Finite element analysis of functionally graded nano-scale films

• Investigating Mindlin bending behavior of ultra-thin FG films.
• Accounting for the position of neutral plane and incorporating surface energy effects.
• Developing a finite element model to study the mechanical behavior of ultra-thin FG films.
• Size-dependent analysis of ultra-thin FG films based on Gurtin and Murdoch surface conditions.

In this paper, a size-dependent finite element model, for Mindlin plate theory accounting for the position of the neutral plane for continuum incorporating surface energy effect, is proposed to study the bending behavior of ultra-thin functionally graded (FG) plates. The size-dependent mechanical response is very important while the plate thickness reduces to micro/nano scales. The classical finite element model is adopted to allow insertion of the surface energy into the total energy of the plate. Bulk stresses on the surfaces are required to satisfy the surface balance conditions involving surface stresses. Therefore, unlike the classical continuum plate models, the bulk transverse normal stress is preserved here. Moreover, unlike most of previous studies in the literature, the exact neutral plane position is pre-determined and considered for FG plates. A series of continuum governing differential equations which include surface energy and neutral plane position effects are derived. A comparison between the continuum analysis of FG ultra-thin plates with and without incorporating surface energy effects is presented.