Nonlinear theory of localized and periodic waves in solids undergoing major rearrangements of their crystalline structure

Aero, E.; Bulygin, A.
January 2011
Continuum Mechanics & Thermodynamics;Jan2011, Vol. 23 Issue 1, p35
Academic Journal
This article analyses the propagation of nonlinear periodic and localized waves. It examines crystals whose lattice consists of two periodic sub-lattices. Arbitrary large displacements of sub-lattices u are assumed. This theory takes into account the additional element of translational symmetry. The relative displacement in a sub-lattice for one period (and even for a whole number of periods) does not alter the structure of the whole complex lattice. This means that its energy does not vary under such a relatively rigid translation of sub-lattices and should represent the periodic function of micro-displacement. The energy also depends on the gradients of macroscopic displacement describing alterations in the elementary cells of a crystal. The variational equations of macro- and micro-displacements are shown to be a nonlinear generalization of the well-known linear equations of acoustic and optical modes of Karman, Born, and Huang Kun. Exact solutions to these equations are obtained in the one-dimensional case-localized and periodic. Criteria are established for their mutual transmutations.


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