The relations of a local gradient non-ferromagnetic electroelastic continuum are used to solve the problem of an axisymmetrical loaded hollow cylinder. Analytical solutions are obtained for tetragonal piezoelectric materials of point group 4 mm for two cases of external loads applied to the body surfaces. Namely, the hollow pressurized cylinder and a cylinder subjected to an electrical voltage V across its thickness are considered. The derived solutions demonstrate that the non-uniform electric load causes a mechanical deformation of piezoelectric body, and vice versa, the inhomogeneous radial pressure of the cylinder induces its polarization. Such a result is obtained due to coupling between the electromechanical fields and a local mass displacement being considered. In the local gradient theory, the local mass displacement is associated with the changes to a material’s microstructure. The classical theory does not consider the effect of material microstructure on the behavior of solid bodies and is incapable of explaining the mentioned phenomena. It is also shown that the local gradient theory describes the size-dependent properties of piezoelectric nanocylinders. Analytical solutions to the formulated boundary-value problems can be used in conjunction with experimental data to estimate some higher-order material constants of the local gradient piezoelectricity. The obtained results may be useful for a wide range of appliances that utilize small-scale piezoelectric elements as constituting blocks.

利用局部梯度非铁磁电弹性连续体的关系解决轴对称加载空心圆柱体问题。获得了点群为 4 mm 的四方压电材料在两种外载荷作用于体表的情况下的解析解。即空心加压圆柱体和承受电压V的圆柱体考虑其厚度。推导出的解表明不均匀的电负载会导致压电体的机械变形，反之亦然，圆柱体的不均匀径向压力会导致其极化。由于机电场和所考虑的局部质量位移之间的耦合，获得了这样的结果。在局部梯度理论中，局部质量位移与材料微观结构的变化有关。经典理论没有考虑材料微观结构对固体行为的影响，无法解释上述现象。还表明局部梯度理论描述了压电纳米圆柱体的尺寸相关特性。公式化的边值问题的解析解可以与实验数据结合使用，以估计局部梯度压电的一些高阶材料常数。获得的结果可能对使用小型压电元件作为构成块的各种器具有用。