Supplementary material for "Modified nonlocal strain gradient theory for static bending, free vibration and buckling analysis of functionally graded piezoelectric nanoplates"

A novel modified nonlocal strain gradient is employed in this research for the comprehensive analysis of functionally graded piezoelectric nanoplates. This is a unique theory that is compatible for analysis of a wide range of small-scale structures varying from nano-scale to micro-scale dimensions. Both nonlocal (softening) and strain gradient (hardening) effects are considered simultaneously in this novel theory. This theory simultaneously incorporates scale-dependent effects into both the stress–strain and electric displacement–electric field relationships, which is a significant advancement over previous approaches. To satisfy the Maxwell equation in the quasi-static approximation, an electric potential field that contains a cosine and linear variation is utilized. The governing equations of motion are then established through the integration of the modified nonlocal strain gradient theory with a higher-order shear deformation theory and Hamilton’s principle. Validation of accuracy and efficiency of the present theory and calculated algorithm is achieved by comparing the calculated results with the available solutions in various examples. Numerical investigations are carried out to explore the influence of material gradation, scale parameters, and electro-mechanical loading on the static bending, free vibration, and buckling behaviors of functionally graded piezoelectric nanoplates. The proposed theory provides a unified and reliable tool for the design and analysis of next-generation piezoelectric nanostructures.