# Mechanics

## Axisymmetric Problem Lemba for the Cosserat Medium

The article deals with elastic homogeneous isotropic half-space filled with the Cosserat medium. At the initial instant

of time and at infinity, there are no perturbations. At the boundary of the half-space, normal pressures are given. All the

components of the stress-strain state are supposed to be limited. A cylindrical coordinate system is used with an axis directed

inward into the half-space. With allowance for axial symmetry, the resolving system of equations includes three hyperbolic equations

## Creep and Long-Term Strength Modeling for Thick-Walled Tubes under Combined Loading with Axial Force, Torsional Moment and Internal Pressure

We have developed a method for solving the boundary-value problem of rheological deformation and creep

rupture of thick-walled tube under combined loading with axial force, torsional moment and internal pressure. Energetic variant

of the theory of creep and long-term strength is used to describe creep process. Experimental verification of proposed method has

been performed using known test data for creep and long-term strength of thick-walled tubes made of D16T alloy and Steel~20.

## Instantaneously not Elongated Directors in Three-Dimensional Kinematics of the Coulomb – Mohr Medium

Three-dimensional flows of perfectly plastic medium are considered within the framework of the Coulomb\,--\,Mohr continuum

model. The model is to be used in applied problems related to limit states and flows of sands, rocks and any other kind of

granular media. The present study is based on a notion of asymptotic directions of the stress tensor and the strain tensor

increment and as well on instantaneously not elongated directors which are orthogonal to the asymptotic directions and lie in the

## Stress and Strain Fields in a Plate of Stress State Dependent Material Properties

The paper analyzes the properties of the constitutive relations proposed to describe the behavior of materials whose

deformation diagrams depend on the type of external forces. In this case, various forms of nonlinearity arise, related to the

dependence of the properties of materials on the type of the stressed state, the nonlinearity of the deformation diagrams, and

the relationship between the shear and volume deformation processes. The influence of these forms of nonlinearity on the

## Low-Frequency Vibration Modes of Strongly Inhomogeneous Elastic Laminates

The dynamic behaviour of thin multi-layered structures, composed of contrasting “strong” and “weak” layers, is considered.

An asymptotic procedure for analysing the lowest cutoffs is developed. A polynomial frequency equation is derived, along with

the linear equations for the associated eigenforms corresponding to displacement variation across the thickness. For a five-layered

laminate with clamped faces two term expansions for eigenfrequencies and eigenforms are compared with those obtained

## On the Complex Dynamics in Simplest Vibrational Systems with Hereditary-Type Friction

The dynamics of a number of vibrational systems, accounting for the forces of hereditary-type dry friction and a

vibration limiter, are studied in the paper. The interaction between the vibration limiter and the vibrational system is

assumed to obey Newton's hypothesis. A general mathematical model has been developed, which is a strongly nonlinear non-autonomous

system with a variable structure. The dynamics of the mathematical model is studied numerically-analytically, using the mathematical

## On Some Models of Indentation for Functionally-Graded Coatings

The paper presents approximate models of deformation for an inhomogeneous elastic strip. Approximate models reflect the

distribution features of the inhomogeneous structure properties over the thickness. The models are based on hypotheses about the

nature of the change in the components of the displacement field, which allows to consider arbitrary laws of heterogeneity:

## Studying of Elastoplastic Properties of Coal Specimens Using Indentation Technique

A numerical study on elsatoplastic properties in problem of coals specimen nanoindentation by Berkovich pyramid is

presented. The stress-strain state of specimen during indentation is calculated using finite element method including complex

elastoplastic behaviour on the basis of Drucker-Prager model. The effective axisymmetrical indenter of cone shape is introduced and

used for the simulation. The influence of basic geometrical and material parameters of the solid model on the indentation curve is

## Approximate Theory of a Laminated Anisotropic Plate Vibrations

The multi-layered plate vibration is investigated. A two-dimensional asymptotic model of the second order accuracy with

respect to the small thickness parameter is proposed with account for the transverse shear and the normal fibre extension. The model is appropriate for a monoclinic plate described by 13 elastic moduli which is heterogeneous in the thickness direction. In particular, the model can be applied to a multi-layered plate

consisting of orthotropic layers of arbitrary orientation. In this case the elastic moduli are piece-wise constant functions. The

## On the Unsymmetrical Buckling of Shallow Spherical Shells under Internal Pressure

This work isdevoted to the numerical study of unsymmetrical buckling of shallow spherical shells and annular plates with varying mechanical characteristics subjected to internal pressure. We suppose that the edge of the shell is clamped but moving freely in the shell’s plane. For the annular plate a roller support is considered for the inner edge of the plate, i.e. the edge that can slide along the figure axes without changing the slope. The