Compatibility of Martensitic Microstructures in Polycrystals
arXiv:2607.12572
The paper analyzes how martensitic microstructures can be compatible across grain boundaries in polycrystals, focusing on cubic-to-tetragonal and cubic-to-orthorhombic phase transformations and introducing bounds for the Taylor set of deformation gradients.
Abstract
The paper studies martensitic microstructures in polycrystals, focussing on their compatibility across grain boundaries. After a reduction to the case of a planar grain boundary, the case when the grain boundary separates two constant gradients of zero energy is considered. It is shown that for cubic-to tetragonal transformations such a configuration can occur when the relative grain rotation is not in the cubic group. Then the case when the grain boundary separates two simple laminates of zero energy is considered, it being shown using a computer-assisted symbolic calculation that in the cubic-to-tetragonal case compatibility is only possible for a closed set of measure zero in the manifold of grain boundary normals and relative grain rotations, and that a similar slightly weaker result holds for cubic-to-orthorhombic transformations. The results suggest why higher-order laminates are often observed for such transformations. The Taylor set of deformation gradients is defined and studied, this set having the property that any deformation whose gradient belongs to it corresponds to a zero-energy microstructure for the polycrystal independent of grain geometry and grain rotations. New upper bounds for the Taylor set are proved for cubic-to-tetragonal and cubic-to-orthorhombic transformations, generalizing those of Bhattacharya and Kohn using the geometrically linearized theory. We give a simple proof of a related result of Peigney characterizing the positive diagonal matrices in the quasiconvex hull of the energy wells for cubic-to-tetragonal transformations.
28 pages. Misleading statement corrected in introduction and abstract