Mixed finite-element formulations in piezoelectricity and flexoelectricity

• Published in: Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Vol. 472; no. 2190; p. 20150879
• Main Authors: Mao, Sheng, Purohit, Prashant K., Aravas, Nikolaos
• Format: Journal Article
• Language: English
• Published: England The Royal Society Publishing 01.06.2016
• Subjects: Finite-Element; Flexoelectricity; Gradient Elasticity; Mixed Formulation; mixed formulation; finite-element; gradient elasticity; flexoelectricity
• ISSN: 1364-5021; 1471-2946
• DOI: 10.1098/rspa.2015.0879

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.