Tuning the pull-in instability of soft dielectric elastomers through loading protocols

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 criterions 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.