Loss of ellipticity and structural transformations in planar simple crystal lattices

This work focuses on investigation of structural (phase) transformations in crystal lattices from continuum and discrete points of view. Namely, the continuum, which is equivalent to a simple lattice in the sense of the Cauchy–Born energy, is constructed using long-wave approximation, and its strong ellipticity domains in finite strain space are obtained. It is shown that various domains correspond to variants of triangular and square lattices, and the number of the domains depends on the interaction potential parameters. Non-convex energy profiles and stress–strain diagrams, which are typical for materials allowing twinning and phase transformations, are obtained on the straining paths which connect the domains and cross non-ellipticity zones. The procedures of the lattice stability examinations and estimation of energy relaxation by means of molecular dynamical (MD) simulation are developed, and experimental construction of the envelope of the energy profiles, corresponding to the energy minimizer, is done on several straining paths. The MD experiment also allows to observe the energy minimizing microstructures, such as twins and two-phase structures.