Elastic mechanical behavior of nano-scaled FGM films incorporating surface energies

A general, global theory is developed for nano-scaled functionally graded films considering surface effects. In addition to the Kirchhoff hypothesis of the classical thin plate theory, the surface layers of the film are modeled by the continuum theory of surface elasticity. Bulk stresses on the surfaces are required to satisfy the surface balance conditions involving surface stresses. Unlike the classical plate theory, the bulk transverse normal stress is preserved here. By incorporating the surface energies into the principle of minimum potential energy, a series of non-classical governing differential equations which include intrinsic length scales are derived. To illustrate application of the theory, a simply supported nano-scaled film in cylindrical bending is investigated. Numerical examples are presented to clarify the effects of surface energies on the bending behavior of FGM films, whose effective elastic moduli are predicted using the Mori–Tanaka method. Finally, the nature of intrinsic length scales, and the effects of gradient index and aspect ratio on the displacements are discussed.