# Mechanics

## Development of Asymptotic Methods for the Analysis of Dispersion Relations for a Viscoelastic Solid Cylinder

Propagation of time-harmonic waves in a viscoelastic solid cylinder is considered. Vibrations of the cylinder are described by three-dimensional viscoelasticity equations in cylindrical coordinates. The stress-free surface boundary conditions are imposed. Viscoelastic properties are described by integral operators with a fractional-exponential kernel. For the case of a rational singularity parameter the method of asymptotic analysis of dispersion relations is proposed, which is based on the generalized power series expansion.

## On Rationally Complete Algebraic Systems of Finite Strain Tensors of Complex Continua

The paper is devoted to the mathematical description of complex continua and the systematic derivation of strain tensors by the notion of isometric immersion of complex continuum in a plane space of higher dimension. Problem of establishing of complete systems of irreducible objective strain and extra-strain tensors for complex continuum immersed in an external plane space is considered. The solution to the problem is given by methods of the field theory and the theory of algebraic invariants.

## Exact Solitary-wave Solutions of the Burgers – Huxley and Bradley – Harper Equations

It is shown that the exact soliton-like solutions of nonlinear wave mechanics evolution equations can be obtained by direct perturbation method based on the solution of a linearized equation. The sought solutions are sums of the perturbation series which can be found using the requirement that the series are to be geometric. This requirement leads to the conditions for the coefficients of the equations and parameters of the sought solutions.

## Bending of Multiconnected Anisotropic Plates with the Curvilinear Holes

An approximate method for determination of the stress state of thin plates with curvilinear holes, consisting in the use of the complex potential theory of bending of anisotropic plates, approximating the contours of holes by ellipse arcs and straight sections, the use of conformal mapping, presentation of complex potentials by Laurent series and determining the unknown series coefficients of the generalized least squares method. Isotropic plates are considered as a special case of anisotropic plates.

## Determination of Attaching Parameters of Inhomogeneous Beams in the Presence of Damping

Characterization of solids by additional data on displacements amplitudes or resonance frequencies have been increasingly attracting attention of researchers in recent years. Among the tasks of this type, the problems associated with definition of parameters describing boundary conditions and characterizing an interaction of the body studied with the surrounding bodies are of particular interest. In this paper, we investigate the problem of determining the parameters of the boundary conditions in a beam.

## Slot of Variable Width in a Hub of Friction Pair

Plane problem of fracture mechanics for a hub of a friction pair is studied. It is suggested that near the rough friction surface, the hub has a rectilinear slot of variable width. The slot width is comparable with elastic deformations. A criterion and a method for solving the inverse problem of mechanics of contact fracture on definition of displacement function of the hub external contour points in a friction pair with regard to the temperature drop and irregularities of the contact surface in friction pair components is given.

## Investigation of the Problem of Optimal Correction of Angular Elements of the Spacecraft Orbit Using Quaternion Differential Equation of Orbit Orientation

In this paper we consider the problem of optimal correction of angular elements of the spacecraft orbit. Control (jet thrust vector orthogonal to the plane of the orbit) is limited by absolute value. The combined quality functional characterizes the amount of time and energy consumption. With the help of the Pontryagin maximum principle and quaternion differential equation of the spacecraft orbit orientation, we have formulated differential boundary value problem of correction of the angular elements of the spacecraft orbit.

## Nonlinear Waves Mathematical Modeling in Coaxial Shells Filled with Viscous Liquid

There exist wave motion mathematical models in infinitely long geometrically nonlinear shells filled with viscous incompressible liquid. They are based on related hydroelasticity problems, described by dynamics and viscous incompressible liquid equations in the form of generalized KdV equations. Mathematical models of wave process in infinitely long geometrically nonlinear coaxial cylindrical shells are obtained by means of the small parameter perturbation method.

## Numerical Study of Stress-Strain State of a Thin Anisotropic Rectangular Plate

Static bending of a thin rectangular anisotropic plate is considered in the framework of Kirchhoff hypotheses. At each point of the plate there is one plane of elastic symmetry parallel to the middle plane of the plate. It is assumed that the type of boundary conditions does not change along each of the straight sides. By applying of a modified method of spline collocation the twodimensional boundary value problem for the determination of deflection is reduced to a boundary value problem for the system of ordinary differential equations, which is solved numerically.

## One-Demential Automodel Problem about Impact of Rigid Body with Elastoplastic Half-Space

The one-demential automodel problem about impact of rigid body with elastoplastic half-space is considered. In the case of plastic deformation is accumulated inside Riman simple wave is presented. The solution with a possible wave picture, when perturbation in the environment propagate by means of two elastic waves and one plastic simple wave, is shown.