Size-dependent direct and converse flexoelectricity around a micro-hole
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O R I G I NA L PA P E R
Xinpeng Tian · Mengkang Xu · Qian Deng · Jan Sladek · Vladimir Sladek · Miroslav Repka · Qun Li
Size-dependent direct and converse flexoelectricity around a micro-hole
Received: 16 December 2019 / Revised: 12 July 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract In this paper, the complete explicit solution for physical fields around a micro-hole is solved with the simultaneous consideration of the strain gradient elasticity, the direct flexoelectricity, and the converse flexoelectricity. First, the higher-order Navier-like governing equations are proposed for an isotropic flexoelectric solid by using an extended linear theory of flexoelectric materials considering the coupling between the strain gradient and the polarization, and conversely between the polarization gradient and the strain. Second, the displacement, the electric potential, and the polarization are successfully obtained by solving the corresponding boundary value problems for a micro-hole model. Finally, the influence of the strain gradients and the flexoelectric effect on the mechanical fields and the electric responses are studied around the micro-hole in flexoelectric solids, and the size effects of the direct flexoelectricity and the converse flexoelectricity are also investigated. The results indicate that an electric response can be obviously induced by a mechanical loading due to the direct flexoelectricity. Conversely, a mechanical strain can be produced by an electric field through the converse flexoelectricity. The size dependence of both the direct flexoelectricity and the converse flexoelectricity is also successfully predicted in this paper, in the form that the flexoelectricity significantly increases with the decrease in the sample size. Keywords Flexoelectricity · Strain gradient elasticity · Micro-hole · Physical fields · Size effects
1 Introduction Flexoelectricity is an electromechanical coupling property of all dielectric materials, whereby an electric polarization is generated by an inhomogeneous deformation with a finite strain gradient. Because of the size dependence of strain gradients, the flexoelectric effect is most conspicuous for microscale dielectrics [1–5]. Flexoelectricity in elastic dielectrics was originally pointed out by Mashkevich and Tolpygo [6] and Toupin [7] in the study of lattice dynamics. Thereafter, Mindlin and Eshel investigated the electromechanical phenomenon in centrosymmetric dielectrics accounting for the polarization gradient [8], the strain gradient, and the surface energy [9]. On the basis of Mindlin’s framework, Sharma and his co-workers [10–12] proposed an extended linear theory for centrosymmetric dielectrics with the direct flexoelectricity (DF), the converse flexoelectricity (CF), and the strain gradient elasticity (SGE). Shen and his collaborators [13–15] presented an electric enthalpy variational principle for nanosized flexoelectric dielectrics concerning with the electrostatic force and the X. Tian · M. Xu · Q. Deng · Q. Li (B) State
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