Dynamics of patterns in ferroelastic-martensitic transformations. I. Lattice model.

Abstract

A lattice model and its nonlinear dynamics for ferroelastic-martensitic transformations is proposed. The lattice model presented involves the necessary interactions required in a cubic-tetragonal transformation for proper ferroelastic materials for which the strain tensor is merely the order parameter. Basically, the lattice model is a two-dimensional system including both nonlinear and competing interactions. The latter are considered as two kinds: (i) interactions by particle pairs and (ii) noncentral interactions or bending forces. A one-dimensional version is derived from the two-dimensional system, with the former possessing the anisotropic nature of the original lattice. The equations of motion are deduced as a set of difference-differential equations placing thus the discrete macroscopic and microscopic stresses in evidence. Moreover, upon investigating homogeneous states of deformation of the lattice, a comparison can be made with the Landau theory for ferroelastic phase transitions. On the basis of this reduced one-dimensional model the softening of the transverse-acoustic-phonon branch is examined, leading to two important results: (i) the partial softening of this branch of dispersion at a nonzero wave number and (ii) the positive curvature of the dispersion curve at the long-wavelength limit. All these effects are usually observed by means of neutron-inelastic-scattering techniques and this suggests pretransitional effects characterized by modulated lattice distortions.