# Mechanics

## 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.

## A Certain Approach to Solving of Some One-Dimensional Contact Problems

The paper deals with the problems of the unbonded contact of beams, strings, circular membranes and plates. A new approach to solving of such problems is suggested. This approach includes the rigorous problem statement, the elementary proof of the uniqueness of solution and the analytical solution construction method. The method is based on the iterative correction of the contact region. A number of examples of this method application are given.

## Simulation of Partial Closure Crack-Visible Cavities in Burning Solid Fuel Under the Influence of Body Forces

On the basis of the theory of elasticity mathematical description of the model for the covering crack-visible cavities with end zones in which cohesive forces of material act, in the burning solid-fuel has been performed. It is accepted that the interaction of crack-visible cavity surfaces under the influence of body and surface loads leads to the appearance of overlap zones of their surfaces.

## Generalized Cross-Coupled Type-III Thermoelastic Waves Propagating via a Waveguide under Sidewall Heat Interchange

The paper is devoted to a study of cross-coupled type-III generalized thermoelastic waves propagation via a long cylindrical waveguide. The sidewall of the waveguide is assumed free from tractions and permeable to heat. The analysis is carried out in the framework of coupled generalized theory of GNIII- thermoelasticity consistent with the basic thermodynamic principles. The theory combines the both possible mechanisms of heat transfer: thermodiffusion and wave.

## The Cortical Bone Tissue Controlled Recovering After Treatment in the External Fixation Apparatuses

The paper represents the new bone tissue rehabilitation method. The rehabilitation control loads are affected to the regenerative lower extremity by the use of a special training device. The adaptive changes of the elasticity modulus, porosity, volumetric mineral content and strengthening of the cortical bone tissue are simulated during the union of a bone using the external fixation apparatus (by example of the Ilizarov’s pin apparatus) and the following rehabilitation time.

## Processes of Supersonic Flows Retardation in Channels

Investigated are unsteady laminar boundary layer interaction and separation processes in supersonic flow. Deduced are equations describing such flow and obtained are numerical solutions of linearysed and nonlinear system of equations. It is supposed that these results will allow explain effects accompanying supersonic flow retardation in channels. For the full description it is needed to investigate flow in the reattachment region where selfoscillating processes arise.

## On the Non-Classic Models of Beams, Plates and Shells

For the problems of statics, of free vibrations, and of buckling of beams, plates and shells the Timoshenko – Reissner’s model with shear is compared with the classic Kirchhoff – Love model and with the 3D theory of elasticity. By using some test examples the formal asymptotic character of 1D and 2D models is established and their field of application is found. The asymptotic expansions based on the small shell or plate thickness compared with the length of wave are used. The special attention is paid to the buckling or vibration modes localized near the free surface.

## Duality Theory and Optimization of Sophisticated Engineering Systems

The design of sophisticated engineering systems (like the ship, airborne vehicle, production complex, etc.) is usually a multilevel process when the object under design is split up into separate sub-systems. At the same time, the quest for the best solutions for separate sub-systems should comply with requirements for optimality of the designed object as a whole. It is suggested to meet this condition with the help of local criteria developed using dual assessments (Lagrangian coefficients) of the duality theory in the non-linear mathematic programming.

## Non-Linear Viscoelastic Behavior of Filled Elastomers

The analysis of the results of experimental studies of relaxation properties of rubber at room temperature under the conditions of tension and compression at different strain levels is carried out. For the description of viscohyperelastic deformation of filled elastomers, the constitutive relations are used that are the generalization of non-linear elasticity and linear viscoelasticity Boltzmann – Volterra. The method for the determination of interrelated hyperelastic and reological characteristics of deformation is proposed.