## No 4 (2015)

**Year:**2015**Articles:**18**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/23**DOI:**https://doi.org/10.15593/perm.mech/2015.4

Dynamics of flexible shaft in rigid tube

#### Abstract

The paper is concerned with equations and numerical methods for calculation of flexible shaft rotation in a rigid tube. In the very general statement the shaft is represented as a Cosserat rod with an arbitrary dependence of properties on the coordinate. The quasi-static motion is considered in the first. Six equations of motion are obtained for the arbitrary bent and curved rod in the tube of arbitrary geometry. The projection of the equation of moments on the tangent to the curved axis of the rod is shown to be sufficient for describing the shaft motion. This differential equation is expressed in terms of the rotation angle of the rod cross-section. The solution for the quasi-static rotation is obtained both analytically and using the shooting method for boundary-value problem for an ordinary differential equation. The closed form expression for the angles of rotation of the shaft in the rigid tube as a function of the axial coordinates is obtained. The jumps occur for some combination of the parameters and they cannot be explained in the framework of quasi-static analysis. In order to explain the instability, the dynamics statement is applied. The nonlinear dynamic problem is solved by means of differential-difference method which is tested by a comparison with a closed form solution. Solution to the dynamic problem allows one to explain the quasi-static jumps obtained. The dynamic formulation shows that instead of quasi-static jumps the initial stage of rotation is a smooth rotation which jumps are abruptly replaced by intensive vibrations. The laws of rotation at different rotational velocities are determined too. The qualitative difference in the quasi-static and dynamic solutions is exposed. The suggested approach for solving nonlinear dynamic problems of rotation of the shaft with an arbitrary geometry is promising for modeling of processes of the directional deep drilling which is vital for the problems of oil production industry.

**PNRPU Mechanics Bulletin**. 2015;(4):7-18

Natural vibrations of heated functionally graded cylindrical shells with fluid

#### Abstract

The paper presents the results of studying the natural vibrations of heated cylindrical circular FGM shells containing a quiescent ideal liquid. The temperature dependent effective properties of the material representing a mixture of zirconium oxide and titanium alloy vary through the thickness of the shell according to the power law. The distribution of temperature along the radial coordinate is determined by solving a quasi-linear one-dimensional heat conduction equation. A mathematical formulation of the problem is based on the classical theory of shells and the principle of virtual displacements. The behavior of the liquid is described in the framework of the potential theory. The corresponding wave equation together with the impermeability condition and boundary conditions are transformed into a system of equations using the Bubnov-Galerkin method. As a result, the solution of the problem, which is sought with the use of semi-analytical version of the finite element method, reduces to calculations of complex eigenvalues of the coupled system of equations. The reliability of the results, obtained by application of the developed algorithm, is verified through a comparison with the known numerical-analytical solutions. The data obtained for circular cylindrical shells with different boundary conditions have revealed the dependence of the minimal vibration frequency on temperature at different volume fractions of FGM. The critical values of temperature have been determined for different heating regimes and geometrical dimensions. The difference between the dynamic properties of empty and liquid-containing shells caused by heating has been analyzed. It has been shown that in the case of cantilevered shells the presence of liquid inside the shell exerts the most notable effect on the vibrational behavior of the system.

**PNRPU Mechanics Bulletin**. 2015;(4):19-35

On account of scale effects in the simulation of mechanical and tribological properties of two-phase microand nanomodified polymer coatings

#### Abstract

This work presents simulated mechanical and frictional properties of polymer composite materials derived from the ultra-high molecular weight polyethylenes which are widely used as units of friction and sealing elements in different types of modern technology and medicine. There authors considered the two-phase polymer composites obtained by modifying the high molecular weight polymers by introducing micro- and nanoinclusions. Such materials tend to clear manifestation of scale effects, consisting of the non-monotonic dependence of the mechanical properties and tribological characteristics on the inclusion concentration and on characteristic size of inclusions. In order to register the scale effects it is proposed in this work to use first order gradient models which describe not only the non-local (gradient) effects in volume but also interfacial effects that affect the formation of transition zones in the vicinity of phase boundaries. Scale and surface parameters of gradient model of mechanical properties allow to take into account the peculiarities of the considered modified polymers, associated with the fact that the effect of fillers on properties is determined not with the own properties of the filler, but with the change of polymer morphology, i.e. with the forming of polymer crystallization zones in the vicinity of particles of filler distributed in volume. In order to simulate scale effects on the frictional properties, we proposed a model based on the analogy of the friction coefficient of inhomogeneous surface structures with mechanical characteristics of compliance of inhomogeneous material, when the "weak" phase defines the effective properties. It is shown that the use of this analogy simultaneously with the use of relations for effective compliances of periodic two-phase composite, found under the gradient model of elasticity in the one-dimensional approximation, allows a good description of typical non-monotonic dependences of the friction coefficient on the concentration of filler for the considered polymer composites. It appears that the proposed equations will be useful for the prediction of the properties of designed anti-friction polymer coatings.

**PNRPU Mechanics Bulletin**. 2015;(4):36-54

Simulation of thermal drift of fiber-optic gyroscope taking into account piezo-optical effects

#### Abstract

The prediction and compensation of the fiber-optic gyro errors caused by the influence of the environmental factors has been the subject of many studies for a long time, but still remains an urgent problem. In the past few years an interest in this problem has been increased due to a need in improving the precision of instruments incorporated into sensitive equipment. The theoretical aspects of thermally induced phase non-reciprocity of fiber-optic gyroscope phase (FOG) are considered in detail by the authors. The operating principle of the fiber-optic gyros is outlined and the main objective of the study is formulated. The paper presents the basic classification of the FOG drifts based on the drift root causes. The main constitutive relations for piezo-optical effects occurring in silica fiber are derived and the procedure of evaluating the design value of the FOG thermal drift is described. The key idea of the computational procedure is the development of solution to the differential equation of motion by the finite difference method. The peizo-optical relations are used to elaborate the iterative scheme to calculate the time of the ray traveling through the FOG optical loop (optical path length). The paper also presents a detailed (step-wise) procedure for the temperature drift computation, which employs both the third-party programs and authors’ codes. The interrelation of computational steps and the precedence of operations to be made for obtaining the required estimates have been determined. Thus, the problem of FOG thermoelasticity was solved based on the STAR-CCM+ applied package, whereas the drift computation was done separately in the MATLAB environment. A qualitative picture of the FOG drift obtained by direct numerical simulation without using the algorithms for signal analysis has been obtained and analyzed. The advantage of the externally connected processor is multiple processing of the initial data obtained from thermoelastic computations by any of the methods and algorithms which have been programmed by users in the processors.

**PNRPU Mechanics Bulletin**. 2015;(4):55-71

Modelling of deformation of wire spiral structures

#### Abstract

Analysis of power transmission lines (PTL) involves the calculations of static states and vibrations of conductors (and cables) together with spiral accessories, vibration dampers and other devices attached on them. Many of these problems can be properly solved only by taking into consideration the internal structure of the conductors, the design of which is formed by wire layers wound on each other at different angles relative to the longitudinal axis. For example, such reference is required in the design of the systems of power safety and reliability of information-telecommunication supply of aerodromes, aircraft and rocket systems, overhead transmission lines, subject to intense wind, especially in icing conditions. Due to the complex design of the wire structures the known issues arise in the estimates of their deformations, stiffnesses, bearing capacity, etc. For example, the bending stiffness of the conductor can sufficiently vary markedly as its deformation, since the wire layers may slip relative to each other, and a separate wire is movable within the wire layer. Consequently, the values of stiffnesses can be varied both along the conductor axis, and in time. The paper proposes a new deformation model of wires structures which are similar to a PTL conductors. These structures include not only conductors and cables, but spiral clamps intended for tension, suspension, joints, protection and repair of conductors. On the basis of energy averaging, each wire layer is considered as an elastically-equivalent anisotropic cylindrical shell, thus a conductor or a spiral clamp are modeled as a system of cylindrical shells nested into each other and interacting by the forces of pressure and friction. Following this approach the formulae for the flexibility and stiffness matrices of spiral structures have been obtained. The problem of interaction of a tension clamp with the external wire layer of a conductor has been formulated and solved. The mechanism of the force transfer from the clamp on the conductor has been investigated.

**PNRPU Mechanics Bulletin**. 2015;(4):72-93

Ballistic damages of GFRP and their repairs using ultrasound matrix impregnation

#### Abstract

The research of type of damages was carried out using the GFRP plates made of STEF (matrix - epoxy resin, filler - plain weave glass fibre fabric, six layers, 2 mm of thickness), which were fired at the special stand. Experiments were conducted with steel spherical impactors, 6.35 mm diameter (weight of 1.05 g) with terminal velocity 50-900 m/s. Experimental data of ballistic curves was fitted by least-square regression according to the classical Lambert-Jonas equation. According to this data, the perforation begins with velocity ~180 m/s (ballistic limit V50). The dependence of delamination area vs. initial velocity V0 was obtained as results of experimental data optical processing. The maximum delamination area is formed when the velocity is close to the ballistic limit. The main reason of it is that the specimens absorb all the kinetic energy of the impactor at this velocity. It was found that at the impact point the maximum of interlayer crack gaps (delamination thickness) reaches ~50 microns. The tests provided on tensile specimens with ballistic damages were performed at INSTRON testing machine to determine the residual strength with such defects. On the basis of the experiments we propose to replace the damaged area by an open hole with the equivalent diameter. For an arbitrary loading the evaluation of residual strength was made by the express-method based on the energy approach. Thus, determining the size of the damaged area and the equivalent diameter holes, we can predict the residual strength of the structural element with a defect. Due to the fact that small damages significantly reduce the fracture load of specimens, an effective method of repairingdelamination was developed. The repair occurs by filling the cracks between the layers using the epoxy resin compound. Epoxy resin links the layers together that provides their further joint work. The factors influencing the rheological properties of the matrix (epoxy compound viscosity and surface tension in terms of capillary effects) were studied. It was shown that a matrix ultrasound heating up to the temperature of 60 °C allows a complete filling of the above-mentioned crack of delamination at the length up to 20 mm for 90 seconds. Further, for complete matrix curing we used the local infrared heater. As a result of the repair, GFRP strength properties run up to 80- 90 % of the initial undamaged one.

**PNRPU Mechanics Bulletin**. 2015;(4):94-109

Formation of divergent testing efforts and experimental evaluation of material strength under biaxial stretching

#### Abstract

A method for forming multidirectional testing efforts during the mechanical testing of laboratory samples using the typical one-driving testing machine is suggested here. In the proposed method, this system of forces acting on the sample is created by the inclined faces of the sample and the resulting contact reactions on them when it interacts with parallel support having appropriate ramps. The authors consider the scheme of supporting and the loading of the prismatic sample which implements the considered formation of the test efforts. The justification of choice of angles of the sample inclined faces based on the numerical simulation of model deformation using the numerical finite element method apparatus and solving the contact problem of deformable solids is provided here. Recommendations for using optimum angles of inclination of the supporting surface for creating circuitry biaxial stretching have been given. The results of computational analysis of VAT prismatic samples depending on its basic geometric parameters are described here. The testing of the proposed models is described in determining the strength parameters of spring-tempered spring steel 50HFA located in a biaxial stretching. The analysis of the results obtained by the destruction of the samples is based on the limit state equation of Pisarenko-Lebedev. The determination of these parameters is based on the numerical analysis of VAT tested series of samples at the time of their destruction. The analysis includes consideration of the nature of the contact interaction of the sample with the support elements and the possible occurrence of plastic deformation in the sample. It became possible to present the experimental evaluation of reducing the limit value of the first principal stress in the hearth of destruction of samples of steel 50HFA under biaxial stretching compared to uniaxial tension.

**PNRPU Mechanics Bulletin**. 2015;(4):110-120

Experimental investigation and numerical modeling of elastic properties and strength of porous ceramics

#### Abstract

Advanced ceramics are widely used in responsible structures that work at conditions of high temperature changes, strong electrical fields and impact loads. Sintered ceramics are usually porous which affects their strength and elastic properties. In the first part of this work the results of experimental and numerical strength investigations of hot-pressed alumina ceramic are presented. The disk-shaped specimens with different porosity (4-23 %) were subjected to Piston-on-Ring bending test up to failure. Ultimate tensile strength is varied in the range of 180…490 MPa. Finite element method was used for stress state analysis of ceramic disk during bending test. Elastic properties of porous ceramic for numerical simulations were determined by using the known approximation of dependences “property - porosity” and some experimental data. In the second part of this work three-dimensional numerical micro-model was created in ANSYS. This model is a cube with set pores up to 160 of spherical forms. The diameter of sphere is given by Weibull distribution with mean value m = 0,139 μm and standard deviation s = 0,075 μm (defined by SEM analysis of fracture surfaces). Scale parameter λ = 0,156 μm and shape parameter k = 1,919 of the Weibull distribution was determined by the least squares method. The authors generated three to six models with a random distribution of pores for each average porosity; and analyzed stress state under axial tension for each case. The maximum normal stress, stress concentration factor, elastic modulus and Poisson's ratio are dependent on the average porosity. The values of the tensile strength were defined for different porosity according to the Rankine criterion (maximum normal stress criterion). These values are in a good agreement with the experimental results.

**PNRPU Mechanics Bulletin**. 2015;(4):121-137

Modeling the treatment of high-energy pulsed electromagnetic field of the micro-cracks in a polycrystalline metal

#### Abstract

The paper studies electroplastic effect in terms of the hypothesis of the healing of defects in the material under the influence of high-energy pulsed electromagnetic fields. The processes occurring in the metallic samples under the impact of electrical current of high density are considered. The electric, temperature fields, stress-strain state and phase transformations in the vicinity of micro-defects with line size 10 microns are studied. Such defects are always present between the grains in a polycrystalline metal after metallurgical casting or appear in it during its deformation under processing. The coupled quasi-stationary model of the impact of high-energy electromagnetic field on the pre-damaged electroplastic material with an ordered system defects are proposed. The model accounts for melting and evaporation of the metal and the dependence of its physical and mechanical properties on the temperature. The problem is solved numerically by finite elements method with adaptive mesh using on the base of alternative Euler-Lagrange’s method. The boundary value problem is solved for a representative element material with a micro-defect in the case of plane strain. The micro-defects in the form of flat cracks with rounded tips are considered. The presence of micro-cracks leads to inhomogeneous physical and mechanical fields in the material. Numerical modeling has shown that in the vicinity of the micro-defects intensive electromagnetic field and current with large fields gradients arise. This causes rapid local heating in the vicinity of the tip of the micro-crack, followed by thermal expansion of metal, and subsequently melting the material. The inhomogeneous heating results in a high compressive stresses and intense plastic flow of the material in the vicinity of micro-crack. The resulting stress field is not only closing the shores of micro-crack, but also reducing its length and ejecting the metal into the crack. Ejection occurs through the formation of the metal jet of molten metal directed into the crack. These processes may be accompanied not only by melting in the vicinity of the tip, but also evaporation of the metal to crack for the micro-cracks located near the outer boundaries of the sample. The simultaneous reduction in the length, the ejection of the molten metal into the cracks and closing of micro-crack shores leads to the fact that the shores of the crack come into contact with the jet stream and finally the jet material completely jams shores cracks. It is the welding of the crack and healing of the micro-defects which takes place.

**PNRPU Mechanics Bulletin**. 2015;(4):138-158

Experimental investigation of defects influence on composites sandwich panels strength using digital image correlation and infrared thermography methods

#### Abstract

The article is devoted to the experimental investigation of the defects influence on the residual strength of composites structure, as well as the possibility of using local repair operations. The objects of research are structurally similar elements of acoustical sandwich panels after a local repair of defects, such as through breakdown. The specimens were produced by serial technology from a fiberglass prepreg. Research was carried out using a universal electromechanical system Instron 5982 and servo-hydraulic system Instron 8801. For the analysis of the stress-strain state of the deformable elements the authors used the non-contact three-dimensional digital optical system Vic-3D, the mathematical apparatus which is based on the method of digital image correlation (DIC). To control the internal geometry of the specimen and assess the possible defect inspection infrared thermal imaging system FLIR SC7000 was used. The techniques of joint use of testing and measuring systems under static and cyclic tests were offered. For comparison, the repaired sandwich panel specimens were tested in tension and in tension under a preliminary cyclic loading with the registration of the deformation fields and thermal images. Their deformation and fracture mechanisms are analyzed, and their loading diagrams are obtained. The experimental data was obtained from the Vic3d system study of the evolution inhomogeneous fields of axial and transverse deformation on the surface of repaired sandwich panels under static loading and cyclic tests. By using infrared thermal imaging system internal structure, processes of the defects development and the temperature distribution on the surface of the test specimen were detected.

**PNRPU Mechanics Bulletin**. 2015;(4):159-170

ITERATIVE FINITE ELEMENT ALGORITHM and its implementation for STRESS STATE of structural elements with singular points

#### Abstract

The target of research is the stress-strain state (SSS) near and directly in the special points of structures with features in the form of flat composite wedges or spatial edges, which are the intersections of the forming bodies surfaces. Boundary conditions, continuity conditions of stresses and strains on the line (surface) connections of structural members and other constraints, posed by the problem statement into structural elements special points, form a mandatory algebraic equality (MAE), which represent a system of linear inhomogeneous algebraic equations. The MAE number, formulated at special points, exceeds the MAE number in ordinary (not special) points of the boundary, which limits the ability to build solutions meeting all of the UAR using usual solid mechanics methods. Therefore, the works purpose is to create an algorithm allowing constructing a solution consistent with all MAE formulated at special points. Substructure iterative mixed finite element method (FEM) is proposed. Substructures are parts of the computational area with continuous state parameters. The main results. The algorithm and software package for the stress state study near and directly in special points of construction elements are suggested. Depending on the considered flat or spatial structure geometric and material parameters properties, problems of elasticity and thermo elasticity are divided into types and subtypes, distinguished by the MAE number. Mixed finite element method version allows calculating nodal stress parameters without the differentiation of the approximate solution or without using any replenishment method. The iterative approach allows building a solution that is consistent with all MAE at special points. The procedure of the proposed algorithm and its Fortran-95 implementation is described. Features, associated with OpenMP technology used in the algorithm implementation are discussed.

**PNRPU Mechanics Bulletin**. 2015;(4):171-187

Complete asymptotic expansion M. Williams near the crack tips of collinear cracks of equal lengths in an infinite plane medium

#### Abstract

The paper is aimed at analytical determination of the coefficients in crack tip expansions for two collinear finite cracks of equal length in an infinite plane medium under mixed mode loading conditions. The study is based on the solutions of the complex variable theory in plane elasticity theory and the complete asymptotic Williams expansion of the stress field in the vicinity of the crack tip. From the practical point of view, it is very important to know: 1) analytical dependence of coefficients on geometrical parameters of specimens and applied loads; 2) the number of higher-order terms in the asymptotic Williams expansions that we need to keep to accurately describe the stress and displacement fields in the neighborhood of the crack tip. In the paper the authors have provided a multiparametric presentation of the stress filed near the crack tips in the infinite plate with two collinear cracks of finite length. We have presented the analytical determination method of coefficients related to the complete asymptotic expansion. The method is based on the complex variable theory and the classical Kolosoff-Muskhelishvili approach. The expansion of the Kolosoff-Muskhelishvili's potential in the vicinity of the crack tip allows to find the analytical presentation for coefficients of the complete Williams asymptotic expansion. The analytical solution gives the dependence of the amplitude coefficients on the lengths of the crack and the applied loads for pure mode I, pure mode II and mixed mode I/II conditions. It is shown that together with the main term including the stress intensity factor and the term called T-stress, it is necessary to hold higher-order terms in the asymptotic solution. The obtained solution permits to construct the asymptotic stress field expansion containing an arbitrary preassigned number of higher-order terms.

**PNRPU Mechanics Bulletin**. 2015;(4):188-225

On delamination of a stipe along the boundary between two elastic layers Part 1, problem formulation, the case of normal crack

#### Abstract

The problem of a strip, composed by two isotropic elastic layers of different elastic properties and thicknesses, separated by a semi-infinite crack located along the line between the layers, is considered. The mechanical load with nonzero total force and moment is supposed to be applied at infinity. By means of Laplace transformation the problem is reduced to a homogeneous Riemann problem. Under the assumption of possibility to neglect the cross-terms related to the influence of the normal stresses to the shier displacements and the shier stresses to the normal displacements the problem is reduced to two scalar Riemann problems. Such a formulation may be considered as an approximation for the general case (which is not worse than the traditional beam or rode approximation) and as the exact one for the case, when the two layers may slide but may not separate due to cohesion, e.g. by van-der-Waals forces. By means of factorization procedure the exact analytical solution has been obtained for one of the formulated scalar problems, namely, the problem of the normal separation. The asymptotical expression has been derived for the relative displacements of the crack faces far from its tip. It is shown that the leading asymptotic terms of these relative displacements correspond to a beam deflection under the boundary condition of the type of generalized elastic clamping. i.e. the proportionality of the displacement and angle of rotation of the clamping point to the total vector and bending moment of the applied load by means of the matrix of coefficients of compliance. The analytical expressions for these coefficients have been obtained. The asymptotical expression for the stress field near the crack tip (stress intensity factor) was also derived.

**PNRPU Mechanics Bulletin**. 2015;(4):226-245

NONAXISYMMETRIC DYNAMIC PROBLEM OF THE DIRECT PIEZOEFFECT FOR AXIALLY POLARIZED SOLID CYLINDER

#### Abstract

The paper considers nonaxisymmetric dynamic problem for the theory of electroelasticity of anisotropic piezoceramic cylinder axially polarized. Mechanical vibrations are carried out by the action of its end surfaces of the non-stationary load as normal stresses; it is an arbitrary function of the radial, angular position and time. At the same end the surfaces are covered with electrodes and fixed membranes. The developed algorithm of calculation allows to meet a variety of mechanical and electrical conditions on the cylindrical surfaces of the electrodes that are not covered. To be specific, in this paper, they are considered free from normal and shear stresses. The new closed solution is built in three-dimensional statement through the consistent use of the method of incomplete separation of variables in the form of integral transformations. The author has consistently applied Fourier transformation with finite limits on the axial coordinate and the final transformation of generalized (FIT) for the radial variable. At each stage of the decision-making, the standardization is utilized that allows one to bring the boundary conditions on the corresponding coordinate in uniform. The calculated ratio the components of the displacement vector and the electric field potential make it possible to determine the frequency of natural oscillations nonaxisymmetric, stress-strain state of the piezoceramic cylinder, and all parameters induced by the electric field. The potential difference between the end faces is determined by a measuring instrument with a high input impedance which corresponds to the “idle”. The numerical results of calculation lead to the conclusion that the use of the constructed algorithm makes it possible, compared with the numerical methods, to obtain more accurate values of natural frequencies of the spectrum, the stress-strain state and the electric field of the piezoceramic cylinder.

**PNRPU Mechanics Bulletin**. 2015;(4):246-258

Contact interaction plates, reinforced by ribs, with gaps under the influence of white noise

#### Abstract

We have investigated the contacting interaction of a sandwich structure in the form of plates and beams with small gaps between them. Such systems are integral elements of modern devices. The created mathematical model is based on the following hypothesis: the system is a multi-layer structure; the materials are isotropic. To solve the problem we used the finite difference method with approximation 0 (h2), 0 (h4) and Bubnov-Galerkin method in higher approximations of spatial coordinates, as well as the Runge-Kutta 0 (h4), 0 (h6), 0 (h8) time. In solving problems associated with random variations, it is necessary to solve the challenge of an error, so you need to use different numerical methods to validate the results in order to distinguish the chaos of the numerical error. For the analysis of chaotic dynamics we have applied all methods of qualitative analysis. We have investigated the spatiotemporal chaos based on wavelet analysis. We have studied the effect of white noise in the contact interaction of elements of the multilayer structure. Also, the analysis of the complex vibrations of plates and beams in different intensities depending on the type of noise and load has been made. It was found that by using an external additive white noise, it became possible to control chaotic oscillations and transfer the system from a chaotic state to a harmonious one and enable or disable the contact interaction.

**PNRPU Mechanics Bulletin**. 2015;(4):259-272

Biharmonic (two-frequency) load frequencies influence on mechanical behavior of solid propellant simulator

#### Abstract

This work is devoted to the engineering unit development for stress-strain state assesses of the viscoelastic structures in aerospace engineering under some biharmonic (two-frequency) loads. Uniaxial monoharmonic (one-frequency) tests of low-modulus viscoelastic polymer composite were conducted under different values of pre-static deformation and initial phase shift angle, as well as uniaxial biharmonic (two-frequency) tests under different values of first (low-frequency) and second (high-frequency) harmonics frequencies. The complex modulus method was used to describe the behavior of viscoelastic material under harmonic loads. Material dynamic deformation properties were determined by a specially developed method. The method allows determining the dynamic modulus and loss angle of the first and the second harmonics. A relative frequency factor was offered for the study of the effect of biharmonic load frequencies on the behavior of viscoelastic material. The method of experiment results under boundary values of frequencies ratio of the first and second harmonics was offered. This approach allows us to determine the behavior of viscoelastic materials under monoharmonic loads in the biharmonic formulation of the problem. Dependencies of the dynamic modulus and the loss angle of the second harmonic on frequencies ratio of the first and second harmonics were shown. It has been found that the initial phase shift change does not impact viscoelastic materials behavior. A mathematical model describing material behavior under different values of frequencies was proposed. This model allows us to describe, with a sufficient accuracy, the material behavior under any frequencies ratio. Biharmonic tests are more preferable for determining the dependence of the dynamic modulus on pre-static deformation than monoharmonic tests under pre-static deformation changing.

**PNRPU Mechanics Bulletin**. 2015;(4):273-292

Stress-strain analysis and strength prediction of composite outlet guide vane for aircraft jet engine

#### Abstract

Today the composite materials are widely used in aviation industry for load bearing elements of aircraft engines. The following problems need to be solved for an effective composite application: firstly, choosing the optimal reinforcing scheme for main working area of construction, and secondly, realizing the stress-strain analysis for the most loaded zones with complex geometry such as joints, layer crossings and area connections. The present work is devoted to detailed stress-strain analysis of composite outlet guide vane (OGV) for aircraft engine. The 3D problem formulation of mechanics of anisotropic laminated composite for strength prediction was given. The technological scheme of laying out anisotropic plies and fastening method were taken into account in the model. The scheme of reinforcing for OGV was determined in the previous researches. The maximum stress criterion was used for estimation of strength margin. The numerical simulation of this problem was carried out by finite element method (FEM) with ANSYS Workbench software. Due to the high dimensionality of FE model, the high-performance computing complex was used. The in-depth layered analysis of stress-strain state of the structure was made. The special focus was made on the areas with twisted layers near the flange of vane where the initiation of high interlaminar stresses is most likely to take place. An estimate of the influence of fastening conditions on the stress state for OGV was obtained. It was shown that interlaminar shear stresses are the most dangerous. It was found that the VKU-39 material and [0°/±45°] reinforcing scheme allow to provide the double strength margin under working loads for developed OGV.

**PNRPU Mechanics Bulletin**. 2015;(4):293-307

Structure and mechanical properties of zirconium oxide modified with carbon nanotubes

#### Abstract

Results of experimental investigation of mechanical properties of synthesized composite “zirconium oxide - multiwalled carbon nanotubes” are presented in this paper. Ceramic composite was produced on the basis of hydrothermal synthesis from zirconium salt solution with dispersed carbon nanotubes followed by critical point drying of hydrogel and thermal treatment of obtained aerogel. Zirconium oxide/carbon nanotubes aerogel fragments were then sintered in order to obtain bulk sample in the form of tablet by hot pressing technique. Mechanical properties of composite were investigated with microindentation system NanoTest. Experiments on single loading with different values of maximum applied load in range 100...500 mN were conducted. Elastic modulus and hardness of the composite were calculated from experimental indentation curves with standard Oliver-Pharr model. Energy dissipated into the material during indentation and energy of elastic recovery during unloading stage were calculated as areas under corresponding parts of indentation curves. It was found that composite microhardness is a power law function of indentation depth, whereas elastic modulus linearly decreases with increasing indent depth. Synthesized composite exhibits huge fraction of energy dissipated into material which increases with increasing load/indentation depth. Moreover, for all investigated values of maximum applied loads there were no cracks in the edges of indentation trace. The obtained results have shown that synthesized composite is highly effective in suppressing the cracks growth. Aforementioned regularities of mechanical behavior of composite can be linked with its phase composition as well as with presence and arrangement of carbon nanotubes in ceramic matrix.

**PNRPU Mechanics Bulletin**. 2015;(4):308-316