Size dependent stability analysis of circular ultrathin films in elastic medium with consideration of surface energies

In this article, analytical framework is developed for size dependent symmetric stability and self-instability of circular nanoplates including surface effects using modified Kirchhoff plate theory. The surrounding elastic medium is modeled as Winkler elastic foundation and its effect is comprehensively studied on self-instability problems. The derived explicit solutions contain Bessel functions with modified arguments reflecting the size dependency of the buckling loads. In order to check the results an inverse formulation is presented for effective Young's modulus using the buckling loads to be verified by previous experimental results for nanowires. Several numerical examples are given for two types of materials with positive and negative surface properties to show the general trends of size dependencies. Some problems and limitations are explored for consistency of results with experiments and suggestions for future works.

Research highlights
► Explicit solutions are presented for stability and self-instability of embedded circular nanoplates including surface effects. For this purpose classical plate theory is generalized to cover the surface effects through analytical approach. In this manner the effect of elastic bed is comprehensively depicted on self-instability of circular nanoplates.
► The general trends of size dependencies are shown with respect to different geometric parameters, elastic foundation and mode numbers. In order to give a complete set of numerical examples, two types of materials with positive and negative surface properties are examined in numerical analyses.
► Some contradictions are explored for the effect of additional surface properties on buckling of macroscaled circular plates. Accordingly, self-instability problem due to surface residual stresses may occur even in macroscale, which must be considered by designers.
► An inverse formulation is proposed to calculate the effective or equivalent Young's modulus of circular nanoplates to compare them with the existing experimental results for nanowires. This idea is taken from a basic concept of macro-mechanics in which the results of unidirectional tension are used even in 3D elasticity analyses.