Highly Non-Linear Theory of Bifurcation Transformations of Crystalline Lattice Structures
A highly-nonlinear theory is elaborated which describes elastic and inelastic phenomena in media with complicated lattice structure consisting of two sublattices. In the framework of this approach, the standard linear theory of acoustic and optic oscillations of a complicated lattice is generalized, taking into account internal translational symmetry of relative shear of the sublattices taken into account. As a result, the interaction between the sublattices is characterized in terms of a non-linear periodic force described, in particular, as sine of relative shear of two atoms belonging to an elementary cell. The corresponding equations in the case of solids without a central symmetry contain terms that describe the interatomic interactions. We have the situation with quasistatic loading of solids. The dependence of effective stresses on macroscopic strains is found which has a bifurcation point responsible for a structural transformation of the twinning type. It is shown that the transformation is related to a transformation of the interatomic interaction potential, the namely occurrence of both an additional minimum of the potential and a new structure (which has mirror symmetry relative to the initial structure.