Nonlinear dynamics and stability of localized modes in lattice and quasi-continuum models

With the view of examining strain instability and micro-texture formation in material, lattice models are considered. More precisely, we want to understand how macroscopic behaviour of material is affected by the underlying microphysics taking place at the crystal scale. For instance, nucleation process, domain growth or phase boundary movement are often responsible for particular behaviours in solids undergoing phase transformations (hysteresis, etc.).
In a previous works the emphasis has been put on a mechanism of formation of elastic microstructures or ferroelastic phase in the lattice model framework. The models thus considered include non-linear (non-convex potential) and competing interactions [1,2]. These interactions are, in fact, intimately related to specific interatomic forces that account for unstable, metastable or stable states of lattice configurations. Moreover, such interactions describe the cubic-tetragonal phase transformation in binary alloys. From 1D and 2D models some interesting, but nonetheless promising results have been obtained, among them (i) the softening of the phonon dispersion at a non-zero wave-number [2], (ii) the propagation of an array of strain solitary waves [3], (iii) the transverse instability of a strain band in a 2D lattice [4] and (iv) the instability mechanism of a modulated (periodic) strain structure [5].
Along with the same model, we are continuing the study and extending it. In particular, the problem of discreteness effects seems to be crucial for phase growth and lattice instability. Accordingly, it is more appropriate to keep the discrete nature of the model (without considering the long wavelength approximation). In particular, the model enables us to examine the stability of periodic phases or the nucleation of small elastic domains. Numerical simulations should be performed directly on the discrete system and exhibiting microtexture of very rich morphology [6].
The analysis of the dynamics of the localized structures in 1D an 2D systems is examined by computing the associated mass, momentum and energy. Conservative laws are deduced for the lattice and quasi-continuum models. The conserved quantities are computed as function of the soliton or localized mode parameters [7] - the velocity or the strain amplitude - in order to characterize the dynamics of the associated quasi-particle. At low velocity we recover the classical newtonian dynamics. For bigger velocities the dynamics of the localized object become more complicated and it can be described by numerical simulations.
A second task for the research works should be the investigation of the influence of applied forces and dissipative effects on the formation and the dynamics of localised structures. This part should lead to a connection between applied stimuli to material and its macroscopic response. We must specify that this kind of approach has a great impact in micro-mechanics or nano-mechanics where low-dimensional materials involve few atomic planes (devices using novel or advanced materials).

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[2] J. Pouget, Phys. Rev. B43, 3575 (1991).
[3] J. Pouget, Phys. Rev. B43, 3582 (1991).
[4] J. Pouget, Phys. Rev. B46, 10554 (1992).
[5] J. Pouget, Phys. Rev. B48, 864 (1993)
[6] J. Pouget, Meccanica 30, 449 (1995).
[7] M.M. Bogdan, A.M. Kosevich and G.A. Maugin, Phys. Rev E, 1999 (in press).