## No 1 (2016)

**Year:**2016**Articles:**10**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/22**DOI:**https://doi.org/10.15593/perm.mech/2016.1

Features of solving technological problems in mechanics of bodies with non-uniform metal structure transformed in thermo-force loading

#### Abstract

Solution for the actual problems of mechanics, materials science and related sciences of formation of the metal body in high-technological processes was proposed. The paper is related to using pulsed electro-mechanical treatment (EMT) as an example of analysis of production and related problems of structure formation and mechanical properties of the treated body. The algorithm for solving the thermo-elasto-plasticity problem taking into account changes in the thermo-structural state of metal, dynamic, mechanical effects and surface deformation changes in the space of intensity of stresses, strains and temperature was developed. The necessity of formulating and solving specific problems of material science of structure, based on existing empirical correlations was shown. The procedure of time grid confirming the correct numerical solution of various-related thermal, structural and mechanical processes was offered. The convergence, stability and adequacy of the proposed numerical method for solving the technological problems for EMT of steels were developed. Particular attention is paid to the problem of physical processes connectivity and inertial effects in a dynamically changing thermal, structural and stress-strain fields during EMT. Calculation examples of the structural areas and elasto-plastic stress distribution during EMT of steel bodies simulated by semi-infinite regions with a homogeneous structure and a non-uniform two-layer composition of the sample with hardened surface layer were shown. Comparison of the results has allowed identifying and describing the effects in the stress-strain distribution in inhomogeneous bodies with the transforming structure during EMT.

**PNRPU Mechanics Bulletin**. 2016;(1):5-25

Application of planar beamforming method to identification of spinning acoustic modes

#### Abstract

Geometrical and mechanical parameters of multi-layer panels used as sound-absorbing liners in the ducts of aircraft engines are determined by acoustic impedance requirements of the liners. The requirements are stated based on the analysis of sound modes propagation in ducts and in particular in the inlet. Therefore, knowledge of the sound field modal structure in an intake duct is essential for the choice of geometrical and mechanical parameters (depth of layers, dimensions and forms of honeycomb chambers, percent open area, material of construction and core, type of glue and etc.) of sound-absorbing liners. Experimental determination of the modal structure can be made with a microphone array mounted inside or outside the duct. In the present work, a planar microphone array placed outside the inlet duct has been used for measuring spinning acoustic modes, and the obtained data was processed with the beamforming method. Spinning modes were produced by a special generator based on the inlet of PS-90 turbofan engine. Sound was generated by 34 acoustic drivers JBL 2451H placed around the circumference under the test rig. Tests were carried out without flow. Experimental investigations were performed in the brand new anechoic chamber of Perm National Research Polytechnic University. The measurement results show that planar beamforming method locates a spinning mode at a point, which position depends on the mode number (this phenomenon is similar to the results obtained with planar beamforming method for propeller or open rotor noise). The conclusion has been made that measurements with planar beamforming method of spinning modes radiated from an inlet allow, in principle, determining noise modal structure in the duct but practical implementation of the method for real aircraft engines requires special research. Localization of spinning modes at a point has to be taken into account in the analysis of aircraft engine noise sources measured with beamforming method.

**PNRPU Mechanics Bulletin**. 2016;(1):26-38

Variant of thermoviscoplasticity theory

#### Abstract

Basic terms and equations of the theory of thermoviscoplasticity (inelasticity) belonging to the class of theories of flow in combined hardening are discussed. The tensor of strain rates is presented as a sum of tensors of the velocities of elastic and inelastic deformations. It should be noted that in this theory there is no conventional separation of the inelastic strain for deformation plasticity and creep. Elastic strain follows the generalized Hooke's law common to non-isothermal loading. The authors introduce a yield surface which isotropically expands or contracts and shifts to the process of loading. The current yield surface is defined by the process loading. The impact of the time factor is the process of loading too. For the radius of the yield surface it became possible to formulate the evolution equation that takes into account additional isotropic hardening under non-proportional (complex) load; and it is also generalized to non-isothermal loading processes and return to mechanical properties after annealing. We accepted parameter Kadashevich-Mosolov (corresponding to the angle between the velocity vectors of strain and stress) as a parameter describing a measure of the complexity of the process of loading,. The displacement yield surface is described on the basis of Novozhilov-Suboshi implying that the total displacement is the sum of displacements; each of them is its evolution equation. The analysis of hysteresis loops of plastic has allowed to allocate three types of microstresses (displacements) and formulate three types of evolution equations generalized to non-isothermal loading processes and relieving of microstresses during annealing. To determine the rate tensor of inelastic deformation we use the associated (gradiently) law of flow. For hard and soft loading regimes, the expressions for determining the rate of the accumulated inelastic deformation were found. The terms of elastic and inelastic states are formulated. To describe the nonlinear process of damage accumulation we introduced the kinetic equation of damage accumulation, where the energy that is required to create damage in the material is taken as the energy equal to the work of microstresses of the second type on the field of inelastic deformations. Here these kinetic equations are generalized to non-isothermal loading and the processes of embrittlement and heal the damage. Material functions closing the theory variant are specified, the basic experiment and method of identification of material functions are formulated. The description of the verification version of the theory of thermoviscoplasticity on a wide range of structural steels and alloys and programmes of experimental research are presented.

**PNRPU Mechanics Bulletin**. 2016;(1):39-56

DETERMINATION OF MECHANICAL CHARACTERISTICS OF TRANSVERSELY ISOTROPIC FIBER COIL ACCORDING TO ISOTROPIC PROPERTIES OF ITS COMPONENTS

#### Abstract

A fiber optic gyroscope is a complex system, which consists of the sensor and electronics unit. The sensing element includes the frameless fiber coil and integrated optical phase modulator. The article discussed the structure of frameless fiber coil. During the exploitation, the gyroscope is exposed to external factors, which influence should be minimized. It is indicated that the mechanical resonance is one of the causes of gyro-out errors in the operating mode. It is proposed to predict the behavior of the sensitive element at the design process of fiber optic gyroscopes. The authors it indicate the problem of great computing resources due the complex internal structure of the fiber coil. It is proposed to make a transition from a multi-component structure of the fiber coil to the transversely isotropic homogeneous material. Рассмотрена модель элементарного объема как ячейки периодической структуры волоконного контура. The authors considered the elementary volume model as the cell with periodic structure of fiber coil. Four tasks of finding the stress-strain state of an elementary volume of the fiber coil were set. The problem was solved by using the finite element method implemented in the software package Creo Simulate 2.0. It defined four independent mechanical characteristics of the transtropic body in the operational temperature range of the fiber-optic gyroscope. The experiment of finding natural frequencies of fiber coil in "free" suspension was carried out. Using finite element method the authors found the natural frequencies and mode shapes of the fiber coil with the transversely isotropic material model. The cylindrical coordinate system was used to set the material properties. Comparing the modal analysis results and experimental data proves that it is possible to use the obtained elastic constants to solve the deformable body mechanics problems.

**PNRPU Mechanics Bulletin**. 2016;(1):57-67

Study of heat source evolution during elastic-plastic deformation of titanium alloy Ti-0.8Al-0.8Mn based on contact and non-contact measurements

#### Abstract

This work is devoted to investigation of the heat source evolution during quasistatic tensile testing of titanium alloy specimens using a contact heat flux sensor and infrared thermography. The purpose of the study is to evaluate the possibility of using two different measurement (contact and non-contact) methods to monitor the state of material by changing the heat source value registered on the specimen surface during deformation. The obvious advantages of infrared thermography are non-contact temperature measurements of the material surface under various conditions and heat source field calculations. However this method has a number of limitations associated with the reflectivity of the tested material, noisy signal caused by external factors, heat transfer conditions between the specimen and environment, and accuracy of heat source calculations. These problems do not allow using infrared thermography under operating conditions in order to evaluate the energy state of materials and structures. The paper attempts to verify the heat source value arising during the elastic-plastic deformation of the material using infrared thermography data. For this purpose, a Seebeck effect heat flux sensor has been developed by the authors. Contact sensor and infrared thermography data give time dependence of the heat flux value. The satisfactory agreement of the results shows that contact and non-contact measurements can be used either in combination (to verify the heat source value, its distribution over the material surface and heat exchange conditions for specimen and environment) or separately (as an express method to evaluate material conditions at different stages of loading).

**PNRPU Mechanics Bulletin**. 2016;(1):68-81

Chaotic dynamics of flexible rectangular panels in white noise field

#### Abstract

The paper studies the impact of a flexible rectangular panel intensity of the external white noise field (which is normal to the panel surface) on the nature of oscillations. A mathematical model of panel oscillations is based on Kirchhoff hypotheses with dissipation. The geometric nonlinearity is taken into account in the form of Karman. We consider a rectangular panel with an aspect ratio in plan, by an external longitudinal load. Equations of motion are joined by inhomogeneous boundary conditions of bearing on flexible incompressible (inextensible) edges and zero initial conditions. The resulting system of nonlinear differential equations in private derivatives is reduced to a nonlinear system of ordinary differential equations by the method of finite differences in spatial variables. As for the time, the system is solved by the Runge-Kutta fourth-order accuracy. The number of degrees of freedom of the mechanical system in the experiment equals to 196. In order to analyze the results obtained in the work, in addition to the Fourier analysis, the authors applied the wavelet transformation unit, which allows a more detailed study of the local time signal features. The experiment has revealed a range of amplitudes outer longitudinal load, where the behavior of the dynamical system is not sustainable. For this range of amplitudes of longitudinal load we studied the influence of the white noise field with varying intensity on the nature of the panel vibrations. The numerical experiments show that the white noise field is capable of reducing the amplitude of the oscillation panel, reducing the number of frequencies in the vibration spectrum of the system and transfering to the asymmetric waveform symmetry. So, it is possible to state that effecting the dynamic system with noise field can lead to safer vibrational modes. That is, using the white noise can be controlled by the nature of vibrations of the mechanical system.

**PNRPU Mechanics Bulletin**. 2016;(1):82-92

A mathematical modeling and experimental study of forming and relaxation of the residual stresses in plane samples made of EP742 alloy after the ultrasonic hardening under the high-temperature creep conditions

#### Abstract

The authors carried out a complex (computational and experimental) research of residual stresses in prismatic samples made of EP742 alloy after ultrasonic hardening and he temperature of 650 °С for 100 hours unloaded. The laws of residual stresses distribution over the thickness of the surface-hardened layer were discovered. The experiments proved that the ultrasonic hardening of the sample caused compression residual stresses in the surface layer. The maximum stresses were observed in the subsurface layer and they decreased on reaching the surface. When the temperature was exposed, the induced compression residual stresses became relaxed, the level of residual stresses decreased by 1.4-1.6 times and its maximum was displaced deep into the sample, but the thickness of the compressed layer was kept to about 200 micron. We developed the mathematical model of residual stresses forming in prismatic samples after surface plastic deformation and relaxation under high-temperature creep conditions. We used the hardened half-space as a model object for the prismatic sample, because the hardened layer was rather thin. We introduced Cartesian coordinate system, where x 0 y plane coincided with the half-space hardened surface, and 0 z axis was directed into the depth of the hardened layer.

**PNRPU Mechanics Bulletin**. 2016;(1):93-112

On strain hardening of cast iron with proportional and complex loading

#### Abstract

The behavior of thin-walled tubular specimens of cast iron SCH 15-32 are modeled using loaded tension with internal pressure. The experimental data for this iron has been obtained by V.M. Zhigalkin and O.M. Usoltseva. This material is orthotropic according to the experimental diagrams of the axial and circumferential strains. Proportional and complex loadings are considered. Biaxial tension which has been carried out in the experiment corresponded to pure shear (characterized by the parameter of Lode-Nadai), where axial or circumferential stress prevailed. The trajectory of complex loading is given as a two-tier sloping in stress space; the first link path is consistent with axial tension, the second one is consistent with the circumferential tensile either under constant axial strain, or at a certain ratio of the increment of axial tension to the circumferential stress. In the latter case it produces an intermediate full unloading, then repeated complex loading at a high value of the first link path. Yield points are defined according to the initial elastic anisotropy, which is detected at a sufficiently small limit for residual maximum principal strain. Strain strengthening is described based on the concept of slipping and loosening. It is shown that the mechanism of flat plastic deformation is realized in the form of an idealized slipping on the plane of main shear stresses due to the initial anisotropy of materials. Priority inclusion in the work of these planes is characterized by the level and kind of stress. Single dependence on the principal stresses is proposed for maximum principal purely plastic component inelastic deformation (independent of the type of stress state). This dependence is a monotonically increasing function. It describes the change planes of slipping when changing the state of stress. The deformation loosening is considered occurring simultaneously and uniformly in all directions in accordance with V.V.Novozhilov’s hypothesis, which is associated with the plastic deformation. The coefficient of loosening is defined by comparing the longitudinal and lateral strain in uniaxial tension. In all the cases of tension beyond the elasticity, the lateral strain is remained close to its corresponding elastic value according to the interaction of local slipping and loosening. The revealed features of cast iron strain hardening are reflected with the proposed defining relations in the form of uniform relationships between the finite values of the stress and strain both under proportional and complex loadings.

**PNRPU Mechanics Bulletin**. 2016;(1):113-128

Fractal analysis of deformation curves of fiber-reinforced fine-grained concretes under compression

#### Abstract

The authors set out the methodology of determining the fractal dimension of deformation curves based on the minimum coverage method, allowing to obtain an integral quantitative estimation of destruction process of building composites under compression; and determine the position of the parametric point of destruction curve. The proposed method was compared with the algorithms of the Hurst exponent determination and fractal dimension by the coverage squares method. It became possible to show the advantage of the method based on the determination of the fractal dimension by the minimum coverage method. For the realization of mechanical tests of compositions of fine-grained fiber-reinforced concretes we used hardware-software complex WilleGeotechnik®, which was additionally equipped by a climatic chamber with the possibility of adjusting the temperature (from - 40 to + 100 °C) and humidity (10 to 96 %) in the process of loading. The change of stresses and deformations of samples in the process of load was fixed with a step of 0.01 sec. The following components were used as basic ones in fiber-reinforced fine-grained concretes: cement of CEM I 42,5R class, river sand, densified condensed microsilica DCM-85, polycarboxylate superplasticizer Melflux 1641 F. The dispersed reinforcement of concretes was provided by a separate injection of three types of fibers: polypropylene multifilament fiber, polyacrylonitrile synthetic fiber FibARM Fiber WВ and basalt microfiber “Astroflex-MBM” modified by astralene. As a result, we defined values of fractality index and fractal dimension of the growth stress and strain curves of deformation of fine-grained concrete by the minimal cover method. Based on fractal analysis of time series, the position and neighborhood of the point of transition of the concrete sample from the resting state to the state of the pronounced trend were determined. The position change of parametric point was detected using the dependence of types in the applied fibers. It was established that the introduction of 1 % polypropylene multifilament fiber or 5 % modified astralene basalt microfiber “Astroflex-MBM” leads to a significant increase of the first “critical” level, respectively, to 54 and 47 % as in the analysis of growth stresses and deformations compared with 19 and 28 % for the formulations containing 1,5 % polyacrylonitrile synthetic fiber. The suggested method of fractal analysis of deformation curves based on the method of minimal coverage allows obtaining valuable information about the process related to destruction of composite materials of different natures.

**PNRPU Mechanics Bulletin**. 2016;(1):129-146

Finite-element modelling of thermal and deformational processes in manufacturing high-strength wire

#### Abstract

The paper considers simulation of thermal strain processes while high-strength wire manufacturing. Thermal strain processing (TSP) technique was studied. Processes model consisting of wire high-speed heating up to temperatures exceeding Ас3, homogenizing soaking, workpiece straining, workpiece transfer to cooling zone under controlled and stabilized temperature providing polygonised structure formation. Simulation of thermal strain induction heating, wire drawing and reduction by running in three-roll running head was performed on the basis of conceptual functional model. Induction heating thermal field is not homogeneous within the section and depends on wire diameter, inductor transfer speed and induction heating frequency. Wire drawing shows complex state of stress, the process is axisymmetric, equivalent plastic strains are slightly different from rated strains; possible tear location on leaving the die is shown. Running equivalent plastic strains are considerably different from the rate one in terms of reduction due to tangential component. Three-roll running head application may result in higher material strengthening than it has beenpredicted during technique development on the account of pre-assigned degree of strain.

**PNRPU Mechanics Bulletin**. 2016;(1):147-165