Analytical model for flexoelectric sensing of structural response considering bonding compliance

Flexoelectricity has generated huge interest as an alternative to piezoelectricity for developing electromechanical systems such as actuators, sensors, and energy harvesters. This article presents a generic theoretical framework for the sensing mechanism of a flexoelectric sensor bonded to a host beam through an adhesive layer. The model incorporates piezoelectric and flexoelectric effects and considers both shear-lag and peel stresses at the sensor-beam interface. The formulation also includes the electric field gradient terms that are often overlooked. Consistent one-dimensional constitutive relations and governing equations of equilibrium are derived from the electric Gibb’s energy density function and extended Hamilton’s principle. The sensor is assumed to follow the Euler–Bernoulli beam-type membrane and bending deformation behaviour. Closed-form solutions are obtained for the interfacial stresses by analytically solving a seventh-order non-homogeneous ordinary differential equation, satisfying the stress-free boundary conditions at the sensor edges. The induced electric potential at the sensor top is derived by solving a fourth-order differential equation obtained from the charge balance equation, satisfying the electric boundary conditions. For validation, the sensor output is compared with the results of the existing non-rigid bonding piezoelectric sensor model. Numerical results show a significant impact of non-rigid bonding and the electric field gradient terms on the induced electric potential. Further, the importance of bonding compliance on the interfacial stress distributions is illustrated. Finally, the effects of adhesive and transducer thicknesses on the peak amplitudes of interfacial stresses and sensory potential are presented.