Non-affine fields in solid-solid transformations: the structure and stability of a product droplet

TL;DR

This paper models the microstructure and stability of product droplets in solid-solid transformations using a Landau theory that incorporates both affine and non-affine deformations, revealing three stable structural solutions.

Contribution

It introduces a variational approach to analyze non-affine effects in solid-solid transformations, identifying three distinct stable microstructures depending on material parameters.

Findings

Three stable solutions for droplet microstructure ({ extbf I}, { extbf II}, { extbf III}) identified.

Non-affine degrees of freedom influence strain incompatibility and elastic interactions.

Width and size relations of twins vary across different solution types.

Abstract

We describe the microstructure, shape and dynamics of growth of a droplet of martensite nucleating in a parent austenite during a solid-solid transformation, using a Landau theory written in terms of conventional affine, elastic deformations and {\em non-affine} degrees of freedom. Non-affineness, $\phi$, serves as a source of strain incompatibility and screens long-ranged elastic interactions. It is produced wherever the local stress exceeds a threshold and anneals diffusively thereafter. A description in terms of $\phi$ is inevitable when the separation between defect pairs, possibly generated during the course of the transformation, is small. Using a variational calculation, we find three types of stable solutions ({\hv I}, {\hv II} and {\hv III}) for the structure of the product droplet depending on the scaled mobilities of $\phi$ parallel and perpendicular to the parent-product interface and the stress threshold. In {\hv I}, $\phi$ is vanishingly small, {\hv II} involves large $\phi$ localized in regions of high stress within the parent-product interface and {\hv III} where $\phi$ completely wets the parent-product interface. While width $l$ and size $W$ of the twins follows $l\propto\sqrt{W}$ in solution {\hv I}, this relation does not hold for {\hv II} or {\hv III}. We obtain a dynamical phase diagram featuring these solutions and argue that they represent specific microstructures such as twinned or dislocated martensites.