# Mechanics

## On a form of the first variation of the action integral over a varied domain

Field theories of the continuum mechanics and physics based on the least action principle are considered in a unified framework. Variation of the action integral in the least action principle corresponds variations of physical fields while space-time coordinates are not varied. However notion of the action invariance, theory of variational symmetries of action and conservation laws require a wider variation procedure including variations of the space-time coordinates.

## The loading parameters calculation of a hollow sphere at the large elastocreep deformations

We presented a model of large elastocreep deformations. Separation of Almansi strain tensor is determined by the quadratic form of reversible and irreversible components. We consider spherically symmetric deformation of a hollow sphere in the steady creep process. Numerical solution of boundary-value problem was obtained. A method for determining loading force on the deformed state was proposed. Functions of the external loading force according to the laws of a given change in the displacement field were constructed.

## Oscillations of shallow shells at abrupt influence of thermal flow

On the basis of the closed integrals of the initial and boundary problems for incoherent thermoelastisity of shallow shells the quantitative analysis of influence of the geometrical parameters on the oscillations of constant rotation and cylindrical shells, which are conditioned by the thermal shock to outbound surface of shallow shell are carried out.

## Analytical Solution of Equations of Near-circular Spacecraft’s Orbit Orientation

The problem of optimal reorientation of spacecraft’s orbit with a limited control, orthogonal to the plane of spacecraft’s orbit, is considered. An approximate analytical solution of differential equations of near-circular spacecraft’s orbit orientation by control, that is permanent on adjacent parts of the active spacecraft’s motion, is obtained.

## Modelling of Cracking in Circular Disk Loaded by Concentrated Forces

An isotropic disk of radius R, loaded on the contour by two concentrated forces P, apllied to the points z1 = R and z2 = −R, is considered. A model of cracking in a circular disk, based on consideration of fracture process zone, is proposed. It is assumed that the fracture process zone is a finite length layer, containing material with partially broken bonds between individual structural elements. Equations for determination of the external load critical value at which the crack is observed are obtained.

## ON WEAK DISCONTINUITIES AND JUMP EQUATIONS ON WAVE SURFACES IN MICROPOLAR THERMOELASTIC CONTINUA

The present study is devoted to problem of propagating surfaces of weak and strong discontinuities of translational displacements, microrotations and temperature in micropolar (MP) thermoelastic (TE) continua. Problems of propagation of weak discontinuities in type-I MPTE continua are discussed. Geometrical and kinematical compatibility conditions due to Hadamard and Thomas are used to study possible wave surfaces of weak discontinuities.

## Analytical Solution of Linear Differential Error Equations of Strapdown Inertial Navigation System, Functioning in the Normal Geographic Reference Frame, for the Case of an Object, Following the Geographical Equator

Analytical solution of linear differential error equations of the strapdown inertial navigation system, functioning in the normal geographic reference frame, for the object, following the Earth equator with constant speed and on the constant height, is derived. The solution is represented in the form, which is convenient for the analysis. The roots of the auxiliary equation are derived in the explicit form. Obtained results can be used, for example, for analysis of the accuracy of strapdown inertial navigation system.

## Explicit Models for Flexural Edge Waves in Thin Orthotropic Plates

Analysis of flexural edge wave propagation in thin plates is presented. Several problems of semi-infinite plates vibrations are solved. These plates are assumed to be orthotropic. Some basic features of flexural edge wave propagation are found using the constructed explicit parabolic-ellipticmodels. They extract the localized wave contribution into the overall solution.

## Explicit Models for Flexural Edge and Interfacial Waves in Thin Isotropic Plates

Exact solutions for problems of vibrations of isotropic thin elastic plates are presented in the work. Some basic principles of explicit dual parabolic-elliptic models for flexural edge and interfacial waves propagation are revealed. The obtained explicit models extract the contribution of the flexural wave into the full dynamic response. Also, these models reveal a dual parabolic-elliptic nature of the flexural edge and interfacial waves.

## Antisymmetric Higher Order Edge Waves in Plates

This paper is concerned with the propagation of surface waves localized near the edge of plate (edge waves). Antisymmetric waves in a plate subject to traction free boundary conditions are considered. To study higher order edge waves three-dimensional equations of theory of elasticity are used. Asymptotic analysis is performed, which shows that there are an infinite spectrum of higher order edge waves. For the large values of wave number asymptotics of phase velocities are obtained.