As is well known, classical continuum theories fail to adequately describe material behavior when the scale of applied loads approaches the scale of the material structure (e.g., atoms, grains, cracks, pores) and when locality requirements are not satisfied, i.e., there are long-range forces affecting the body (e.g., atomic interactions). A thin film possessing a columnar structure is an example of such a physical system since any deformation of the film results in interaction between the columns. These interaction forces are transferred to the film substrate and certainly introduce a non-local affect on the substrate, although the substrate may be considered homogeneous when compared with the length scale of the columns.
The analysis in this work begins by establishing the kinematics relationships for a two-dimensional discrete unit cell mode. The total strain energy of the unit cell is calculated and used to develop Lagrange's equations for the quasi-static case. These equations provide the basis for the continuum governing equations, which are determined through a homogenization process by applying Taylor series expansions of the displacement fields. Numerical solutions of the resulting micropolar continuum model comparing the deformation of the columnar thin film with a traditional Euler-Bernoulli beam are presented. The goals of this work are to use micropolar theories to account for non-local affects and to develop models that can be used in finite element calculations to simulate and predict the mechanical properties of columnar thin films both during deposition and in service.
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