Nonlinear Displacement Analysis of Functionally Graded Flexoelectric-Piezoelectric Nanobeams

The integration of flexoelectricity with piezoelectricity has recently attracted considerable interest due to its ability to enhance electromechanical coupling at the nanoscale. This paper presents a nonlinear finite element framework for the static analysis of functionally graded piezoelectric--flexoelectric nanobeams undergoing nonlinear displacements. The governing equations are derived using Hamilton’s principle and incorporate nonlinear von Kármán strains, flexoelectric and piezoelectric effects, and power-law distributions of constituent materials along the nanobeam's longitudinal and transverse axes. The weak form is discretized using a mixed finite element formulation with the penalty method. Validation against homogeneous nanobeam cases demonstrates the accuracy and robustness of the proposed formulation. Comprehensive parametric studies are conducted to explore the influence of gradient indices, boundary conditions, and material composition on beam deflections. Results reveal that compliant polymer--ceramic combinations, such as PVDF--BaTiO$_3$, exhibit significantly larger deflections compared with purely ceramic configurations, highlighting the role of low-modulus constituents in enhancing flexibility. In contrast, stiffer ceramics such as SrTiO$_3$ and PbTiO$_3$ increase overall rigidity and reduce deflections. The developed framework provides new insights into the coupled electromechanical behavior of graded piezoelectric--flexoelectric nanobeams, offering valuable guidelines for the design of next-generation nanoelectromechanical systems. Applications include highly sensitive sensors, ultra-precise actuators, and nanoscale energy harvesters, where optimized material distributions can maximize electromechanical efficiency.