## No 4 (2016)

**Year:**2016**Articles:**21**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/20**DOI:**https://doi.org/10.15593/perm.mech/2016.4

Numerical analysis of dynamic strength of composite cylindrical shells under multiple-pulse exposures

#### Abstract

The target of this research is studying the fiberglass cylindrical shellswith exposed ends resulted from the cross-winding of tapes made of an unidirectional composite material. The aim of the study was to develop a numerical technique to model a progressive fracture of laminated composite cylindrical shells under multiple-pulse loading with an internal pressure of a various intensity. Kinematic model of deforming the laminate package is based on the applied geometrically nonlinear theory of shells. The formulation of geometric dependencies is based on the relations of the simplest quadratic variant of the nonlinear elasticity theory. The physical relations of the elementary layer are formulated based on the generalized Hooke's law for the orthotropic material based on the hypotheses of the applied shells theory. The process of a progressive shell failure is described within the degradation model of stiffness characteristics in elementary layers of a multilayer package which is based on Hoffman's criteria for composite materials and on the criterion of maximum stresses for the fibers. The process of damage accumulation in the shell material due to a multiple application of impulse load is takeninto account by means of the computational scheme in which the calculation of the current stress strain state is carried out with stiffness characteristics obtained in the model of their degradation under previous loading. Energetically consistent system of motion equations of the applied shell theory is deduced from the stationarity condition related to the functional of the shell total energy. A numerical method for solving the formulated initial-boundary value problem is based on an explicit variational difference scheme. The accuracy of the considered techniqueis proved by comparing the obtained results with the known experimental data. The results of how the number of loadings affects the value of marginal circumferential deformationsare presented. It is established that the level of maximalring deformations is approximately ten times less than their limit values compared to a single loading.

**PNRPU Mechanics Bulletin**. 2016;(4):7-19

Stress-strain state of an elastic soft functionally-graded coating subjected to indentation by a spherical punch

#### Abstract

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.

**PNRPU Mechanics Bulletin**. 2016;(4):20-34

CORRELATION FUNCTIONS OF STRESS AND STRAIN FIELDS IN MICRO-HETEROGENEOUS MEDIA

#### Abstract

The paper is devoted to the development of a method for calculation of microstructural stresses and strains fields in the multi-phase media based on the calculation of the statistical characteristics of the local stress and strain fields in the components, which takes into account the geometrical and mechanical properties of components. Representative volumes ofstructurally mulit-phase heterogeneous materialswere investigated. It is assumed that the components are homogeneous and isotropic. The internal geometry of the structure as well as the assessment of spatial interaction is described by the moment functions of different orders. The behavior of individual components of the microstructure during loading ofthe representative volume is estimated using the statistical characteristics of the local stress and strain fields. The characteristics of deformation processes are the statistical moment functions of the stress and strain fields in the components of the material. Analytical expressions for the statistical moments and correlation functions of the stress and strain fields are obtainedusing statistical averaging of integral-differential equations that contain moment functions, and derivedfromthe solution of the stochastic boundary value problems in elastic and elastoplastic formulation. Some special cases of typical heterogeneous media with a random microstructure were considered. The correlation functions of stress and strain for sparse structures with spherical and ellipsoidal hollow inclusions in the elastic and elastoplastic cases were built. The study and the selectionof approximating dependences obtained for the correlation functionswere performed. The numerical results can be used to evaluate the mechanical behavior of the inhomogeneous medium microstructural component under different loading conditions and to predict fracture initiation.

**PNRPU Mechanics Bulletin**. 2016;(4):35-51

Analysis on cyclic deformation and low-high-cycle fatigue in uniaxial stress state

#### Abstract

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.

**PNRPU Mechanics Bulletin**. 2016;(4):52-71

Anisotropy of material fatigue properties and its effect on durability of structural elements

#### Abstract

The generalization of the criterion of multiaxial fatigue failure is proposed for the case of the alloy with the anisotropy of fatigue properties. The second invariant of stress deviator is replaced by the Hill’s function that is usually used to describe anisotropic plasticity of metals. In various studies the dependence of fatigue limit on the axis of loading is investigated for samples with texture at uniaxial fatigue tests. The texture is typically induced during manufacturing semi-finished products (primarily - in rolling). In present study we develop a method for express calculation of stress-strain state (SSS) for the elastic annular disk of variable thickness (disk of the gas turbine engine compressor) for the mode of low-cycle fatigue (flight cycles). Simplified representations are used for dependence of solution on coordinates along the disk thickness (the power series) and in the circumferential direction (the Fourier series). For the radial distribution of stresses and displacements the systems of ordinary differential equations have been derived and solved by the orthogonal sweep method. Proposed criteria of multiaxial fatigue and results are used to calculate the distributions of fatigue durability for typical disk of gas turbine taking into account the centrifugal forces. Location and time to appearance of fatigue failure zones determined accounting the influence of the anisotropy of fatigue properties. The calculations indicated that the significant decrease of the fatigue durability is in the vicinity of the disk rim.

**PNRPU Mechanics Bulletin**. 2016;(4):72-85

Algorithms for numerical simulation of structures deformation and fracture within relations of damaged medium mechanics

#### Abstract

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.

**PNRPU Mechanics Bulletin**. 2016;(4):86-105

Experimental and numerical research of the dynamic response of composite outlet guide vane for aircraft jet engine

#### Abstract

The object of this research is composite outlet guide vane (OGV) for advanced aircraft engine. The weight reduction due to using of polymer composite materials (PCM) instead of the metal in OGV can reach 40 %. The vane is exposed to intense aerodynamic loads during operation. The modal analysis is needed for the detuning a structure’s resonance frequency. The results of such analysis are presented in this work. Experimental technique of modal analysis for composite OGV in frequency range up to 6,4 kHz is described in paper. Experimental study was carried out for three full-scale OGV samples with the help of three-component scanning laser vibrometer using PSV-400-3D hardware. As the results the mean values and coefficients of variation of natural frequencies were obtained and the main natural modes were shown. The numerical simulation of this problem was carried out by finite element method (FEM) with ANSYS Workbench software using high-performance computing complex. The technological scheme of laying out of anisotropic plies was taken into account in the developed OGV model. The results of numerical simulations of natural frequencies and modes were compared with the test data. Good correlation was found. This fact confirmed that the stiffness of a full-scale OGV, manufactured with various possible fluctuations of process parameters and mechanical properties of materials, meets the required conditions. For further researches the developed numerical model allows to study the effect of reinforcing scheme and other design parameters changing on OGV frequency response. Laboratory modal analysis can be used to control the dimensional stability and material mechanical properties.

**PNRPU Mechanics Bulletin**. 2016;(4):106-119

SPECTRAL DECOMPOSITIONS IN DYNAMICAL VISCOELASTIC PROBLEMS

#### Abstract

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.

**PNRPU Mechanics Bulletin**. 2016;(4):120-150

Nucleation recrystallization mechanisms in metals at thermomechanical processing

#### Abstract

In the last 15-20 years, mathematical models are the most important “tool” in the design and creation of technology of thermomechanical processing of metals and alloys. This is the result of appearance of new class of models based on physical theories. Single-level macrophenomenological models based on macro experiments are replaced by crystals plasticity. Founders of physical theories of plasticity are G.I.Taylor, G.H.Bishop , R.Hill, T.G.Lin. Many others researchers from the Soviet Union and Russia made a significant contribution to the development of this direction: R.Z. Valiev, Y.D. Vishnyakov, S.D. Volkov, O.A. Kaybishev, V.A. Likhachev, V.E. Panin, V.V. Rybin, T.D. Shermergor et al. Physically based approach requires deep understanding of the internal mechanisms and processes that accompany thermomechanical effects caused by inelastic deformation at different scale levels. The important one for the microstructure formation and mechanical properties of finished products obtained by thermomechanical processing methods is the process of recrystallization. At that point in this article provides an review of the existing theories of recrystallization nucleation mechanisms. Basic physical mechanisms of nucleation recrystallized grains are classified: 1) the mechanism, based on the classical theory of fluctuations, proposed by J.E.Burke and D.Turnbull; 2) R.W. Cahn’s mechanism of nucleation and growth subgrains polycrystal, formed as a result of the process polygonization; 3) P.A. Beck’s and P.R. Sperry’s mechanism grain boundary migration, initially present in the polycrystal - strain induced boundary migration (SIBM); 4) the mechanism of nucleation and growth of new grains as a result of coalescence subgrains (H.Hu, J.C.M.Li, H.Fujita ). Analysis of existing models describing the inelastic deformation at high temperatures demonstrates the need for consideration and inclusion in the models description of physical mechanisms of high-temperature processes accompanying plastic deformation.

**PNRPU Mechanics Bulletin**. 2016;(4):151-174

Dispersive characteristics of flat longitudinal elastic waves extending in porous liquid-saturated medium with cavities

#### Abstract

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.

**PNRPU Mechanics Bulletin**. 2016;(4):175-186

A modifiedBouc-Wen model to describe the hysteresis of non-stationary processes

#### Abstract

A number of known phenomenological models are considered, which are used to describe a variety of hysteresis effects in nature. In this case, the system is considered as a "black box" with known experimental values of input and output parameters. Correlations between them are established by mathematical functions, whose parameters are identified using experimental data. Amongthe phenomenological models there are marked the Bouc-Wen model andits analogsthat have been successfully usedin variousscientific and technical fieldsdue to the possibilityof the analytical description ofvarious hysteresis loopsof non-stationary processes. The conditions are formulatedwhich must be satisfied by the Bouc-Wen model. The main ones are the model adequacy of the physical process and stability. To describe the hysteresis, a mathematical model is suggested, according to which the force and kinematic parameters are bound by a special differential equation of the first order. In contrast to the Bouc-Wen model, the right side of this equation is chosen in the form of a polynomial of two variables determining the trajectory of a hysteresis in the process diagram. It is stated that this presentation provides the asymptotic approximation of the solution to the curves of the comprehending (including) hysteresis cycle.This cycle is formed by curves of direct and reverse processes ("loading-unloading" processes), which are based on experimental data for the maximum possible or permissible intervals of parameter changes during the steady vibrations. Coefficients in the right part are determined from experimental data for the comprehending hysteresis cycle under conditions of steady-state oscillations. Approximation curves of the comprehending cycle are constructed using the methods of minimizing the discrepancy of analytical representations to the number of experimental points. The proposed approach allows by one differential equation to describe the trajectory of hysteresis with a random starting point within the area of the comprehending cycle.

**PNRPU Mechanics Bulletin**. 2016;(4):187-199

Theoretical-experimental method for determination of aerodynamic damping component of test samples with diamond-shaped cross-section

#### Abstract

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.

**PNRPU Mechanics Bulletin**. 2016;(4):200-219

The influence of fluid filtration on the strength of porous fluid-saturated brittle materials

#### Abstract

The paper is devoted to the study on how the strength of fluid-saturated permeable brittle materials depends on strain rate. The study has been carried out by means of a numerical simulation using a hybrid cellular automata method and a coupled model, which takes into account the interplay of deformation of solid skeleton, pore pressure change and fluid filtration. It has been found that the influence of pore fluid on the material strength is determined by the competition of fluid pore pressure change (due to the volume deformation of solid skeleton) with filtration. On the basis of a parametric study we obtained the combinations of physical and mechanical characteristics of solid skeleton and fluid as well as of linear dimensions of the samples, which uniquely defines the dependence between the strength of the deformed fluid-saturated sample and the strain rate. By the examples of uniaxial compression and constrained shear tests we have shown that the character of influence of the fluid filtration on the sample strength is determined by the sign and magnitude of pore volume change during the course of deformation. Under the loading accompanied by a decrease in the pore volume and increase in the pore pressure, fluid redistribution reduces the local maxima of pore pressure and thereby provides an increase in strength of the samples. Under the loading conditions which determine an increase in the pore volume and pore pressure drop, the filtration maintains the fluid pressure and thereby reduces the strength of the samples. Based on the simulation results, we have constructed the generalized logistic dependences between the samples strength of brittle permeable materials and strain rate, mechanical properties of liquid and solid skeleton and sample dimensions. These results show that the non-stationary character of the related deformation and filtration processes determines a significant variation of the strength in the samples of permeable materials even at low strain rates.

**PNRPU Mechanics Bulletin**. 2016;(4):220-247

Numerical analysis of poroviscoelastic prismatic solids and halfspaces dynamics via boundary element method

#### Abstract

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.

**PNRPU Mechanics Bulletin**. 2016;(4):248-262

Methodology of numerical modelling of mechanical properties of the porous heat-shielding material based on ceramic fibers

#### Abstract

We propose a method to predict the compression strength and elastic modulus of high porous ceramics based on fibers or whiskers. The method is based on the direct numerical simulation of material microstructure using a finite element approach. The representative volume elements of material samples are created using the random algorithm taking into account the given sample size, fibers mean size and orientation and porosity volume fraction. The fiber structure is assumed to consist of long rods as fibers and short rods as links (contacts) between fibers. For the considered structures we proposed the formulation of the strength criterion, in accordance with which the destruction of the material occurs due to the failure of connections between the fibers. It is proposed to consider the ultimate strength of the fiber contacts as unknown model parameter. Its value should be determined using the fitting of the estimation results to the experimental data. Predicted values of effective stiffness and strength of material are based on the analysis of representative element stress state under mechanical pressure. In this paper, we studied the repeatability of the numerical calculations results for the same type of representative elements with the same average microstructural characteristics. The convergence of the effective properties values with the increasing of the fragments size is also studied. Test results of the mechanical properties modeling of fibrous materials with different porosity and fibers orientation are presented in this article.

**PNRPU Mechanics Bulletin**. 2016;(4):263-274

Finite element investigation of the effectiveness of the tubular piezoelectric vibratory gyroscope depending on the type of polarization and boundary conditions

#### Abstract

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.

**PNRPU Mechanics Bulletin**. 2016;(4):275-288

Mathematical modeling of piezo-electro-luminescent effect and diagnostics of pressure distribution along fiber optic sensor

#### Abstract

The algorithm of finding the distribution function of pressure along the three-phase fiber optic sensor based on the results of light intensity proceeding from a fiber optic phase measured on the edge section of the sensor is developed for a case of nonlinear "function of a luminescence" which is a dependence between the intensity of light and voltage acting on the electrophosphor. The problem is reduced to the solution of the Fredholm integral equation of the 1st kind with the differential kernel depending on the calculated effective parameters of the sensor and on the derivative set function of a luminescence of an electrophosphor. The analytical solution has been obtained for the function of probabilities density of pressure distribution for a special case when the kernel is expressed by the delta-function; and the Fredholm integral equation is reduced to an algebraic one. "Direct" and "reverse" problems of the Fredholm integral equation for a case of nonlinear function of a luminescence of an electrophosphor are solved as an illustration of the algorithm. The light intensity derivative at the optical fiber output of control voltage for the set uniform law of probability-density function with regard to pressure distribution is found in a direct problem. As for the reverse problem, the probability-density function is determined in comparison with the known exact solution using the direct problem solution for the derivative of light intensity. The numerical solution of the reverse problem is carried out in different approximations in which the distribution of nodal points in intervals and required nodal function values of probability-density pressure are found from a condition of minimizing summary discrepancies based on the values of the light derivative intensity which have been set and calculated on each step based on control voltage at the output of the optical fiber.

**PNRPU Mechanics Bulletin**. 2016;(4):289-302

Packaging and deployment of large shell structures by internal pressure loading

#### Abstract

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.

**PNRPU Mechanics Bulletin**. 2016;(4):303-316

A new algorithm for generating a random packing of ellipsoidal inclusions to construct composite microstructure

#### Abstract

The subject of the work is a microstructure of a composite which consists of a continuum matrix and a set of isolated particles homogeneously distributed inside the matrix. It is assumed that the reinforcing particles have ellipsoidal shapes, while distribution and orientation are random. The main point of the work is a new computationally-efficient algorithm to generate microstructure of such a composite. In the algorithm the existing “concurrent” method based on an overlap elimination is extended to ellipsoidal shapes of the particles. It begins with randomly distributed and randomly oriented ellipsoidal particles which can overlap each other. During the performance of the algorithm intersections between particles are allowed and at each step the volumes of intersections are minimized by moving the particles. The movement is defined for each pair of particles based on the volume of the intersection: if two particles are overlapped, then the reference point inside the intersection is chosen and then two particles are moved in such a way that the reference point becomes the tangent point for both particles. To define the relative configuration of two particles (separate, tangent or overlapping) and to choose reference point inside the intersection volume the technique based on formulating the problem in four dimensions and then analyzing the roots of the characteristic equation are applied. The algorithm is able to generate close packed microstructures containing arbitrary ellipsoids including prolate and oblate ellipsoids with high aspect ratios (more than 10). The generated packings have a uniform distribution of orientations.

**PNRPU Mechanics Bulletin**. 2016;(4):317-337

DYNAMICS OF MULTILINK ROD SYSTEM WITH CONSTRAINTS: A PLANE PROBLEM IN FINITE ELEMENT FORMULATION

#### Abstract

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.

**PNRPU Mechanics Bulletin**. 2016;(4):338-363

Dynamics of unbalanced flexible rotor with anisotropic supports during contact with the stator

#### Abstract

One of the important problems of rotor systems mechanics is rotor dynamics when it contacts stator elements. There are well-known cases when rotor - stator rubbing led to serious accidents. The main aim of the work is to gather data about movement of rotor upon contact with the stator suitable for verification of mathematical models. Also diagnostic features of rubbing in the rotor system with anisotropic stiffness were determined. Vibration behavior of the unbalanced flexible rotor with two ball bearings when it contacts the stator was experimentally investigated. The influence of support stiffness anisotropy, unbalance value, contact properties and stator elastic flexibility were taken into consideration. The calculation method of rotor dynamics with the stator rubbing was created and identified via experimental data. This method is based on Jeffcott model and allows taking into consideration the support stiffness anisotropy, unbalance value, contact properties, rpm and stator elastic flexibility. The model explains the mechanism of additional harmonics appearing in Campbell diagram and the presence of frequency response when the rotor contacts the stator. It makes possible to use phenomena of additional harmonics appearing as a diagnostic feature. The created experimental method and gathered data can be used for the verification and tuning of mathematical models. The suggested mathematical method of rotor-stator interaction modelling is suitable for the detection and elimination of rotor stator contacting situations. Also, it can be used as a basis for more complicated rotor system models.

**PNRPU Mechanics Bulletin**. 2016;(4):364-381