On the elastic problem of representative volume element for multiphase thin films

Multiphase thin films exhibit unique physical functionalities stemming from their dimensions and interactions among phases. In these materials, elasticity plays an important role both in their growth and physical performance. An outstanding problem in this regard is the elastic formulation of representative volume element (RVE) of thin film systems. As thin films RVEs lack translation invariance in the direction perpendicular to the film free surface, the boundary value problem of the RVE involves integral kinematic boundary constraints that must be satisfied together with the governing elastic boundary value problem. These constraints were developed here as a part of a homogenization scheme designed to deliver the elastic solution in a heterogeneous thin film, with both eigenstrain and modulus mismatch within the phases. We formulated this problem together with an iterative solution scheme based on Fast Fourier Transform with an augmented Lagrangian fixed-point iteration algorithm. The numerical solution was benchmarked with the analytical solution of the famous Eshelby problem for the case of homogeneous and inhomogeneous cylindrical inclusions. Diffuse interface and discrete Green's operator methods were tested to investigate the attenuation of Gibbs oscillations at interfaces. Examples of thin film morphologies generated using kinetic Monte Carlo simulations at different growth conditions were used as microstructure input to test the current approach. We show that the elastic energy tends to be concentrated near the pillar/matrix interface. This approach is expected to enable the on-the-fly coupling of elasticity solution with thin film growth models to account for the elastic strain effects on diffusion, bonding, and interfacial energies.