Modelling coupled electro-mechanical phenomena in elastic dielectrics using local conformal symmetry

Abstract

The local scaling symmetry of the Lagrange density is exploited to investigate the myriad electro-mechanical coupling effects observed in the elastic dielectrics. In contrast to most known flexoelectricity theories, this approach has also explicated on the geometric underpinnings of the induced polarization and electric potential. Due to the inhomogeneous scaling of the metric, the invariance of the Lagrange density will be lost. To restore the gauge invariance of the Lagrange density, we introduce the notion of minimal replacement, viz., define a gauge covariant operator in place of the ordinary partial derivative. Minimal replacement introduces gauge compensating one form field. We relate the exact part of the gauge compensating one form field with electric potential. However, the gauge compensating one form field's anti-exact part is correlated with the electric polarization vector. We constructed different components of the gauge invariant energy density using the geometric objects like scale invariant gauge curvature. We appropriately contracted the skew symmetric gauge curvature tensor with the metric to construct the scale invariant energy associated with the gauge field. Finally, we derive the governing coupled equations employing Hamilton's principle. In order to assess how the theory performs, we carry out a few numerical simulations and validation. Model prediction of the pressurized flexoelectric disk with a central hole shows good agreement with the analytical solution. We also investigated flexoelectric cantilever beam subjected to a point load and different earthing conditions. Explorations of this kind of coupling may have notable implications in many industrial and laboratory applications.