Geometry of polycrystals and microstructure

Abstract

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-totetragonal 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.

Interaction of Martensitic Microstructures in Adjacent Grains

“…We thus assume that 0 < α < π 2 , since this covers all nontrivial cases. As remarked in [5], by a result from [9] there always exists a zero-energy microstructure constructed using laminates, with gradient Young measure ν x = ν satisfying (1.2) that is independent of x and has macroscopic deformation gradientν =…”

“…In general one can make a linear transformation of variables in the reference configuration which turns the corresponding energy wells into the form (1.1). However, in [4, Section 4.1] and the announcement of the results of the present paper in [5] it was incorrectly implied that the analysis based on K as in (1.1) applies to a general orthorhombic to monoclinic transformation. This is not the case because the linear transformation in the reference configuration changes the deformation gradient corresponding to austenite in [4] and to the rotated grain in the present paper.…”

“…This extra condition, typically satisfied in practice, was accidentally omitted in the announcement in[5].…”

On the interplay of anisotropy and geometry for polycrystals in single-slip crystal plasticity

“…Compared with the single-crystal case, the analysis of polycrystals holds additional challenges related to the geometry and orientation of the different grains, as well as to the compatibility of microstructures across grain boundaries; in the context of linear and nonlinear elasticity, the latter has been studied in [11,12] and [6,7], respectively. The description of polycrystal geometry in [6] by Ball & Carstensen in combination with the modeling of [16,22] constitutes the basis for our framework of polycrystalline finite crystal plasticity with one active slip system under the assumption of elastically nonlinear but rigid behavior; the detailed setup is given in Section 1.1. Let us remark that the stress is not well-defined in this elastically rigid strain-based setting; for an analysis of a stress-based formulation of polycrystal perfect plasticity, see [31].…”

Variational analysis of multiscale problems with differential constraints

“…(1. 19) The solvability of (1.19) depends on the topological properties of both A and I (or the restricted…”