Analysis of a two-dimensional invariant line interface for the case of a general transformation strain and application to thin-film interfaces

An explicit solution is obtained for a general two-dimensional deformation matrix between two phases containing any combination of expansion/contraction and/or shear. Application of the solution to various cases of deformation shows that in the presence of a shear, an invariant line can be obtained even when the deformations along orthogonal directions are both contractions or expansions. Utilization of the solution in the design of thin-film overgrowths on substrates and the relationship between the solution eigenvectors in real and reciprocal space are discussed.