The influence of dislocation distribution density on curvature and interface stress in epitaxial thin films on a flexible substrate

We consider the problem of relaxation of coherency stresses by lattice misfit dislocations, which can be represented as edge dislocations uniformly distributed along the interface, and can be associated with the concept of dislocation density according to the mathematical theory of continuous distributions of dislocations. The orientation of dislocation line density is constrained within the plane of epitaxial layer and the Burgers vector density within the same plane is considered as a local variable of the problem. Under the external stress-free conditions the system is then enforced 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 is then connected with the deformation and rotation tensors and subsequently with the curvature of the system. The closed form solution of the problem is given for a generic one-dimensional case.