Computational evaluation of the flexoelectric effect in dielectric solids

Abstract

Flexoelectricity is a size-dependent electromechanical mechanism coupling polarization and strain gradient. It exists in a wide variety of materials, and is most noticeable for nanoscale objects, where strain gradients are higher. Simulations are important to understand flexoelectricity because experiments at very small scales are difficult, and analytical solutions are scarce. Here, we computationally evaluate the role of flexoelectricity in the electromechanical response of linear dielectric solids in two-dimensions. We deal with the higher-order coupled partial differential equations using smooth meshfree basis functions in a Galerkin method, which allows us to consider general geometries and boundary conditions. We focus on the most common setups to quantify the flexoelectric response, namely, bending of cantilever beams and compression of truncated pyramids, which are generally interpreted through approximate solutions. While these approximations capture the sizedependent flexoelectric electromechanical coupling, we show that they only provide order-of-magnitude estimates as compared with a solution fully accounting for the multidimensional nature of the problem. We discuss the flexoelectric mechanism behind the enhanced size-dependent elasticity in beam configurations. We show that this mechanism is also responsible for the actuation of beams under purely electrical loading, supporting the idea that a mechanical flexoelectric sensor also behaves as an actuator. The predicted actuation-induced curvature is in a good agreement with experimental results. The truncated pyramid configuration highlights the critical role of geometry and boundary conditions on the effective electromechanical response. Our results suggest that computer simulations can help understanding and quantifying the physical properties of flexoelectric devices. (c) 2014 AIP Publishing LLC.Postprint (published version

Revisiting pyramid compression to quantify flexoelectricity: A three-dimensional simulation study

“…Introducing these expansions into the continuum total electromechanical enthalpy 35 , we obtain its discrete representation as…”

“…We refer to Ref. 35 for a complete description of the theory and its Galerkin numerical discretization. For the reader's convenience, we provide in A details about the numerical implementation in 3D.…”

“…On rectangular or brick geometries, finite difference calculations have been applied to flexoelectricity [31][32][33] . To deal with more general geometries with non-uniform grid refinement, we have recently resorted to meshfree methods, relying on smooth basis functions 34 , to solve numerically the continuum equations of flexoelectricity in 2D 35 . Surprisingly, we found that previous simplified calculations on beam and truncated triangle configurations provided only rough order-of-magnitude estimations of the flexoelectric response.…”

An immersed boundary hierarchical B-spline method for flexoelectricity

“…The equations are a coupled system of 4th-order partial differential equations, which renders analytical solutions difficult to obtain and precludes the use of conventional C 0 finite elements. Several numerical alternatives have been proposed in the literature, based on smooth approximations with at least C 1 continuity [22][23][24][25][26][27][28] or on mixed formulations [29,30]. The first self-consistent numerical solution of the linear flexoelectric problem was provided by Abdollahi et.…”

Modelling the flexoelectric effect in solids

“…Many of the aforementioned methods to generate C 1 -continuous finite element approximations have been used to model the problem of flexoelectricity. Abdollahi et al [27] chose a meshless method. The analysis was restricted to two dimensions and to the linearised theory.…”