Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics最新文献

{"title":"Differential equations of continuum equilibrium for plane deformation in cartesian axials at biquadratic approximation of closing equations","authors":"S. V. Bakushev","doi":"10.17223/19988621/76/6","DOIUrl":"https://doi.org/10.17223/19988621/76/6","url":null,"abstract":"The subject under analysis is construction of differential equations of equilibrium in displacements for plane deformation of physically and geometrically nonlinear continuous media when the closing equations are biquadratically approximated in a Cartesian rectangular coordinate system. Proceeding from the assumption that, generally speaking, the diagrams of volume and shear deformation are independent from each other, six main cases of physical dependences are considered, depending on the relative position of the break points of biquadratic diagrams of volume and shear deformation. Construction of physical dependencies is based on the calculation of the secant module of volume and shear deformation. When approximating the graphs of volume and shear deformation diagrams using two segments of parabolas, the secant shear modulus in the first segment is a linear function of the intensity of shear deformations; the secant modulus of volume expansion-contraction is a linear function of the first invariant of the strain tensor. In the second section of the diagrams of both volume and shear deformation, the secant shear modulus is a fractional (rational) function of the intensity of shear deformations; the secant modulus of volume expansion-contraction is a fractional (rational) function of the first invariant of the strain tensor. The obtained differential equations of equilibrium in displacements can be applied in determining the stress-strain state of physically and geometrically nonlinear continuous media under plane deformation the closing equations of physical relations for which are approximated by biquadratic functions.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":"30 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74297487","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Analysis of a dynamic structure of secular resonances in circumlunar orbital space","authors":"N. Popandopulo, A. Aleksandrova, T. Bordovitsyna","doi":"10.17223/19988621/77/9","DOIUrl":"https://doi.org/10.17223/19988621/77/9","url":null,"abstract":"When developing new circumlunar satellite systems for various purposes, it is necessary to know the dynamic features of various regions of circumlunar space. It is noted that one of the main features of the dynamics of artificial Moon satellites (AMS) is the increase in the eccentricity of their orbits. It is known that such orbital evolution of objects can be caused by resonant perturbations. In this regard, the purpose of this paper is to study the influence of secular and half-secular resonances occurring on the AMS. The results are obtained by means of the processing of data from an extensive numerical experiment to study the orbital evolution of the uniformly distributed circumlunar objects moving in the range of large semi-axes from 1911.8 to 26070 km and with inclinations from 0 to 180°. The study of the influence of secular and half-secular resonances on the orbital evolution of the AMS is carried out using numerical and analytical techniques. Both techniques have advantages and disadvantages and are used in the study. Maps of secular and half-secular resonances are of great practical importance. They clearly demonstrate the areas of influence of secular resonances on the orbital evolution of circumlunar objects depending on the initial values of the semimajor axis and inclination of the orbits of objects. The analysis of each particular resonance consists in the revealing of a resonant characteristic type, which is determined by the nature of the change in a critical argument: libration - stable resonance, libration/circulation - unstable resonance, and circulation - no resonance. To estimate the whole range of the influence of resonances on the AMS dynamics, maps of their overlaps are built. The maps allow one to determine areas with significant influence of the resonances on the orbital evolution of circumlunar objects. The obtained results show that the reason for a large increase in the eccentricity in a vast area of circumlunar orbital space is the presence of the stable apsidal resonance of the Kozai-Lidov type.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":"121 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75485326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Formal derivation of mechanical motion magnitudes","authors":"V. Pavlov","doi":"10.17223/19988621/78/11","DOIUrl":"https://doi.org/10.17223/19988621/78/11","url":null,"abstract":"Quantum-mechanical differential equations are considered, which are formal analogues of the Schrödinger equation. Their differences from each other and from the Schrödinger equation lie in the orders of partial derivatives. A characteristic feature of these equations is the presence of dimensional coefficients, which are the product of integer powers of mass and velocity, which allows us to consider them as quantities of mechanical motion. The logical regularity of the formation of these values is established. The applied nature of two of them - the integral Umov vector for kinetic energy and backward momentum - is considered.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":"30 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73198662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A new processing algorithm for photoelasticity method data","authors":"Aleksey V. Likhachev, M. Tabanyukhova","doi":"10.17223/19988621/79/9","DOIUrl":"https://doi.org/10.17223/19988621/79/9","url":null,"abstract":"The photoelasticity method is a reliable tool for studying the stress state of flat elements in building structures using the models made of optically sensitive materials. In this paper, the classical photoelasticity is considered. The experimental data obtained with the use of the method are presented as interferograms. A decoding procedure implies the obtaining of some normal and tangential stress values in the plane of the model. The polarization-projection installations that are used in optical methods are rather simple. However, the digital processing of the images obtained during the loaded model transmission requires high-intelligent software. Nowadays, national and international laboratories, working with polarization-optical methods, strive to develop digital photoelasticity. For some reasons, the authors of the presented work needed to develop their own algorithms for decoding experimental data of the photoelasticity method. This work is mainly devoted to a formulation of the problems to be solved. Some of them have already been solved, and the results obtained are presented here. The authors place special emphasis on the description of the algorithm for tracing of interference fringes based on the analysis of the image gradient.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79256429","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Prediction of durability of soldered contact joints of chips","authors":"A. Azin, Nikolay Maritsky, S. Ponomarev, Sergey V. Rikkonen","doi":"10.17223/19988621/76/4","DOIUrl":"https://doi.org/10.17223/19988621/76/4","url":null,"abstract":"The study is aimed at developing a nondestructive testing method for electronic equipment and its components. This method allows one to identify critical design defects of printed circuit boards (PCB) and to predict their service life taking into account the nature of probable operating loads. The study uses an acoustic emission method to identify and localize critical design defects of printed circuit boards. Geometric dimensions of detected critical defects can be determined by X-ray tomography. Based on the results of the study, a method combining acoustic emission and X-ray tomography has been developed for nondestructive testing of printed circuit boards. The stress-strain state of solder joints containing detected defects is analyzed. Durability is predicted using the damage function of the material, experimental fatigue curve with allowance for rheological properties of materials, the temperature effects, and complex stress-strain state. The results of using the developed method for estimating the degree of damage of the electronic board have been verified based on the experimental results of studies carried out in accordance with IPC-9701. The prediction error does not exceed 5%. Contribution of the authors: the authors contributed equally to this article. The authors declare no conflicts of interests.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":"54 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86952176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Simulation of the stress state in barriers made of anisotropic materials","authors":"M. Krivosheina","doi":"10.17223/19988621/79/8","DOIUrl":"https://doi.org/10.17223/19988621/79/8","url":null,"abstract":"To study the properties of anisotropic materials, a mathematical model is proposed that accounts for the anisotropy of elastic and plastic properties, as well as the anisotropy of “thermal” and “cold” components of pressure. The model is applied in a threedimensional simulation of the deformation of an HCP-single-crystal barrier under impact loading by an aluminum impactor. The numerical simulation results are obtained using the dynamic finite element method with a difference scheme modified to account for the anisotropy of “cold” and “thermal” pressure components. To simulate the anisotropy of the stress deviator in the region of elastic deformations, generalized Hooke's law is used, while in the region of plastic deformations, the Mises-Hill plasticity function (Hill48) is used with account for the anisotropy of elastic properties and anisotropy of the Gruneisen coefficient. The experimentally and numerically obtained velocity profiles of the back surfaces of single-crystal zinc barriers during the spall fracture are compared with each other. When the impact loading direction coincides with [0001] axis, the elastic precursor is not observed on the velocity profile calculated numerically, which is the same for the one derived experimentally. This effect may be explained only with the use of anisotropic pressur.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":"70 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79183305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Investigation of acoustic characteristics of a single supersonic jet flowing into a flooded space","authors":"Adil A. Askerov, A. V. Chervakova, K. Kostyushin","doi":"10.17223/19988621/78/4","DOIUrl":"https://doi.org/10.17223/19988621/78/4","url":null,"abstract":"In this paper, the acoustic characteristics of a single supersonic jet flowing from a nozzle of a rocket engine into a flooded space at different pressure ratios are studied. A system of the Favre-averaged Navier-Stokes equations is used to describe the unsteady flow of a viscous compressible heat-conducting gas in a supersonic Laval nozzle and an outflowing jet. The system is enclosed by the ideal gas law. The implementation of the physical and mathematical model and the numerical studies are carried out using the OpenFOAM Extended open platform based on the modified dbnsTurbFoam solver. A conical nozzle with an opening angle of 45° at the Mach number of 3 at the nozzle exit is considered in this study. Air is used as the working gas. Amplitude-frequency spectra of acoustic radiation at the point located at a distance from the nozzle outlet are obtained at different pressure ratios of the outflowing supersonic jet. Analysis of the amplitude-frequency characteristics of the jet under study shows that the maxima occur mainly at low frequencies. The maximum oscillation amplitude for the considered jet configurations is revealed at a pressure ratio of 1 on a frequency of 787 Hz. The maximum sound pressure level is 149 dB.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":"35 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89333075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Determination of the elastic fields induced by body forces in transtropic bodies of revolution","authors":"D. A. Ivanychev","doi":"10.17223/19988621/77/7","DOIUrl":"https://doi.org/10.17223/19988621/77/7","url":null,"abstract":"The paper presents a method for determining the stress-strain state of transversely isotropic bodies of revolution under the action of non-axisymmetric stationary body forces. This problem solution involves the use of boundary state method definitions. The basis of the space of internal states is formed using the fundamental polynomials. The polynomial is placed in any position of a displacement vector of the plane auxiliary state; the spatial state is determined by transition formulas. A set of such states forms a finite-dimensional basis, in which after orthogonalization, the desired state is expanded into Fourier series with the same coefficients. The series coefficients are scalar products of the vectors of given and basic body forces. Finally, the determination of the elastic state is reduced to solving quadratures. The solutions to problems of elasticity theory for a transversely isotropic circular cylinder are analyzed in terms of the action of body forces given by various cyclic laws (sine and cosine). Recommendations are given for constructing the basis of internal states depending on the type of the function of the given body forces. The analysis of the series convergence and the estimation of the solution accuracy are given in a graphical form.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75244667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Simulation of gas aerodynamics and particle trajectories in the interaction of two opposing swirling flows","authors":"R. R. Turubaev, Aleksander V. Shvab","doi":"10.17223/19988621/75/12","DOIUrl":"https://doi.org/10.17223/19988621/75/12","url":null,"abstract":"The paper presents an original geometry of a vortex chamber in which aerodynamics of a gas flow is simulated during the interaction of two opposing swirling flows in order to equalize the centrifugal forces in the central region of the apparatus as applied to the tasks of separating powders by fractional composition or intensifying heat and mass transfer in chemical reactors. Based on the results obtained, the distributions of the trajectories of motion of single particles are determined, based on the discrete-trajectory approach, and the reliability of the results obtained for the aerodynamics of swirling flow in the proposed geometry of the vortex chamber was shown. As a result of mathematical modeling of the dynamics of motion of a single heavy particle in a swirling flow, the influence of swirl on the quality of the particle classification process is shown; for example, an increase in swirl leads to a greater influence of the centrifugal force which picks up heavy particles and throws them to the peripheral sector. In addition, an increase in the centrifugal force leads to a displacement of the boundary particle size. However, since this work presents a laminar formulation of the problem, the difference will amount to tens of microns; when updating to the turbulent problem, the difference will already be calculated in units of microns.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":"26 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83726657","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Features of an experimental study of the stable and unstable displacement in the Hele-Shaw cell filled with glass balls","authors":"A. Rakhimov, A. A. Valiev","doi":"10.17223/19988621/77/11","DOIUrl":"https://doi.org/10.17223/19988621/77/11","url":null,"abstract":"The paper presents the results of an experimental study of immiscible displacements: displacement of water by oil (stable) and displacement of oil by air or water (unstable) at constant pressure drop in the Hele-Shaw cell filled with glass balls, which is a physical model of a porous fracture in an oil reservoir. The presented data allows one to draw a logical conclusion: the higher the inlet pressure, the higher the displacement efficiency. With the same oil displacement time in the Hele-Shaw cell, the passed volume of the gas significantly exceeds that of the water. The displacement by water is more complete, though. Displacements are highly dependent on inhomogeneity of the model skeleton, most significantly for the flow at low pressure drop. At the edges of the model, where the volume of the dispersed medium is greater, the motion is faster. When displacement in the model with balls is stable, the flow spreads out towards the outlet and takes a \"funnel\" shape that gradually expands; the motion is directed through the channel passed. For unstable displacement at low pressure drop, the motion occurs along the edges of the cell; with an increase in the pressure drop, the motion becomes more uniform, and the \" viscous fingers\" begin to \"branch\" and grow even at a distance from the outlet.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":"495 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83775449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}