Abstract
Flexoelectricity is characterised by the coupling of the second gradient of the motion and the electrical field in a dielectric material. The presence of the second gradient is a significant obstacle to obtaining the approximate solution using conventional numerical methods, such as the finite element method, that typically require a C1-continuous approximation of the motion. A novel micromorphic approach is presented to accommodate the resulting higher-order gradient contributions arising in this highly-nonlinear and coupled problem within a classical finite element setting. Our formulation accounts for all material and geometric nonlinearities, as well as the coupling between the mechanical, electrical and micromorphic fields. The highly-nonlinear system of governing equations is derived using the Dirichlet principle and approximately solved using the finite element method. A series of numerical examples serve to elucidate the theory and to provide insight into this intriguing effect that underpins or influences many important scientific and technical applications.
Highlights
In a piezoelectric material, an applied uniform strain can induce electric polarisation
Flexoelectricity1 is the property of an insulator whereby it polarises when subjected to an inhomogeneous deformation
Flexoelectricity can occur in materials of any symmetry, broadening the range of materials for use as actuators and sensors [1]
Summary
An applied uniform strain can induce electric polarisation (or vice versa). Romeo [24] directly accounted for electromagnetic contributions at the microscale in a micromorphic framework by accounting for electric dipole and quadrupole densities This theory was extended to account for dielectric multipoles [25,26] and thereby describe the piezoelectric and flexoelectric effects. A central impediment to developing finite element models for flexoelectricity, or gradient elasticity, is the requirement that the basis functions used to approximate the displacement field must be piecewise smooth and globally C1-continuous. This constraint arises as the partial differential equation governing the mechanical problem is of fourth-order.
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