Geometry of polycrystals and microstructure

TL;DR

This paper explores the geometric properties of polycrystals, the microstructures resulting from intergrain contact, and the constraints on microstructures in martensitic transformations, providing new insights into grain geometry and microstructure formation.

Contribution

It establishes geometric properties of convex and general polycrystals and analyzes microstructure constraints in martensitic transformations, extending the understanding of intergrain microstructures.

Findings

Interior grains in convex polycrystals are polyhedral.

Triple points are small in general grain geometries.

Pure phase microstructures at grain boundaries are impossible under certain conditions.

Abstract

We investigate the geometry of polycrystals, showing that for polycrystals formed of convex grains the interior grains are polyhedral, while for polycrystals with general grain geometry the set of triple points is small. Then we investigate possible martensitic morphologies resulting from intergrain contact. For cubic-to-tetragonal transformations we show that homogeneous zero-energy microstructures matching a pure dilatation on a grain boundary necessarily involve more than four deformation gradients. We discuss the relevance of this result for observations of microstructures involving second and third-order laminates in various materials. Finally we consider the more specialized situation of bicrystals formed from materials having two martensitic energy wells (such as for orthorhombic to monoclinic transformations), but without any restrictions on the possible microstructure, showing how a generalization of the Hadamard jump condition can be applied at the intergrain boundary to show that a pure phase in either grain is impossible at minimum energy.