A flexoelectric spherical microshell model incorporating the strain gradient effect

Abstract

• The size-dependent theoretical model of flexoelectric spherical microshell is proposed. • The analytical solutions of direct and converse flexoelectric responses are obtained. • Flexoelectric responses in spherical microshells with and without a central opening are discussed. • There is optimal central angle for both direct and converse flexoelectric responses. • Strain gradient effect will reduce the total flexoelectric responses. In this paper, a size-dependent flexoelectric spherical microshell model is proposed considering flexoelectric effect and strain gradient effect. By means of the variation principle, explicit expressions of the governing equations and the boundary conditions are deduced. Solving corresponding governing equations, analytical solutions of both direct and converse flexoelectric responses in static axisymmetric bending problem are obtained. Then, the flexoelectric responses in barium strontium titanate spherical microshells with and without a circular top opening are numerically investigated. Both the direct and converse flexoelectric responses are found to vary non-monotonically as the central angle increases. The converse flexoelectric bending is examined to exist even in clamped spherical microshells, which is different from the case of flat structures. In addition, for all cases, the strain gradient effect will highly reduce the flexoelectric responses, particularly when the thickness approaches the material internal scale constants.