title: Mixed finite-element formulations in piezoelectricity and flexoelectricity.
abstract: Flexoelectricity, the linear coupling of strain gradient and electric polarization, is inherently a size-dependent phenomenon. The energy storage function for a flexoelectric material depends not only on polarization and strain, but also strain-gradient. Thus, conventional finite-element methods formulated solely on displacement are inadequate to treat flexoelectric solids since gradients raise the order of the governing differential equations. Here, we introduce a computational framework based on a mixed formulation developed previously by one of the present authors and a colleague. This formulation uses displacement and displacement-gradient as separate variables which are constrained in a 'weighted integral sense' to enforce their known relation. We derive a variational formulation for boundary-value problems for piezo- and/or flexoelectric solids. We validate this computational framework against available exact solutions. Our new computational method is applied to more complex problems, including a plate with an elliptical hole, stationary cracks, as well as tension and shear of solids with a repeating unit cell. Our results address several issues of theoretical interest, generate predictions of experimental merit and reveal interesting flexoelectric phenomena with potential for application.
authors.affiliation: Department of Mechanical Engineering and Applied Mechanics , University of Pennsylvania , Philadelphia, PA 19104, USA.
authors.forename: Sheng
authors.identifier: 0000-0001-9468-5095
authors.initials: S
authors.lastname: Mao
authors.affiliation: Department of Mechanical Engineering and Applied Mechanics , University of Pennsylvania , Philadelphia, PA 19104, USA.
authors.forename: Prashant K
authors.identifier: 0000-0003-3087-6233
authors.initials: PK
authors.lastname: Purohit
authors.affiliation: Department of Mechanical Engineering, University of Thessaly, 38334 Volos, Greece; International Institute for Carbon Neutral Energy Research (WPI-I2CNER), Kyushu University, 744 Moto-oka, Nishi-ku, Fukuoka 819-0395, Japan.
authors.forename: Nikolaos
authors.identifier: 0000-0001-6894-3716
authors.initials: N
authors.lastname: Aravas
keywords: finite-element; flexoelectricity; gradient elasticity; mixed formulation