Multi-Phase Equilibrium of Crystalline Solids

TL;DR

This paper develops a continuum model for crystalline solid equilibrium that explicitly incorporates lattice structure, defects, and interfacial energies, providing a comprehensive framework for multi-phase equilibrium analysis.

Contribution

It introduces a novel spatial formulation accounting for defects and interfacial energies without relying on a fixed reference configuration, extending previous elastic models.

Findings

Derives bulk and interfacial conditions for multi-phase equilibrium.

Recovers previous elastic models as special cases.

Highlights the role of configurational forces in phase transitions.

Abstract

A continuum model of crystalline solid equilibrium is presented in which the underlying periodic lattice structure is taken explicitly into account. This model also allows for both point and line defects in the bulk of the lattice and at interfaces, and the kinematics of such defects is discussed in some detail. A Gibbsian variational argument is used to derive the necessary bulk and interfacial conditions for multi-phase equilibrium (crystal-crystal and crystal-melt) where the allowed lattice variations involve the creation and transport of defects in the bulk and at the phase interface. An interfacial energy, assumed to depend on the interfacial dislocation density and the orientation of the interface with respect to the lattices of both phases, is also included in the analysis. Previous equilibrium results based on nonlinear elastic models for incoherent and coherent interfaces are recovered as special cases for when the lattice distortion is constrained to coincide with the macroscopic deformation gradient, thereby excluding bulk dislocations. The formulation is purely spatial and needs no recourse to a fixed reference configuration or an elastic-plastic decomposition of the strain. Such a decomposition can be introduced however through an incremental elastic deformation superposed onto an already dislocated state, but leads to additional equilibrium conditions. The presentation emphasizes the role of {configurational forces} as they provide a natural framework for the description and interpretation of singularities and phase transitions.