Progress of Phase-transition Fronts: Approaches at Different Scales (Research Projects at Paris and Montpellier in Solid Phase Transitions)
The presentation deals with three aspects developed by the team of Paris-Montpellier:
Four approaches at different scales or levels of understanding are proposed to study the propagation of phase boundaries and domain walls in thermo-deformable solids: (i) a microscopic one relying on a lattice model that yields the representation of the thin transition zone as a solitonic structure - ``the condensed-matter physicist's vision''; (ii) a thermodynamics macroscopic one that is based on the use of the jump relations associated with the canonical formulation of continuum mechanics on the material manifold - the mixed ``mathematical-physicist - engineer'' vision; (iii) a mesoscopic continuum approach using directly a mixed viscosity-gradient modelling - the ``applied-mathematician'' approach; and (iv) a global quasi-particle viewpoint which, although accounting for the structure of the transition zones, is also canonical and allows for the study of transient motions of these zones - the ``theoretical physicist's viewpoint''. The notion of ``soliton complex'' is evoked on this occasion, yielding complicated internal structures for transitions zones or walls. The common points and respective interest of the approaches are discusssed.
On the one hand, starting with a 2D-discrete model and studying the long-time evolution and stability of strain bands one obtains a Kadomtsev-Petviashvili type of evolution equation which does exhibit the lateral instabilities and the formation of islands. On the other hand, studying modulated structures by means of a semi-discrete analysis (modulation of discrete variable and then consideration of slowly varying amplitudes) one shows that the 2D problem is governed by a 2D nonlinear dispersive wave equation (not a Ginzburg-Landau equation of evolution) for the amplitude. Numerical simulations exhibit the formation of localized nonlinear structures, e.g., elastic micro-domains. Future works include the consideration of domain growth, nucleation, and direct work on a microscopic (discrete) model by use of the notion of pseudo spins.
Three different points receive special attention:
a. Mesoscopic phenomenological analysis: This considers the case of the monocrystal and a thermomechanical approach involving internal variables related to the variants of martensite. The coupled thermo-mechanical 2D evolution is studied by means of a finite-element method.
b. Numerical experiments at micro- and meso-scales: This considers the monocrystal and quasi-convexified energies. Two scales are considered, a microscale at which a finite-element computation is performed using the quasi-convexified energy; then at each mesoscopic point, a minimization or the real energy in a domain of micro observation is performed, the matching condition being that the mean of the gradient of displacement be the same at the gradient of the meso displacement. This yields hopelessly long computations and using Young measures as variables seems to be a more efficient methodology.
c. Homogenization modeling of polycrystals, i.e., aggregates of SMA grains. In this modeling the dissipation at grain boundaries is neglected but one can see the emergence of the influence of texture on the pseudoelasticity of polycrystals. The post-doctoral positions offered correspondning to the three aspects will be concerned with:
1. Cellular automata using the material Eshelby stress (the relevant configurational force) and the thermodynamics of discrete systems to simulate the progress of 2D phase transition fronts. 2. Study of domain growth and nucleation on discrete atomic systems;
3. Numerical approximation of configurational energies defined by relaxation procedure (quasi-convexification); development of numerica schemes based on splitting into a difference of two convex functions.