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Reconstructive phase transformations, maximal Ericksen-Pitteri neighborhoods, dislocations and plasticity in crystals
Sergio Conti and Giovanni Zanzotto
We study the reconstructive phase transformations in crystalline solids (i.e. transformations in which the parent and product lattices have arithmetic symmetry groups admitting no finite supergroup), the best known example of which is the bcc-to-fcc transformation in iron. We first describe the maximal Ericksen-Pitteri neighborhoods in the space of lattice metrics, thereby obtaining a quantitative characterization for weak transformations. Then, focussing for simplicity on a two-dimensional setting, we construct a class of strain-energy functions which admit large strains in their domain and are invariant under the full symmetry group of the lattice; in particular, we give an explicit energy suitable for the square-to-hexagonal reconstructive transformation in planar lattices. We present a numerical scheme based on atomic-scale finite elements and use it to analyze the effects of transformation cycling on a planar crystal, by means of our constitutive function. This example illustrates the main phenomena related to reconstructive phase changes: in particular, the formation of dislocations, vacancies and interstitials in the lattice.