Analysis of Flexoelectric Solids with a Cylindrical Cavity

Flexoelectricity, a remarkable size-dependent effect, means that strain gradients can give rise to electric polarization. This effect is particularly pronounced near defects within flexoelectric solids, where large strain gradients exist. A thorough understanding of the internal defects of flexoelectric devices and their surrounding multiphysics fields is crucial to comprehend their damage and failure mechanisms. Motivated by this, strain gradient elasticity theory is utilized to investigate the mechanical and electrical behaviors of flexoelectric solids with cylindrical cavities under biaxial tension. Closed-form solutions are obtained under the assumptions of plane strain and electrically impermeable defects. In particular, this study extends the Kirsch problem of classical elasticity theory to the theoretical framework of higher-order electroelasticity for the first time. Our research reveals that different length scale parameters of the strain gradient and bidirectional loading ratios significantly affect the hoop stress field, radial electric polarization field, and electric potential field near the inner cylindrical cavity of the flexoelectric solid. Furthermore, we validate our analytical solution by numerical verification using mixed finite elements. The congruence between the two methods confirms our analytical solution's accuracy. The findings presented in this paper provide deeper insights into the internal defects of flexoelectric materials and can serve as a foundation for studying more complex defects in flexoelectric solids.