Peierls-Nabarro landscape for martensitic phase transitions

We propose a simple one-dimensional model which can be used to reconstruct the Peierls-Nabarro energy landscape and compute the limits of lattice trapping for martensitic phase transitions. The model represents a nontrivial analog of the discrete Frenkel-Kontorova model. We find the explicit expression for the critical driving force which sets phase boundaries in motion and delimits the width of the rate-independent hysteresis. The theory is applied to β-martensitic phase transition in Cu-Al-Ni alloy. Rate-independent hysteresis is commonly observed in materials undergoing martensitic phase transitions, e.g. Ag-Cd [1], In-Tl [2], Cu-Al-Ni [3, 4] and Ni-Ti [5, 6]. When subjected to a quasistatic tensile loading, these materials deform elastically until the load reaches a critical value. At the critical load one or several phase boundaries nucleate and propagate through the specimen. Upon unloading the reverse transformation takes place at a lower critical load. Quantitative prediction of the width of the resulting quasistatic hysteresis and the associated kinetics of the moving interfaces represents an important challenge for the theory [7, 8]. While it is well understood that the rate-independent hysteresis in martensites reflects the presence of multiple metastable configurations, the physical origin of the critical load has been a subject of rather different theories emphasizing: disorder [9], elastic incompatibility [10], surface energy [11] and material discreteness [12]. The dissipation in these materials is typically attributed to an unavoidable development of fast internal instabilities (Barkhausen jumps) associated with the overall ruggedness of the energy landscape [13, 14, 15]. The martensitic phase boundaries represent highly mobile planar defects of a crystal lattice, and the physical situation is reminiscent of the one in metal plasticity where the analogous carriers of inelastic deformation are linear defects (dislocations). The complex interaction of both kinds of defects with a crystal lattice leads to an intense radiative damping and results in dry-friction type kinetics [16, 17]. Both martensitic phase transformations and plastic flow take place at nearly constant yield load, and in both cases nucleation is typically associated with a small load drop [18, 6]. The driving forces responsible for the release of the transformation-generating defects are known as Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, email: [email protected] Corresponding author, Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, email: [email protected]