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<p>Holes of different shapes usually appear in piezoelectric devices for functional or design purposes. This paper aims to obtain exact three-dimensional (3D) analytical solutions for a transversely isotropic and functionally graded piezoelectric material (FGPM) plate containing a circular hole under mechanical loading. Based on the extended England–Spencer plate theory, this study advances the framework by replacing elastic materials with piezoelectric materials. Within the context of 3D piezoelectricity, the expression for the electric potential function is constructed by referencing the form of the displacement field. Using the complex function method, four complex potential functions of the plate’s mid-plane are derived to describe the 3D electromechanical responses. Two typical loading conditions are considered: mechanical loading applied at infinity and on the hole edge, respectively. In numerical examples, the proposed analytical solution is first verified by comparison with existing solutions in the literature. Subsequently, the influence of material gradation and loading conditions on the 3D stresses and electric displacements near the circular hole is investigated, with a focus on comparing the responses of FGPM and functionally graded material (FGM) plates. The present analytical method provides an effective 3D approach for solving hole problems in FGPM plates. The derived 3D solution can also serve as a reliable benchmark for validating solutions obtained from various approximate methods.</p>

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Three-Dimensional Analytical Solution for Functionally Graded Piezoelectric Plates with a Circular Hole

  • 期刊论文

  • 发表日期：

    2026年4月21日

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  • Youyuan Zhang,
  • Lulu Shen,
  • Weiqiu Chen,
  • Bo Yang

摘要

Holes of different shapes usually appear in piezoelectric devices for functional or design purposes. This paper aims to obtain exact three-dimensional (3D) analytical solutions for a transversely isotropic and functionally graded piezoelectric material (FGPM) plate containing a circular hole under mechanical loading. Based on the extended England–Spencer plate theory, this study advances the framework by replacing elastic materials with piezoelectric materials. Within the context of 3D piezoelectricity, the expression for the electric potential function is constructed by referencing the form of the displacement field. Using the complex function method, four complex potential functions of the plate’s mid-plane are derived to describe the 3D electromechanical responses. Two typical loading conditions are considered: mechanical loading applied at infinity and on the hole edge, respectively. In numerical examples, the proposed analytical solution is first verified by comparison with existing solutions in the literature. Subsequently, the influence of material gradation and loading conditions on the 3D stresses and electric displacements near the circular hole is investigated, with a focus on comparing the responses of FGPM and functionally graded material (FGM) plates. The present analytical method provides an effective 3D approach for solving hole problems in FGPM plates. The derived 3D solution can also serve as a reliable benchmark for validating solutions obtained from various approximate methods.