
tuning the actuation method.
