The article is devoted to the construction of fields of displacements, stresses and strainsarising in a linearly elastic half-space with a functionally-graded coating subjected to indentation by a punch with a spherical tip. Calculations of displacements, stresses and strains at the inner point of the coating andthe substrate is reduced to the integration on an infinite interval. The integrand is dependent on an unknown function of stresses distribution in the contact region. Contact stresses arising due to the indentation of a rigid spherical punch into an elastic half-space with a functionally-graded coating have earlier been constructed by the authors by solving the problem with mixed boundary conditions. For this purpose, the problem was reduced to the solution of a dual integral equation using the integral transformation technique. For a general case of independent arbitrary variation of Young’s modulus and Poison’sratioin the depth of the coating, the kernel transform of the integral equation can be calculated only numerically from the solution of a Cauchy problem for a system of ordinary differential equations with variable coefficients. Using approximations for the kernel transform of the integral equation by a product of fractional quadratic functions, approximated analytical expressions for the contact stresses and unknown radius of the contact area were constructed. The expressions obtained are asymptotically exact for both small and big values of relative coating thickness and high accuracy of intermediate values can be reached. The method is effective for an arbitrary variation of elastic properties and makes it possible to consider values of Young’s modulus of the substrate with more than two orders of magnitude higher than that in the coating. Series of numerical calculations of elastic displacements and stresses inside the coating and the substrate are provided for a case of soft homogeneous or functionally-graded layer lying on an elastic half-space (foundation). Young’s modulus of the layer is assumed to be constant or linearly varying (increasing or decreasing) in depth. At the layer-foundation interface Young’s modulus of the layer is 100 times as much as that of the foundation. This approach makes it possible to avoid the use of assumption about the non-deformability of the foundation for modeling soft homogenous or functionally-graded layers.

Having analyzed the hysteresis loop of plastic, the authors have formulated the evolution equations for three types of backstresses which are responsible for the shift of yield surface; and based on them the equations of the theory of plastic flow under combined hardening have been formulated. By integrating the evolution equation for the backstresses of the second type under rigid symmetric cyclic loading with a constant magnitude of plastic deformation in uniaxial stress state,we have obtained the expressions for the backstresses on the first half cycle and a stable maximum and minimum values of backstresses. After it, we have examined the work of backstresses of the second type on the field of plastic deformations; andbased on the experimental dataitis shown that the value of this work is a constant feature of fracture in the conditions of low-high-cycle fatigue (from 101 to 106 cycles). Based on these results we formulatedthelow-high cycle fatiguecriterion. Its asymptotes at a small and large number of cycles beforefailure have been obtained. The computational and experimental results for fatigue have been compared. The computational and experimental behavior of the accumulated plastic strain underlow-high-cycle fatigue has been analyzed. Kinetic equation of damage accumulation describing the nonlinear process of damage accumulation has been formulated based on the analysis of experimental data with regard to damageaccumulation under cyclic loadings.A comparison of the calculated and experimental results in multi-block cyclic loading is considered. The ratcheting and landing processes with respect to plastic hysteresis loops under asymmetric cyclic loading have been analyzed as well as a parameter and its functional relationship to describe these processes have been determined. Thecomputational and experimental research results of processes in soft and hard asymmetrical cyclic loadings have been compared.

The article describes the algorithms implemented in computational complex UPAKS (VC UPAKS) which allow studying the start and development of fatigue cracks under low-cycle thermopower loadings by a direct numerical simulation based on the finite element method within the time acceptable for engineering computations. The studies are carried out in the framework of damaged medium mechanics and use the hypotheses on the multi-stage nature of damage development in the process of material failure. The paper presents an algorithm of predicting deformation processes and damage accumulation in structural elements at low-cycle thermopower loadings, which combines the possibility to have a detailed description of deformation and damage accumulation in the early fracture stages while minimizing the number of computations in the numerical modeling of these processes on the basis of FEM.To investigate the third stage of fracture process, the authors have developed an algorithm for crack propagation modeling in structural elements on the basis of results of the process simulation in the first two stages without changing the initial topology of finite elements of the studied structure. The results of simulation of elastic-plastic fracture of the experimental sample with a concentrator under a plain bending which have been made using VC UPAKS are given. The comparison of numerical results with experiments has shown their good compliance which proves that the efficiency of the proposed algorithm. By using the simulation examples/cases of low-cycle fracture of a cylindrical recessedsample the authors verified the software that implements the proposed algorithms as part of VC UPAKS software. It proved that they can be effectively used to simulate low-cycle fracture processes of structure elements.

Theoretical relations obtained from solutions of dynamic problems of viscoelasticity represent an effective framework for experimental identification of dynamic rheological properties of materials. For the construction of such relations, closed solutions of boundary value problems (i.e. written in the form of convergent series or integrals) are preferred, because they(unlike solutions obtained by numerical methods) allow strict error estimates. However, the construction of analytical solutions is associated with the following difficulties. 1. As usual, the hypothesis of proportionality is accepted for relaxation operators corresponding to the first and second Lamé moduli, which is equivalent to the hypothesis of a constant Poisson’s ratio. This significantly reduces the generality of consideration. 2. Representation of solutions for three-dimensional problems in the form of expansions in eigenfunctions makes it necessary to taking into account the large eigenvalues which in the vast majority of problems can be found only numerically, as the roots of transcendental equations, thus, it is likely to skip closely spaced and multiple roots. 3. Constructed series converge slowly. In this paper we suggest ways to overcome these difficulties. Solutions of initial boundary value problems are presented in the form of spectral expansions, but in contrast to the classical method of Fourier decomposition they are expanded over biorthogonal system of eigenfunctions of mutually conjugate pencils of differential operators. This pencils define generalized Sturm-Liouville problem with a polynomial spectral parameter. This eliminates the hypothesis of proportionality relaxation operators. Effective relations for the terms of spectral(in particular normalization factors) coordinate functions and asymptotic formulas for the initial approximations of eigenvalues excluding their omission in calculations are obtained. Power related ranking of elements of the spectral decomposition is proposed which allows achievingthe required accuracy of calculations on the partial sums of a low order.

At first sight, many continuous media possess numerous micropores which contain or do not contain liquid. These pores are much less than the macroscopic sizes of medium, but they are bigger than nuclear or molecular ones. Such models of porous medium as soil model are widely usedin geophysics. Liquid distribution (oil, water) in soil is explained by this model. This model is also used in biology, in particular, to describe the penetration of liquid through plants, for example, wood. In recent years artificial porous materials which are widely applied in everyday life, in equipment and other areas of human activitieshave been created. The present work considers the distribution of flat longitudinal waves in porous liquid-saturated medium with cavities. It is supposed that the energy dissipation of a wave in the medium can be neglected. The behavior of linear waves in porous media with cavities is studied. It is known that in the porous medium (Biot's medium) two longitudinal waves can extend, one of them is slow and the other one is fast. In our problemthree longitudinal waves are extending: two waves do itlike in the Biot medium and the third one does it due to medium cavities. If the mediumhad neither pores, nor cavities, then one fast wave would extend. The study of linear wave’s behavior is conducted by receiving and analyzingthe dispersive equation, phase speed and group speed characterizing the wave energy transfer.The density of spectral frequency is considered to determinethe range of the dispersion degree.The work presents the formation and analysis of dispersive dependences for the considered system. Areas of strong and weak dispersion, areas of normal and abnormal dispersion at certainvalues of system parameters are found.

A numerical method for processing of experimental vibration data has been developed to find the lowest experimental frequency and amplitude dependences of the logarithmic decrement which are used to determine damping properties of test-samples. The logarithmic decrement (LD) is determined by the experimental decay curve obtained from the tip point amplitude measurements of test-samples during their flexural vibrations and approximated by the sum of two exponents with four parameters determined by a direct search of the objective function depending on these parameters. The conducted numerical experiments confirmed the reliability of the developed method. It is shown that the material of the test samples with a diamond-shaped cross-section must have stable and low damping properties for a reliable determination of the experimental aerodynamical damping component. Duralumin alloys absolutely meet these requirements. The damping matrix of the finite element model of the test-sample with an arbitrary cross-sectional shape is constructed in the case of the amplitude-independent internal friction in the material. The internal damping parameter which specifies the material damping properties is obtained. The experimental aerodynamic component of damping is obtained from the series of test-samples with the diamond-shape cross section. It has been noted that the elasticity modulus of duralumin D16 AT is frequency dependent. An iterative algorithm is developed to determine the lowest vibration frequency of the test-sample considering this dependence. The conducted numerical experiments using the test-samples with the specified cross-section confirm the reliability of the developed algorithm. The theoretical and experimental method is developed to construct the structural formulae to determine the aerodynamic component of damping for the test-samples with the diamond-shaped cross-section. The method is based on the modification of the basic formulae for thin plates with the constant thickness and the experimental data on the damping properties obtained for the series of test samples with the specified cross-sectional shape. The reliability of the obtained structural formulae has been confirmed by the performed numerical experiments.

Dynamic behavior of poroelastic and poroviscoelastic solids is considered. Poroviscoelastic formulation is based on Biot’s model of fully saturated poroelastic media. The elastic-viscoelastic correspondence principle is applied to describe viscoelastic properties of elastic skeleton. Viscoelastic constitutive equations are introduced. Classical viscoelastic models are used, such as Kelvin-Voigt, Standard linear solid and model with weakly singular kernel of Abel type. Differential equation system of full Biot’s model in Laplace transform and formulas for elastic modules are given. Original problem’s solution is built in Laplace transform and numerical inversion is used to obtain the solution in time domain. Direct boundary integral equation (BIE) system is introduced. Regularized BIE system is considered. Mixed boundary element discretization is introduced to obtain discrete analogues. Gaussian quadrature and hierarchic integrating algorithm are used for integration over the boundary elements. Numerical inversion of Laplace transform is done by means of modified Durbin’s algorithm with a variable integrating step. The described numerical scheme is verified by a comparison with analytical solution in a one-dimensional case. Isotropic poroviscoelastic solids and halfspaces are considered. Results of numerical experiments are presented. Problems of axial force acting on the end of prismatic solid and vertical force acting on a halfspace are solved. Viscous parameter influence on dynamic responses of displacements and pore pressure are studied. Surface waves on poroviscoelastic halfspace are modelled with the help of boundary element method.

In the present paper, a dynamic behavior of piezoelectric vibratory gyroscope in the form of a hollow piezoelectric cylinder with two pairs of electrodes placed crosswise on the outer side surface has been analyzed. In the case of one fixed end, two variants of the piezoceramic material polarization have been considered, namely, full radial polarization and partial radial polarization only under the electrodes on the outer side surface. For a completely polarized material (in addition to the case of a cantilevered end) two other variants of the fixations simulating the conditions of hinged opening were also considered. The behavior of the harmonic oscillation of gyroscope has been studied in the framework of linear theory of piezoelectricity (electroelasticity) which takes into account mechanical damping and rotational effects in the relative coordinate system. All of the investigated variants admit the availability of electrically active bending oscillation modes in two perpendicular planes which can be controlled by given electric potentials on the two pairs of electrodes. In such configurations when the gyroscope operates near to the corresponding resonance frequencies the primary flexural vibrations are generated in one plane and the secondary flexural motions are created due to the axial gyroscope rotation in the perpendicular plane. These secondary oscillations can be a measure to determine the value of the rotation frequency. The finite element method, ANSYS finite element package and specially designed computer programs written in the macro language APDL ANSYS were used for numerical calculations. The results of computational experiments have shown that the variant with one fixed end and with a full radial polarization of the piezoceramic material gives the largest maximum of the output potential in the presence of gyroscope rotation. It has been found that the variant of the gyroscope with the fixations simulating the conditions of hinged opening is also quite promising for application.

The packaging of large composite shell structures (corrugation, a cylinder and a truncated cone) and their deployment by internal pressure loading are explored. It is believed that the medial surfaces of the constituent elements have involutes which coincide with them in a packed state. The corrugation consists of the ring components, the cylinder and cone consist of trapezoidal plane components. These components are made of carbon fiber with orthotropic or transversely isotropic elastic properties and stapled by joints. The joints do not perceive resistance to rotation about the tangent to the weld line. The contemplated structures perceive bending loads (unlike pneumatic ones) made of soft materials (fabrics, films). Geometrically nonlinear solid mechanics problems with the internal pressure loading are solved by using the engineering computing system ANSYS. The deployment pressure dependence on the shell material structure, shell thickness and amount of constituent elements are investigated. It is shown that the deployment pressure of the large shell is commensurate with the pressure of pneumatic structures of soft materials. It was found that the stresses in the corrugation shells can reach critical values but in the cylinder and the truncated cone the stresses are insignificant. The task formulation and its solution on the thermodynamic state of the injected gas under quasi-static internal pressure loading of the shell are suggested. It is shown that in the beginning of deployment the gas temperature will drop by about 50-80 degrees Celsius according to gas composition, and then its temperature is tending to increase to the injected gas temperature. These results enable to expand the choice of materials for the pneumatic products manufacturing including space applications design.

In this paper, the dynamic of a structure composed of flexible rod elements connected via hinges is modeled. It is assumed that the hinges have constraints - rigid and non-rigid, controlled and uncontrolled ones. Mathematically, they are considered as differential ones in integrable or non-integrable forms. Mathematical model is formulated based on the finite element method taking into account finite deformations and the nonlinearity of the inertial forces. The rod element ends are considered to be connected with rigid bodies whose dimensions are small relative to the element length. Each finite element is associated with a local coordinate system for which the displacements, angles of rotation, the translational and rotational speed are strictly considered. Shape functions are taken as quasi-static approximations of local displacement and rotation angles of element cross-sections. Absolute displacements and rotation angles of element boundary cross-sections are taken as generalized coordinates of the problem. The dynamic equations are obtained using d'Alembert-Lagrange principle. It is considered that the generalized coordinates are subjected to the linear relations relative to the generalized velocities. Variation of the problem functional for which to look for the steady-state value is transformed by the addition of the constraint equations multiplied by the undefined Lagrange multipliers. The variational problem for the transformed functional is solved as a free. The stationarity conditions together with the differential equations of constraints determine the desired values of the generalized coordinates. This paper proposes an approach that allows to avoid cumbersome calculations of the nonlinear inertial members without simplification of the physical model and (or) changing the original structure of equations. An example of deploying rod system consisting of three flexible rods connected in series via hinges is considered. The solution of nonlinear dynamic equations is obtained numerically using the integral curve length parameter as a problem argument. This transformation gives a system of resolving equations the best conditioning of the numerical solution process.