Landau theory of a constrained ferroelastic in two dimensions

The Landau expansion of the elastic energy in powers of the strains and their derivatives is applied to the ferroelastic transformation of a grain constrained so that the displacement vanishes on the boundaries of the grain; the model applies strictly only to the square-rectangular transformation, but some results may apply also to the tetragonal-orthorhombic transformation. The displacement and the strains are obtained by numerical minimization of the elastic energy (with respect to the displacement) for a square column with edges parallel to the 100 and 010 planes of the tetragonal phase. The structure obtained is a sequence of twin boundaries (parallel to the 110 planes of the parent phase) with nonzero dilatational and shear strains near the boundaries. The mean-field transformation temperature ${\mathit{T}}_{\mathit{c}}$(L) is depressed from the bulk value due to the finite width L of the grain, behaving roughly as ${\mathit{T}}_{\mathit{c}}$(L)=${\mathit{T}}_{\mathit{c}}$(∞)-const/L.