A theoretical formulation for flexoelectric membranes
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Flexoelectricity is electric polarization induced by a strain gradient. This phenomenon has been observed in different types of materials. It has been studied and documented that biological membranes possess this flexoelectric property. Research by Brownell et al. has shown that the inner-ear hair cells elongate or shrink as a result of external stimuli. This shrinking and elongation of the cells is due to the wrinkling and flattening of the cell membrane surrounding the hair cells. To study this biomechanical response, a soft, elastomeric membrane under loading by in-plane forces, moments and voltage across the membrane is considered. This membrane is assumed to have the flexoelectric property. Using a thermodynamic approach, a set of constitutive equations are derived that relate the physical quantities (forces, moments, voltage) to the state variables (strains, curvatures, electric displacement). The accuracy of these equations is tested by using them to estimate the flexoelectric coefficient of certain materials following a procedure established by Cross et al. Additionally, a critical membrane thickness is found which ensures a positive-definite Helmholtz free energy for the membrane, guaranteeing a stable system. Finally, a model for the wrinkling and flattening of the cell membrane surrounding the inner-ear outer hair cells is developed using the derived constitutive equations.