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Porubov A.V.,Institute of Problems in Mechanical Engineering | Andrianov I.V.,RWTH Aachen
Wave Motion | Year: 2013

A possible improvement of a continuum model for diatomic crystals is examined using continuum limit of a discrete diatomic model. For this purpose, various discrete models of diatomic lattice are compared at the linearized and weakly nonlinear levels. The suitable numbering of the atoms in the lattice is found which is better adopted for continualization than the familiar pair numbering introducing two sub-lattices. The coupled governing partial nonlinear differential equations for longitudinal strain and relative distance between the atoms are obtained in the continuum limit that allows us to describe localization of the strains due to the presence of the atoms of two kinds. It is found, that the equations obtained possess two kinds of localized wave solutions, one related to the acoustical branch and the other one related to the optical branch. © 2013 Elsevier B.V. Source

Porubov A.V.,Institute of Problems in Mechanical Engineering | Andrianov I.V.,RWTH Aachen | Danishevs'Kyy V.V.,Prydniprovska State Academy of Civil Engineering and Architecture
International Journal of Solids and Structures | Year: 2012

Nonlinear strain wave propagation along the lamina of a periodic two-component composite was studied. A nonlinear model was developed to describe the strain dynamics. The model asymptotically satisfies the boundary conditions between the lamina, in contrast to previously developed models. Our model reduces an initial two-dimensional problem into a single one-dimensional nonlinear governing equation for longitudinal strains in the form of the Boussinesq equation. The width of the lamina may control the propagation of either compression or tensile localized strain waves, independent of the elastic constants of the materials of the composite.© 2012 Elsevier Ltd. All rights reserved. Source

Porubov A.V.,Institute of Problems in Mechanical Engineering | Maugin G.A.,University Pierre and Marie Curie
International Journal of Non-Linear Mechanics | Year: 2011

A non-linear dynamic model is developed to account for material inhomogeneities in a growth plate in long bones. The governing equations are obtained to account for non-linear dispersive, viscoelastic and inhomogeneous features of the growth plate. The evolution of non-linear strain waves over the material inhomogeneities is obtained via the asymptotic solutions. It is shown that variations in the amplitude and the width of both the bell-shaped and kink-shaped waves reflect the position and the size of the inhomogeneity. This may be used for a detection of the growing plate features and in the development of the reactiondiffusion equation for the stimulus of the growth of long bones. © 2010 Elsevier Ltd. All rights reserved. Source

Filippenko G.V.,Institute of Problems in Mechanical Engineering
Proceedings of the International Conference Days on Diffraction 2011, DD 2011 | Year: 2011

Shells are the elements of various constructions, partially submerged into the liquid. It is important to estimate the level of vibration fields in these composite systems and analyze energy and energy flow in them. The results represented here are based on the formulas obtained in [1], [2], [3]. © 2011 IEEE. Source

Aero E.L.,Institute of Problems in Mechanical Engineering | Bulygin A.N.,Institute of Problems in Mechanical Engineering
Continuum Mechanics and Thermodynamics | Year: 2011

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. © 2010 Springer-Verlag. Source

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