Stability of microstructure for tetragonal to monoclinic martensitic transformations

We give an analysis of the stability and uniqueness of the simply laminated microstructure for all three tetragonal to monoclinic martensitic transformations. The energy density for tetragonal to monoclinic transformations has four rotationally invariant wells since the transformation has four variants. One of these tetragonal to monoclinic martensitic transformations corresponds to the shearing of the rectangular side, one corresponds to the shearing of the square base, and one corresponds to the shearing of the plane orthogonal to a diagonal in the square base. We show that the simply laminated microstructure is stable except for a class of special material parameters. In each case that the microstructure is stable, we derive error estimates for the nite element approximation. AMS Subject Classi cation. 49J45, 65N15, 65N30, 73C50, 73G05, 73K20, 73V05. The dates will be set by the publisher.