Role of nonlinear elasticity in mechanical impedance tuning of annular dielectric elastomer membranes

We use finite elasticity to examine the behavior of a lightweight mechanism for rapid, reversible, and low-power control of mechanical impedance. The device is composed of a central shaft suspended by an annular membrane of prestretched dielectric elastomer (DE), which is coated on both sides with a conductive film. Applying an electrical field across the thickness of the membrane, attractive Coulombic forces (so-called ‘‘Maxwell stresses’’) are induced that (i) squeeze the annulus, (ii) relieve the membrane stress, and (iii) reduce the mechanical resistance of the elastomer to out-of-plane deflection. This variable stiffness architecture was previously proposed by researchers who performed an experimental implementation and demonstrated a 10× change in stiffness. In this manuscript, we generalize this approach to applications in aerospace and robotics by presenting a complete theoretical analysis that establishes a relationship between mechanical impedance, applied electrical field, device geometry, and the constitutive properties of the dielectric elastomer. In particular, we find that the stiffness reduction under applied voltage is non-linear. Such decay is most significant when the Maxwell stress is comparable to the membrane prestress. For this reason, both the prestretch level and the hyperelastic properties of the DE membrane have a critical influence on the impedance response.