The problem of hydroelasticity of the plate forming at wall of the slot-hole channel with a pulsing layer of viscous incompressible liquid at the set harmonious law of a pulsation of pressure at its end face in flat statement is put and analytically solved. The set regional task represents nonlinear related system of the equations of Navier-Stokes for a layer of viscous incompressible liquid and the equation of dynamics of a plate (beam strip). Conditions of sticking of liquid act as regional conditions to impene- trable walls of the channel, a condition of the free expiration of liquid at end faces of the channel and a condition of a hinged supporting of a plate wall of the channel. The complex of dimensionless variables of a considered task is created and small parameters of a task are allocated. As small parameters we have chosen the relative thickness of a layer of liquid and relative amplitude of a deflection of a plate. Considering asymptotic decomposition in the allocated small parameters of a task we have carried out its linearization by a method of indignations. The solution of the linearized task is obtained by a method of the set forms for a mode of the established harmonic oscillations. Thus, proceeding from boundary conditions for a channel plate wall, the form of its deflection is set in the form of ranks on trigonometrical functions from longitudinal coordinate. The law of a deflection of an elastic wall of the channel and dis- tribution of hydrodynamic parameters are found in liquid. We have obtained frequency dependent func- tions of distribution of amplitudes of a deflection and dynamic pressure along the channel and frequency dependent functions of distribution of phase shift of a deflection of a wall and pressure in the channel of rather initial indignation at an end face. On the basis of calculations it is shown that resonant fluctua- tions of an elastic wall of the channel, pressure excited by insignificant pulsations at its end face, can cause essential changes of dynamic pressure and be the main reason of vibration cavitation in liquid.

The variant of the classical theory of shells (CTS) built on the basis of Lagrange analytical me- chanics is under analysis. The direct approach to shells as material surfaces, the elements of which are material normals with five degrees of freedom - three translations and two rotations, is used. The sys- tem of equations and boundary conditions is derived from the principle of virtual work with direct tensor calculus. Such approach makes it possible to discard the problems and controversies characteristic of conventional concepts. This paper is aimed at comparing this theory of shells (CTS) with widely known variants, as well as with the solution of the spatial problem.Problems for the thin-walled infinite cylinder have been formulated and solved on the basis of three theories: CTS, the well-known theory of A.L. Goldenweiser and three-dimensional elasticity the- ory. For the shell-based models, we have linear algebraic systems, for the three-dimensional models - the ordinary differential equation (ODE) over the thickness. Exponential solutions of static problems with different variability are built analytically. Numerical solutions using computer mathematics have been found.In comparing exponents of solutions with the boundary load, it was found that for small values of the wave number and the shell thickness, both shell theories agree well with the three-dimensional the- ory. As the wavelength decreases relative to the shell thickness, their uncertainty increases, though the area of CTS applicability has turned to be somewhat wider than that in the theory of A.L. Goldenweiser.According to both theories, the detected displacements of the shell under the load rapidly changing by the coordinates are well coordinated with each other. The coordination with the three- dimensional theory is suitable for small values of wave numbers. The calculations have shown that, under external load having the axial and circumferential components, CTS predicts a normal displace- ment component with a greater accuracy.

This paper considers characteristics, features and corresponding boundary value problems of gradient theories of elasticity. A brief description of one-parametric applied model, which is one of the several variants of the gradient elasticity theories is given here. In relation to that, we represent a con- tinuum gradient model of two-phase composite structures that allow evaluation of the influence of scale parameters on their effective mechanical properties.In identifying the additional physical parameters of gradient elasticity models, a new method is introduced where a comparison of the results of continuum and discrete-atomistic modelling for specific tested heterogeneous structures is made. As a result we suggested a procedure and the respective algorithm defining the additional parameter of applied gradient continuum model of heterogeneous me- dia; and in such procedure, an interphase zone is characterized at the contact surface of a two-phase composite and the scale effects represented by cohesions-interaction fields, which are localized near to the boundaries of contact surfaces. This additional physical parameter of gradient model is found through parameters of potentials, which are used to describe the specific studied structure in the dis- crete-atomistic modelling.To justify and validate the proposed method, a numerical investigation is conducted and com- parison is made between the results of continuum and discrete-atomistic modelling. The examination reveals that a high degree of accuracy of prediction can be provided by the continuum one-parametric gradient theory when describing the effective properties of countable multiple set of two-phase hetero- geneous studied structures, which are formed by atomic substructures with various properties (various parameters of potentials).Finally, it is demonstrated that the identification method of parameters in gradient elasticity theo- ries for heterogeneous structures is well described by Leonard-John potential and Morse potential. Fur- thermore, we consider that when the parameters of potentials are known, the various types of cross interactions of atoms can be treated as ‘ideal’ or ‘damaged’ interactions as per Lorentz-Berthelot’s rule.

The paper is devoted to the stress-strain analysis near the crack tip in a power-law material un- der mixed-mode loading. In the paper by the use of the eigenfunction expansion method the stress- strain state near the crack tip under plane stress conditions is found. The type of the mixed - mode loading is specified by the mixity parameter which is varying from 0 to 1. The value of the mixity pa- rameter corresponding to Mode II crack loading is equal to 0 whereas the value corresponding to Mode I crack loading is equal to 1. It is shown that the eigenfunction expansion method results in the nonlinear eigenvalue problem. The numerical solution of the nonlinear eigenvalue problem for all the values of the mixity parameter and for all practically important values of the strain hardening (or creep) exponent is obtained. It is found that the mixed-mode loading of the cracked plate gives rise change of the stress singularity in the vicinity of the crack tip. The mixed - mode loading of the cracked plate results in the new asymptotics of the stress-strain fields which is different from the classical Hutchinson - Rice - Rosengren stress field. The approximate solution of the nonlinear eigenvalue problem is either obtained by the perturbation theory technique (small parameter method). In the framework of the small parameter method the small parameter presenting the difference between the eigenvalue of the nonlinear problem and the undisturbed linear problem is introduced. The analysis carried out shows clearly that the stress singularity in the vicinity of the crack tip is changing under mixed-mode loading in the case of plane stress conditions. The angular distributions of the stress and strain components (eigenvalue functions) in the full range of values of the mixity parameter are given.

Discontinuous deformation as a phenomenon of plastic deformation instability, observes for a variety ductile materials in some range of strain rates and temperature. It is known fact that the tem- perature and strain rate are the most important parameters of the inelastic deformation. For the majority of polycrystalline materials, in the absence of phase transitions, temperature increase and decrease of strain rate leads to a reduction in plastic deformation resistance. At the same time for the majority of alloys in some ranges of temperatures and strain rates observed inverse dependence of the flow stress. Most researchers considered that the main cause of the anomalous behavior is a diffusion process and dislocation-impurity interaction. Portevin-Le Chatelier effect is the best known manifestation of the influ- ence of diffusion processes on the behavior of deformable material. Establishing the ranges of impacts which implement discontinuous yielding and eliminate them from the technological regimes is a very urgent problem today.Various methods and approaches based on mathematical modeling, are most preferred for the analysis of discontinuous yielding, determine the effective processing conditions and design of new materials. Experimental methods for studying this phenomenon resource intensive and applicable only for existing materials. Construction of mathematical models which accurately reproduce investigated processes is impossible without studying the available empirical information to establish the leading physical mechanisms.In the first part of the review considered the works devoted the description of the physical mechanisms and experimental studies of serrated yielding. The main mechanism is considered the interactions between mobile dislocations and diffusing solute atoms. Three main types of Portevin-Le Chatelier effects has been allocated based on the experimental data of uniaxial loading, in real experi- ments could experience different combination of these three types. Different approaches and models (macro phenomenological, structure-mechanical, physical) used for the theoretical description of the discontinuous deformation, in the present review we analyze only phenomenological models.