Lattice model for nonlinear patterns in ferroelastic-martensitic materials

Abstract

On the basis of a two-dimensional lattice model, nonlinear dynamics of elastic domain structures for martensitic-ferroelastic transformations is presented. The lattice model is particularly devoted to a cubic-tetragonal transformation characterized by a shear deformation of the close-packed atomic planes. The microscopic model involves nonlinear and competing interactions emerging from interactions by pairs and non-central interatomic forces. The latter play an important role in the softening of the acoustic phonon dispersion and two crucial results are shown: the partial softening of the transverse acoustic phonon branch at a nonzero wave-number and the positive curvature of the dispersion branch at the long wave-length limit. The emphasis is especially placed on the quasi-continuum approximation in order to incorporate the leading discreteness effects due to the lattice description. The quasi-continuum model allows one to examine the nonlinear dynamics of elastic twinning and domain interfaces in terms of strain solitary waves. Solutions to the quasi-continuum model correspond to periodically modulated structures made of martensitic band- or stripe-type domains. A particular case representing an elastic or martensitic domain moving in the parent phase is described as a martensitic solitary wave. The complete two-dimensional system is examined next in order to place some instability processes in evidence. Numerical simulations show that instabilities with respect to transverse disturbances are developed while the elastic structures are moving. A bifurcation takes place, followed by localized elastic structures. A criterion of stability based on the energy of the system tells us that the system is stable if the total energy does not increase with the amplitude of the shear in the martensitic domains. Also pointed out is the role of the competing interactions for the existence and stability of the nonlinear structures. Approximate models deduced from the two-dimensional lattice system and its quasi-continuum version revealing some similarities with other nonlinear problems in plasma physics or fluid mechanics are discussed.

Key Words:

• Martensitic transformations
• ferroelasticity
• nonlinear patterns
• domain structures