Application of the method of effective potentials to a model for twinning in elastic materials.

Abstract

We extend the method of effective potentials to systems with next-nearest-neighbor interactions, and apply it to a one-dimensional discrete model for twins in elastic materials. The energy of the system is given by $H=\mathbf{\ensuremath{\Sigma}}n\ensuremath{\alpha}{({u}_{n+1}\ensuremath{-}{u}_{n})}^{4}\ensuremath{-}\ensuremath{\beta}{({u}_{n+1}\ensuremath{-}{u}_{n})}^{2}+(\frac{\ensuremath{\gamma}}{2}){({u}_{n+1}\ensuremath{-}{2u}_{n}+{u}_{n\ensuremath{-}1})}^{2}\ensuremath{-}cos{u}_{n}$, where the first two terms model the elastic strain-dependent energy which we take to be in the form of a double well in the strains and the third term gives its dependence on the discretized strain gradients. The periodic potential in the last term is introduced to allow for additional interactions with a background such as a parent phase, grain boundaries, or another array of twins. We obtain the phase diagram and show that it consists of various modulated commensurate as well as incommensurate ground-state configurations. We find continuous phonon-driven transitions between the homogeneous and any modulated phase, an incomplete devil's staircase in a narrow region close to the homogeneous phase and first-order soliton-driven transitions between commensurate phases. The first-order transition lines end at triple points where three commensurate phases coexist. In contrast to other nonconvex models we do not find here any superdegenerate points. We give general arguments which exclude the existence of such points in the present model. Preliminary results obtained by driving the system are discussed. These consist of various metastable configurations exhibiting strong hysteretic variation with the driving force.