Dynamic lattice models for elastic phase transitions and dislocations

Martensitic phase transitions can dissipate energy, but are often described by conservative (Hamiltonian) equations on the lattice scale. How can conservative lattice models generate dissipation on the continuum scale? To understand this, we consider a model problem, namely travelling waves in a one-dimensional chain of atoms with nearest neighbour interaction. The elastic potential is piecewise quadratic and the model is thus capable of describing phase transitions. A solution which explores both wells of the energy will then have a phase boundary, moving with the speed of the wave. We show that for suitable fixed subsonic velocities, there is a family of "heteroclinic" travelling waves (heteroclinic means here that these solutions connect both wells of the energy). Though the microscopic picture is Hamiltonian, we derive non-trivial so-called kinetic relations on the continuum scale. Kinetic relations in turn can be related to the dissipation generated by a moving phase boundary. We then investigate the question of when the kinetic relation does not vanish (dissipation is generated). It turns out that a microscopic asymmetry determines here the macroscopic dissipation.

The technique we developed for lattice problem above can also be applied to the model of disclation dynamics proposed by Frenkel and Kontorova in 1939 (for piecewise quadratic on-site potential), and this results and some implications will be discussed in the second part of the talk

The first part is joint work with Hartmut Schwetlick (Bath), the second one with Carl-Friedrich Kreiner (Munich).