Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory

A generalized refined theory including surface effects is developed for functionally graded ultra-thin films with different surface properties. The classical generalized shear deformable theory is adopted to model the film bulk, while the bulk stresses along the surfaces of the bulk substrate are required to satisfy the surface balance equations of the continuum surface elasticity. As a result, the shape function also shows size-dependence on the film thickness. Since the film is non-homogeneous through the thickness, the state space method and approximate laminate model are employed to derive the variation of shape function through the thickness direction. A simply supported thin film in cylindrical bending is considered as an example to illustrate the application of the present theory. By comparing to the Kirchhoff plate theory including surface effects, the necessity of the present theory for FGM thin films is solidly validated. It is established that the present FGM thin films exhibit significant size-dependence when the thickness approaches to micro-scale. As the gradient index changes, the extent of size-dependence varies accordingly.