Homogenization of electrets with ellipsoidal microstructure and pathways for designing piezoelectricity in soft materials

True piezoelectricity in soft materials is rare if not virtually non-existent. This impedes applications where both large deformation and a strong electromechanical coupling are desirable e.g. soft robotics, biomedical sensors and actuators, a class of energy harvesting devices among others. The widely used soft dielectric elastomers rely on the electrostatic Maxwell stress effect for electromechanical coupling — a one-way quadratic effect that requires extremely large voltage for actuation and does not allow for the facile conversion of mechanical deformation into electricity. Prior research has shown that embedding (and stabilizing) immobile charges or dipoles in soft matter i.e. creating so-called electrets, can lead to an emergent piezoelectric effect. In this work, using a recently developed homogenization theory for soft electret materials, we derive closed-form expressions to design soft apparently piezoelectric materials with an ellipsoidal microstructure. Specifically, we determine both effective longitudinal (d33) and transverse (d31) piezoelectric coefficients of the material and study the impact of the material properties on these two coefficients. Conventional electrets exhibit a rather weak d31, which is quite disadvantageous for applications where flexure is important (e.g. energy harvesting). Either an elastic, or a dielectric contrast is essential for the emergence of piezoelectricity in electrets and, depending on the microstructural details, these two effects can either strengthen or diminish the other. Our results indicate that the microstructure and material properties which lead to an optimum d33 effect are different from the conditions underlying the optimal d31 response. The maximum d31 effect is observed in electrets where the inclusions are mechanically harder but dielectrically softer than the matrix material. Finally, we find that a significantly large d33 piezoelectric response is possible for spheroid inclusion microstructures with large aspect ratios.