Phoebus Rosakis
A two-dimensional, continuum theory of twinning and phase transition
dynamics is presented, aimed at describing complex morphological
evolution of curved interfaces. Twinning is modelled using a
nonconvex elastic stored energy with multiple wells, determined by
lattice symmetry. Equilibrium twinning deformations are briefly
considered; they involve discontinuous shear localized within narrow
twin zones. Their shape is restricted by exclusion of instability:
twin boundaries, which may be curved, must be closely alligned with
special planes and terminate into cusps. This prediction is in
agreement with observed twin morphology.
Various dynamic twin growth problems will be discussed. A kinetic
relation governing twin boundary evolution is imposed; it involves
boundary orientation dependence, whose form is obtained by appealing
to the dislocation model of twinning.
Steady growth of a semi-infinite twin lamella under remotely applied
stress is considered. The steady-state tip velocity may be
subsonic, sonic or supersonic. The steady dynamic problem is solved
explicitly. Under a broad class of kinetic laws we find that
subsonic growth cannot be steady, while supersonic motion is
possible under high enough applied stress, in agreement with recent
developments of dislocation dynamics.
The problem of quasi-steady growth is addressed, concerning the case
when shape changes are slow compared with the average constant
growth speed. Interface evolution is governed by a nonlocal
evolution equation. The stability and large-time dynamics of the
evolving interface curve are determined.
In order to describe more complex morphologies, an adaptation of the
level-set method, ordinarily used to describe curvature driven
motion, is employed. This method allows imposition of an independent
kinetic relation, which governs interface motion through a
Hamilton-Jacobi equation, coupled with momentum balance.
Finite-difference computations exhibit complex morphological
dynamics, including cusp formation, needle growth, tip splitting and
topological changes resulting in microstructure refinement, in
qualitative agreement with experimental observations.