# Modeling

#### Maxwell Stress

When a voltage is applied to a dielectric elastomer actuator, an electric field arises in the dielectric membrane. Generally this electric field is complex and its determination requires solving Maxwell's equations.  However, when the membrane is thin enough, the active area can be modeled as a deformable plate capacitor.

The electric field E inside a plate capacitor is

where Φ is the applied voltage and  is the distance between plates in the plate capacitor. The electric field causes an effective compressive Maxwell stress

in the thickness direction of the membrane, where ε is the permittivity of the membrane. This Maxwell stress causes the active area to decrease in thickness and increase in area.

#### Modes of Failure

When testing dielectric elastomer actuators, various modes of failure have to be considered:

#### Mechanical Failure

Mechanical failure is one of the simplest modes to understand. When the membrane is stretched too much, the polymer chains tear and the material ruptures. If there are any pre-existing tears or notches in the material, this will lead to faster mechanical failure at those locations.

#### Electric Breakdown

When the electric field within the membrane becomes too high, the material can lose its insulating property. A conductive channel builds between the electrodes and burns a hole in the membrane. This phenomenon is called electric breakdown. It can be compared to lightning during a thunderstorm. When the voltage difference between a cloud and the ground becomes too high the air is suddenly conductive for a moment and lightening happens. See Testing section for a video of this.

#### Electromechanical instability

Electromechanical instability is a mode of failure that is critical for dielectric elastomer actuators. When an elastomer membrane is stretched only by a mechanical force, it behaves soft at low stretches but stiffens close to mechanical failure.

To increase stretch membrane one has to continuously increase the applied force. When the membrane is deformed by applying a voltage, the Maxwell stress is coupled with the thickness of the membrane. This means that the thinner the membrane is, the higher the electric field and therefore the higher then effective compressive stress applied to the membrane. This coupling can lead to a voltage-stretch curve shown below:

When the voltage is ramped up from zero, the elastomer stretches until the peak of the curve is reached. When the voltage is increased above the peak the actuator becomes unstable and jumps to a very large stretch on the right side of the curve. During this jump the actuator most of the time becomes so thin that electric breakdown occurs.

Prestretching of the membrane causes the peak to be moved to higher stretches or even be totally removed. For this reason we use a prestretched circular actuator in the fabrication section.

#### Loss of tension

When a prestretched dielectric elastomer actuator deforms, it often reduces its prestretch until it is completely lost. This can either lead to electromechanical instability and destruction of the material or to wrinkling of the material. Since the membrane is very thin it cannot sustain any compressive stresses from the sides (similar to a sheet of paper that wrinkles when you compress it in plane). When the actuator wrinkles it loses its function. See Testing section for a video of this.