A directed continuum model of a columnar thin film

by Randow, Charles L.

As is well known, classical continuum theories fail to adequately describe material behavior as long-range loads or interactions begin to have a significant effect on the overall behavior of the material. For example, when the presence of atoms, grain boundaries, cracks, inclusions, or pores must be considered, the material may no longer conform to the locality requirements of classical continuum theories. One particular example of such a system is a columnar thin film (CTF), which consists of regularly spaced columns (on the scale of tens of nanometers or more in diameter and one or more microns in height) attached to a substrate. This structure may experience loading conditions due to a variety of sources, including the manufacturing process or in use. As a result of the heterogeneous nature of a CTF, the film is influenced by non-local phenomena. A directed continuum theory (also known as Cosserat, micromorphic, or micropolar theory) will be used to capture the non-local behavior of the film, although the directed continuum theory is itself a local theory. The analysis in this work begins by establishing the kinematics relationships for a discrete model, inspired by the physical structure of a CTF, and determining the discrete form of the governing equations. A Taylor series expansion of the displacement terms used in the discrete governing equations is used to obtain a continuous form of the governing equations. This work proposes a strain energy density that, following established directed continuum formulations, yields both the identical set of governing equations (found via the Taylor series) as well as the boundary conditions, i.e., a linear homogeneous boundary value problem (BVP). The BVP is analyzed to gain insight into the relationship between the behavior of the model and the input parameters. It is also solved to demonstrate the variety of deformations that may result from different boundary conditions. In addition, a non-linear discrete model is also introduced and compared with the continuum model. iii