Isogeometric benchmark solutions for 3D geometrically nonlinear analysis of piezoelectric smart structures

To overcome the limitations of the current mainstream numerical analysis methods for piezoelectric smart structures based on the two-dimensional (2D) equivalent single-layer theory – which have limited capability in accurately capturing geometrically nonlinear behaviors and electric field distributions in thick structures and multilayered configurations – this paper presents a geometrically nonlinear isogeometric analysis (IGA) model based on three-dimensional (3D) elasticity theory. The geometrically nonlinear equations of motion for piezoelectric smart structures are systematically derived through the full Green–Lagrange strains and Hamilton’s principle within the framework of the Total Lagrangian (TL) formulation. In the established model, the displacement and electric potential fields are simultaneously obtained, via interpolation using 3D non-uniform rational B-splines (NURBS) basis functions. To verify the accuracy, effectiveness, and scope of application of the present methodology, several widely used nonlinear benchmark examples involving both conventional and piezoelectric smart structures are evaluated in detail. All numerical results, including static bending, dynamic, and shape control characteristics, are presented graphically and in tabular form for clear visualization and comparison. The maximum average error of the proposed method for nonlinear static analysis of conventional and piezoelectric smart structures is merely 0.8%. The numerical examples demonstrate that the proposed formulation can accurately capture the electromechanical coupling effects in the geometrically nonlinear static and dynamic responses of piezoelectric laminated structures, as well as their through-thickness field distributions. The present work establishes a reliable 3D nonlinear numerical framework for the analysis of piezoelectric smart structures and also lays a theoretical foundation for future extensions to more complex multiphysics coupling problems.