Emergent electromechanical coupling of electrets and some exact relations - The effective properties of soft materials with embedded external charges and dipoles

Soft robotics, energy harvesting, large-deformation sensing and actuation, are just some of the applications that can be enabled by soft dielectrics that demonstrate substantive electromechanical coupling. Most soft dielectrics including elastomers, however, are not piezoelectric and rely on the universally present electrostriction and the Maxwell stress effect to enable the aforementioned applications. Electrostriction is a one-way electromechanical coupling and the induced elastic strain scales as (∝E2) upon the application of an electric field, E. The quadratic dependence of electrostriction on the electric field and the one-way coupling imply that, (i) A rather high voltage is required to induce appreciable strain, (ii) reversal of an applied bias will not reverse the sign of the deformation, and (iii) since it is a one-way coupling i.e. electrical stimuli may cause mechanical deformation but electricity cannot be generated by mechanical deformation, prospects for energy harvesting are rather difficult. An interesting approach for realizing an apparent piezoelectric-like behavior is to dope soft dielectrics with immobile charges and dipoles. Such materials, called electrets, are rather unique composites where a secondary material (in principle) is not necessary. Both experiments and supporting theoretical work have shown that soft electrets can exhibit a very large electromechanical coupling including a piezoelectric-like response. In this work, we present a homogenization theory for electret materials and provide, in addition to several general results, variational bounds and closed-form expressions for specific microstructures such as laminates and ellipsoidal inclusions. While we consider the nonlinear coupled problem, to make analytical progress, we work within the small-deformation setting. The specific conditions necessary to obtain a piezoelectric-like response and enhanced electrostriction are highlighted. There are very few universal, microstructure-independent exact results in the theory of composites. We succeed in establishing several such relations in the context of electrets.