# Mechanics

## Technique of Definition Areas Requiring a Quantum Description Within of the Hybrid Method (Quantum Mechanics / Molecular Mechanics)

The new model, which determines the active area (the region for which high-precision quantum methods must be used) of the structure,was developed within the of the hybrid method (QM/MM). Problem of determining atoms with the critical tension values is the basis of this model. The potential energy of these atoms and its nearest neighbours was calculated by quantum-chemical method. The potential energy of the rest structure was calculated by molecular mechanical method.

## The One-dimensional Problem of Unsteady-related Elastic Diffusion Layer

The problem of determining the stress strain state of an elastic medium, taking into account the structural changes caused by the presence of diffusion fluxes. The influence of diffusion processes on the stress-strain state of the environment is taken into account by using the locally equilibrium model of thermoelastic diffusion, which includes the coupled system of equations of motion of an elastic body and the equations of heat and mass transfer.

## Determination of the Wall Temperature Change for a Cavity in a Solid as a Result of the Temperature Change of the Gas Flow in a Cavity

The wall temperature change for a cylindrical cavity in a solid was found as a response to the temperature change of the gas flowing in a cavity. Three important special cases of the gas temperature dependence on time are considered: temperature is constant; temperature changes according to the linear law; temperature changes according to the harmonic law. The plots of five «µ-functions» used to denote solutions are submitted.

## One-dimensional equations of motion of a viscous incompressible fluid in flexible tubes

This paper describes a new variant of the averaging of the Navier–Stokes equations for axisymmetric flow of a viscous incompressible fluid with a minimum number of simplifying hypotheses. The complete system is spatially one-dimensional differential equations describing the dynamics of blood flow in the large arteries.

## Chaotic motion of top with displaced mass center

The motion of solid body with a small displacement mass center from the axis of dynamic symmetry has been studied. Analytical conditions for the existence of a hyperbolic singular point in the phase portrait of the system and the analytical solution for the separatrices have been obtained. Body makes a chaotic motion near separatrices under the influence of small perturbations caused by the asymmetry of the body.

## Parametrical synthesis of stabilization systems

Method of feedback parameters selection for gas jet stabilization systems with elastic roads, based on minimizing the mean square deviation of the real frequency response of the designed system with respect to the real desired frequency response, was implemented. The results of analysis of transient errors stabilization functions, taking into account the effect of time delay in gas jet executive stabilization systems are given.

## Asymptotic integration of dynamic elasticity theory equations in the case of multilayered thin shell

Asymptotic integration of elasticity theory 3D equations is fulfilled for the case of multilayered arbitrary-shaped thin-walled shells. The tangential and the transverse long-wave low-frequency approximations are constructed. The governing 2D equations are derived.

## Percolation of spheres in continuum

The model of the continuum percolation of hard spheres with permeable shells, which describes phase transition sol-gel, has been investigate. Spheres have hard parts in radii r, which can't be blocked with each other, and permeable shells in width d, which can be blocked. Such spheres of the equal size have been randomly packing in the cub with linear size L. The probability of joining the spheres in a cluster is proportional to the volume of overlapping of permeable shells.

## Non-stationary vibration of growth circular cylindrical shell

Small forced vibrations of growing cylindrical shell fixed on circular boundaries is studied in the framework of Kirchhoff–Love shell theory. The process of the accretion are characterized by the continuous adherence of material particles to its facial surface. Since the shell bends during the accretion, its stressed-strained state depends not only on loading, but also on the history of the process of accretion, i.e. the schedule of accretion.

## Rotational Invariance of Non-Linear Lagrangians of Type-II Micropolar Thermoelastic Continuum

The paper contains new results related to extension of the field theoretical approach and its formalism to non-linear coupled micropolar thermoelastic media. A mathematical model of micropolar (MP) type-II (GNII) thermoelastic (TE) continuum is considered. A formulation of the least thermoelastic action principle is discussed. Partial differential equations subsequent to the least action principle are derived. The translational symmetries of non-linear Lagrangians are adopted. Those include an additional symmetry: translations of the thermal displacement.