A note on the solution of the electro-elastic boundary-value problem for rank-two laminates at finite strains

Recent research has shown that hierarchical laminated composites can be profitably employed to improve the actuation performance of electrically-activated soft dielectric transducers. This note focuses on two types of rank-two layouts composed of ideal dielectric phases which follow nonlinear hyper-elastic mechanical behaviour and aims at providing a simplified set of solving equations for voltage-controlled actuation. We obtain such equations by analytical manipulations allowing to partly uncouple the set of equations usually employed within this theoretical framework. By focusing on neo-Hookean hyper-elasticity, we validate the proposed methodology with the results available in literature for one layout. For the other layout, we obtain new configurations by maximising the axial stretch. In both cases, we study the sensitivity of the optimal actuation stretch to changes of the parameters characterising the rank-two meso- and micro-structures. In average, the computational time required to reach a convergent solution with the new methodology is one order of magnitude lower than that necessary to solve the whole set of nonlinear coupled equations.