A directed continuum model of micro- and nano-scale thin films

As is well known, classical continuum theories cease to adequately model a material's behavior as long-range loads or interactions begin to have a greater effect on the overall behavior of the material, i.e., as the material no longer conforms to the locality requirements of classical continuum theories. It is suggested that certain structures to be analyzed in this work, e.g., columnar thin films, are influenced by non-local phenomena. Directed continuum theories, which are often used to capture non-local behavior in the context of a continuum theory, will therefore be used. The analysis in this work begins by establishing the kinematics relationships for a discrete model based on the physical structure of a columnar thin film. The strain energy density of the system is calculated and used to formulate a directed continuum theory, in particular a micromorphic theory, involving deformations of the film substrate and deformations of the columnar structure. The resulting boundary value problem is solved analytically to obtain the deformation of the film in response to applied end displacements.