# Mechanics

## Modification for the Chisnell’s Method of Approximate Analytic Solution of the Converging Shock Wave Problem

The self-similar problem about a convergence to the centre of a strong shock wave is discussed. The approximate analytical solution which has the same form as the Chisnell’s solution is proposed. The simple expressions for definition of self-similar representers of the velocity, density and square of the sound speed are written down. The self similar exponent is determined by solving the algebraic equation. The achived results correlate better with the exact solution of the classical numerical method.

## An Optimal System Constructing Algorithm for Symmetry Algebra of Three-Dimensional Equations ofthe Perfect Plasticity

The present study is devoted to study of a natural 12-dimensional symmetry algebra of the three-dimensional hyperbolic differential equations of the perfect plasticity, obtained by D.D. Ivlev in 1959 and formulated in isostatic co-ordinate net. An optimal system of onedimensional subalgebras constructing algorithm for the Lie algebra is proposed. The optimal system (total 187 elements) is shown consist of a 3-parametrical element, twelve 2-parametrical elements, sixty six 1-parametrical elements and one hundred and eight individual elements.

## Mode-Series Expansion of Solutions of Elasticity Problems for a Strip

Oscillations of a strip are considered as a plane problem of elasticity theory. Description of oscillation modes is provided. Properties of eigenvalues and eigenfunctions are studied for a boundary value problem for their amplitudes. Green’s function is constructed as a kernel of the inverse operator. Completeness and expansion theorems are proved which allow one to solve problems for finite and infinite membranes under arbitrary boundary conditions.

## The Constitutive Equations for the Bone Tissue Structural Adaptation

The constitutive relationships for cortical and trabecular bone tissue structural adaptation are offered. These constitutive equations connect the rate of change of the porous radius with the strain adaptive stimulus and the bone cells activation. The used approach takes account of bone cells activation and it is alternative to the known experimental Frost’s Basic Multicellular Units method. That approach allows spreading the cellular remodeling mechanism on the functional adaptation process.

## Mathematical Modeling of Interaction Between Layer of Viscous Liquid and Elastic Walls of Channel, Which Was Installed on Vibration Foundation

The article solves the problem of mathematical modeling dynamic processes in hydrosupport with elastic stator. The dynamic problem of hydroelasticity is found and amplitude and phase frequency characteristics of hydrosupport was built.

## Lie Symmetry Analysis and Some New Exact Solutions for a Variable Coefficient Modified Kortweg – De Vries Equation Arising in Arterial Mechanics

In this paper, a variable-coefficient modified Korteweg – de Vries equation is considered. By using the classical symmetry analysis method symmetries for this equation are obtained. Then, the generalized Jacobi elliptic function expansion method is used to solve the reduced ODE. Some new exact solutions for the considered PDE are obtained.

## The Feature of Non-Isothermal Viscoelastic Flows Around Sphere at Obstruction Condition

The numerical study is performed for study of the viscoelastic flow characteristics and heat transfer around sphere. The flow of liquid is described by equations of conservation of mass, momentum and thermal energy with rheological constitutive equation of Phan-Thien Tanner(PTT). Thismodelrepresents generalizedMaxwell typemodel with two additional parameters developed from kinetic theory of polymers. The nonlinear behaviour of fluid velocity behind body (<<negative wake>>) is observed.

## Static Bending and Steady-State Vibrations of Thin Cylindrical Shells Under Local Load

The spline collocation method is being used for solving static bending and steady-state vibrations ploblems for thin cylindrical shell under local loads. Maximum displacement values and first three resonance frequencies are given.

## A Boundary-Value Problem with Shifted for a Mixed Type Equation with Fractional Derivative

A non-local problem for a mixed type equation with partial fractional derivative of Riemann – Liouville is studied, boundary condition of which contains linear combination of generalized operators of fractional integro-differentiation. Unique solvability of the problem is then proved.

## Analytical Solution of Differential Equations of Circular Spacecraft Orbit Orientation

The problem of optimal reorientation of spacecraft’s orbit with a limited control, orthogonal to the plane of spacecraft orbit is being investigated. We have found an analytical solution of differential equations of circular spacecraft orbit orientation by control that is permanent on adjacent parts of the active spacecraft’s motion.