Size-dependent free vibration analysis of Mindlin nano-plates with curvilinear plan-forms by a high order curved hierarchical finite element

New analytic bending, buckling, and free vibration solutions of rectangular nanoplates by the symplectic superposition method

“…Equations (19)(20)(21)(22) are written in matrix form as w h e re Z = [w, θ x , V x , M x ] T i s re fe r re d t o a s t h e s t at e ve c t or ; H = F G Q −F T , w i t h…”

“…Bahu and Patel 21 developed an improved quadrilateral finite element for nonlinear second-order strain gradient elastic Kirchhoff plates based on the nonlocal theory. Necira 22 developed the hierarchical finite element method for size-dependent free vibration analysis of Mindlin nano-plates with curvilinear plan-forms. Akgöz and Civalek 23 employed the discrete singular convolution method for the free vibration and bending analysis of nano-scaled graphene sheets having sector shape.…”