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

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.