The problem of the relaxation of coherency stresses by lattice misfit dislocations was considered, which latter could be represented as edge dislocations uniformly distributed along the interface, and associated with the concept of dislocation density according to the mathematical theory of continuous distributions of dislocations. The orientation of dislocation line density was constrained within the plane of epitaxial layer and the Burgers vector density within the same plane was considered to be a local variable. Under external stress-free conditions the system was then forced to satisfy the boundary conditions according to the geometry of the epitaxial film and the substrate together with the corresponding compatibility conditions. The dislocation density tensor was then connected with the deformation and rotation tensors and subsequently with the curvature of the system. A closed-form solution was given for a generic one-dimensional case.

The Influence of Dislocation Distribution Density on Curvature and Interface Stress in Epitaxial Thin Films on a Flexible Substrate. I.Dobovšek: International Journal of Mechanical Sciences, 2010, 52[2], 212-8