Pub Date : 2023-12-21DOI: 10.3103/S0027133023050023
A. A. Bobylev
A boundary value problem is considered in a functionally graded elastic strip. A three-term asymptotic expansion of a transfer function is obtained for the Poincaré–Steklov operator that maps normal stresses to normal displacements on a part of the strip boundary. Padé approximations are determined for the obtained asymptotic series. An approach to computing the transfer function using the asymptotic series and the Padé approximations is proposed, which reduces computational costs.
{"title":"Computing a Transfer Function of the Poincaré–Steklov Operator for a Functionally Graded Elastic Strip","authors":"A. A. Bobylev","doi":"10.3103/S0027133023050023","DOIUrl":"10.3103/S0027133023050023","url":null,"abstract":"<p>A boundary value problem is considered in a functionally graded\u0000elastic strip. A three-term asymptotic expansion of a transfer\u0000function is obtained for the Poincaré–Steklov operator that\u0000maps normal stresses to normal displacements on a part of the\u0000strip boundary. Padé approximations are determined for the\u0000obtained asymptotic series. An approach to computing the transfer\u0000function using the asymptotic series and the Padé approximations\u0000is proposed, which reduces computational costs.</p>","PeriodicalId":710,"journal":{"name":"Moscow University Mechanics Bulletin","volume":"78 5","pages":"134 - 142"},"PeriodicalIF":0.3,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139027067","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}
Pub Date : 2023-11-13DOI: 10.3103/S0027133023040027
V. A. Banko, D. V. Georgievskii
The initial-boundary value problems of acceleration from a state of rest of a two-constant viscoplastic medium (Bingham body) in a half-plane is investigated when the tangential stress is given at the boundary as a piecewise continuous monotonically nondecreasing function of time. As an additional condition at an unknown interface between a flow zone that increases with time in thickness and a stationary semi-infinite rigid zone, the requirement is chosen that the solution of this problem with a tendency to zero of the yield strength of the material at each point and at each moment of time tends to the solution of the corresponding viscous flow problem known as the generalized vortex layer diffusion problem. The exact analytical solutions are found for tangential stress and velocity profiles in nonstationary one-dimensional flow. The cases of self-similarity and so-called quasi-self-similarity are distinguished. The nature of the tendency at (ttoinfty) of the thickness of the layer, in which the shear is realized, to infinity is of particular interest.
{"title":"Quasi-Self-Similar Solutions to Some Parabolic Problems in the Theory of Viscoplastic Flow","authors":"V. A. Banko, D. V. Georgievskii","doi":"10.3103/S0027133023040027","DOIUrl":"10.3103/S0027133023040027","url":null,"abstract":"<p>The initial-boundary value problems of acceleration from a state of rest of a two-constant viscoplastic medium (Bingham body) in a half-plane is investigated when the tangential stress is given at the boundary as a piecewise continuous monotonically nondecreasing function of time. As an additional condition at an unknown interface between a flow zone that increases with time in thickness and a stationary semi-infinite rigid zone, the requirement is chosen that the solution of this problem with a tendency to zero of the yield strength of the material at each point and at each moment of time tends to the solution of the corresponding viscous flow problem known as the generalized vortex layer diffusion problem. The exact analytical solutions are found for tangential stress and velocity profiles in nonstationary one-dimensional flow. The cases of self-similarity and so-called quasi-self-similarity are distinguished. The nature of the tendency at <span>(ttoinfty)</span> of the thickness of the layer, in which the shear is realized, to infinity is of particular interest.</p>","PeriodicalId":710,"journal":{"name":"Moscow University Mechanics Bulletin","volume":"78 4","pages":"102 - 109"},"PeriodicalIF":0.3,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134796259","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}
Pub Date : 2023-11-13DOI: 10.3103/S0027133023040040
S. A. Lurie, P. A. Belov
We consider the classical problem of elasticity theory concerning the conditions of strain compatibility, which ensure the determination of a continuous field of displacements of an elastic body by the strain field. We construct generalized Cesàro representations that allow defining the displacement field through integrodifferential operators on the components of the strain tensor deviator with an accuracy up to quadratic polynomials. It has been established that the quadratures both for the pseudovector of local rotations and for the bulk strain are completely determined by the strain deviator field. We present the conditions for the existence of the listed quadratures, which are written in the form of five third differential order compatibility equations for the five components of the strain deviator tensor.
{"title":"Generalized Cesàro Formulas and Third-Order Compatibility Equations","authors":"S. A. Lurie, P. A. Belov","doi":"10.3103/S0027133023040040","DOIUrl":"10.3103/S0027133023040040","url":null,"abstract":"<p>We consider the classical problem of elasticity theory concerning the conditions of strain compatibility, which ensure the determination of a continuous field of displacements of an elastic body by the strain field. We construct generalized Cesàro representations that allow defining the displacement field through integrodifferential operators on the components of the strain tensor deviator with an accuracy up to quadratic polynomials. It has been established that the quadratures both for the pseudovector of local rotations and for the bulk strain are completely determined by the strain deviator field. We present the conditions for the existence of the listed quadratures, which are written in the form of five third differential order compatibility equations for the five components of the strain deviator tensor.</p>","PeriodicalId":710,"journal":{"name":"Moscow University Mechanics Bulletin","volume":"78 4","pages":"110 - 113"},"PeriodicalIF":0.3,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134796260","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}
Pub Date : 2023-11-13DOI: 10.3103/S0027133023040052
A. V. Romanov
In this paper, a variational principle of Lagrange, the Ritz method, and piecewise polynomial serendipity shape functions are used to obtain a stiffness matrix and a system of linear algebraic equations in the micropolar theory of elasticity for anisotropic, isotropic, and centrally symmetric material in case of a nonisothermal process.
{"title":"On the Variational Principle of Lagrange in the Micropolar Theory of Elasticity at Nonisothermal Processes","authors":"A. V. Romanov","doi":"10.3103/S0027133023040052","DOIUrl":"10.3103/S0027133023040052","url":null,"abstract":"<p>In this paper, a variational principle of Lagrange, the Ritz\u0000method, and piecewise polynomial serendipity shape functions are\u0000used to obtain a stiffness matrix and a system of linear algebraic\u0000equations in the micropolar theory of elasticity for anisotropic,\u0000isotropic, and centrally symmetric material in case of a\u0000nonisothermal process.</p>","PeriodicalId":710,"journal":{"name":"Moscow University Mechanics Bulletin","volume":"78 4","pages":"114 - 118"},"PeriodicalIF":0.3,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134878323","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}
Pub Date : 2023-11-13DOI: 10.3103/S0027133023040039
A. V. Khokhlov
We continue the systematic analytical study of a nonlinear Maxwell-type constitutive equation for shear flow for thixotropic viscoelastic media accounting for interaction of deformation process and structure evolution, namely, the influence of the kinetics formation and breakage of chain cross-links, agglomerations of molecules and crystallites on viscosity and shear modulus and deformation influence on the kinetics. We formulated it in the previous article and reduced it to the set of two nonlinear autonomous differential equations for two unknown functions (namely, the stress and relative cross-links density). We examine the phase portrait of the system for arbitrary (increasing) material function and six (positive) material parameters governing the model and prove that the (unique) equilibrium point is stable and the only three cases are realized: the equilibrium point is either a stable sink, or a degenerated stable sink, or a stable spiral sink. We found criteria for every case in the form of explicit restrictions on the material function and parameters and shear rate.
{"title":"Equilibrium Point and Phase Portrait of a Model for Flow of Tixotropic Media Accounting for Structure Evolution","authors":"A. V. Khokhlov","doi":"10.3103/S0027133023040039","DOIUrl":"10.3103/S0027133023040039","url":null,"abstract":"<p>We continue the systematic analytical study of a nonlinear Maxwell-type constitutive equation for shear flow for thixotropic viscoelastic media accounting for interaction of deformation process and structure evolution, namely, the influence of the kinetics formation and breakage of chain cross-links, agglomerations of molecules and crystallites on viscosity and shear modulus and deformation influence on the kinetics. We formulated it in the previous article and reduced it to the set of two nonlinear autonomous differential equations for two unknown functions (namely, the stress and relative cross-links density). We examine the phase portrait of the system for arbitrary (increasing) material function and six (positive) material parameters governing the model and prove that the (unique) equilibrium point is stable and the only three cases are realized: the equilibrium point is either a stable sink, or a degenerated stable sink, or a stable spiral sink. We found criteria for every case in the form of explicit restrictions on the material function and parameters and shear rate.</p>","PeriodicalId":710,"journal":{"name":"Moscow University Mechanics Bulletin","volume":"78 4","pages":"91 - 101"},"PeriodicalIF":0.3,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134796258","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}
Pub Date : 2023-08-15DOI: 10.3103/S0027133023030044
E. B. Zavoychinskaya, A. S. Plotnikov
A numerical-analytical method for three-axial inhomogeneous elastic residual stress determination based on the data of the displacement optical measurement during the incremental hole drilling method is presented. The constitutive relations for the displacements as the three variable functions (in plane of the hole and along its depth) are represented by the Volterra integral operators. A method for finding the basic functions is given. The stress tensor components recovered by the proposed method are in good agreement with the well-known solution of a problem where the residual stresses are formed by bending an elastic-perfectly plastic beam.
{"title":"On the Method for Identifying Inhomogeneous Fields of Residual Stresses","authors":"E. B. Zavoychinskaya, A. S. Plotnikov","doi":"10.3103/S0027133023030044","DOIUrl":"10.3103/S0027133023030044","url":null,"abstract":"<p>A numerical-analytical method for three-axial inhomogeneous elastic residual stress determination based on the data of the displacement optical measurement during the incremental hole drilling method is presented. The constitutive relations for the displacements as the three variable functions (in plane of the hole and along its depth) are represented by the Volterra integral operators. A method for finding the basic functions is given. The stress tensor components recovered by the proposed method are in good agreement with the well-known solution of a problem where the residual stresses are formed by bending an elastic-perfectly plastic beam.</p>","PeriodicalId":710,"journal":{"name":"Moscow University Mechanics Bulletin","volume":"78 3","pages":"63 - 70"},"PeriodicalIF":0.3,"publicationDate":"2023-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4599425","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}
Pub Date : 2023-08-15DOI: 10.3103/S0027133023030032
I. N. Molodtsov
A new class of constitutive equations for complex loading processes is obtained. It has three state functionals. A new method of mathematical modeling and mathematical principle is formulated. According to them, physically correct equations of state are changed by introducing gyroscopic terms that do not perform mechanical work. The constitutive equations of complex loading processes with two state functionals under conditions of force and kinematic loading are constructed. Their connection with the Il’yushin formula and modern theories of plastic flow is obtained. A mathematical model of the domain mechanism of plasticity is formulated. It represents a real deformable continuum as a mixture of an elastoplastic continuum and a Cosserat continuum—flat veinlets (domains of plastic deformation that are the zones of large relative rotations). A physical justification for the inclusion of the asymmetric part of the stress and rotation tensor in the composition of the thermodynamic parameters of the model is given.
{"title":"Constitutive Equations with Dissipative Stresses","authors":"I. N. Molodtsov","doi":"10.3103/S0027133023030032","DOIUrl":"10.3103/S0027133023030032","url":null,"abstract":"<p>A new class of constitutive equations for complex loading processes is obtained. It has three state functionals. A new method of mathematical modeling and mathematical principle is formulated. According to them, physically correct equations of state are changed by introducing gyroscopic terms that do not perform mechanical work. The constitutive equations of complex loading processes with two state functionals under conditions of force and kinematic loading are constructed. Their connection with the Il’yushin formula and modern theories of plastic flow is obtained. A mathematical model of the domain mechanism of plasticity is formulated. It represents a real deformable continuum as a mixture of an elastoplastic continuum and a Cosserat continuum—flat veinlets (domains of plastic deformation that are the zones of large relative rotations). A physical justification for the inclusion of the asymmetric part of the stress and rotation tensor in the composition of the thermodynamic parameters of the model is given.</p>","PeriodicalId":710,"journal":{"name":"Moscow University Mechanics Bulletin","volume":"78 3","pages":"71 - 79"},"PeriodicalIF":0.3,"publicationDate":"2023-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4598392","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}
Pub Date : 2023-08-15DOI: 10.3103/S0027133023030020
Yu. V. Bolotin, A. V. Bragin
A pedestrian navigation system consisting of two foot-mounted strapdown inertial navigation systems (SINS)s is considered. The zero velocity conditions of the foot in the stance phase of a step and a limited distance between the feet are used for SINS corrections. The aim of the work is to study some consistency properties of the extended Kalman filter. It is shown that this consistency depends on a form of the error equations.
{"title":"On Some Properties of the Kalman Filter in the Pedestrian Navigation Problem","authors":"Yu. V. Bolotin, A. V. Bragin","doi":"10.3103/S0027133023030020","DOIUrl":"10.3103/S0027133023030020","url":null,"abstract":"<p>A pedestrian navigation system consisting of two foot-mounted strapdown inertial navigation systems (SINS)s is considered. The zero velocity conditions of the foot in the stance phase of a step and a limited distance between the feet are used for SINS corrections. The aim of the work is to study some consistency properties of the extended Kalman filter. It is shown that this consistency depends on a form of the error equations.</p>","PeriodicalId":710,"journal":{"name":"Moscow University Mechanics Bulletin","volume":"78 3","pages":"80 - 85"},"PeriodicalIF":0.3,"publicationDate":"2023-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4595688","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}
Pub Date : 2023-08-15DOI: 10.3103/S0027133023030056
A. V. Zvyagin, D. D. Novov
The paper is devoted to the development of the displacement discontinuity method for plane problems of fracture mechanics in consideration of the curvature of crack lines. In this paper, some new representations of biharmonic functions are found. This is necessary to obtain the analytical solutions of problems for an elastic plane weakened by a crack in the form of a circular arc. A numerical method is proposed on the basis of these analytical solutions. The numerical values of the stress intensity factor are compared with its known analytical value.
{"title":"Displacement Discontinuity Method Taking into Account the Curvature of the Crack","authors":"A. V. Zvyagin, D. D. Novov","doi":"10.3103/S0027133023030056","DOIUrl":"10.3103/S0027133023030056","url":null,"abstract":"<p>The paper is devoted to the development of the displacement discontinuity method for plane problems of fracture mechanics in consideration of the curvature of crack lines. In this paper, some new representations of biharmonic functions are found. This is necessary to obtain the analytical solutions of problems for an elastic plane weakened by a crack in the form of a circular arc. A numerical method is proposed on the basis of these analytical solutions. The numerical values of the stress intensity factor are compared with its known analytical value.</p>","PeriodicalId":710,"journal":{"name":"Moscow University Mechanics Bulletin","volume":"78 3","pages":"86 - 90"},"PeriodicalIF":0.3,"publicationDate":"2023-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4600469","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}
Pub Date : 2023-06-22DOI: 10.3103/S0027133023020024
S. D. Algazin, A. A. Sinitsyn
Free oscillations of a conical shell of finite length are considered. This problem was formulated in the 1960s. A modern algorithm without saturation is given in this paper, and specific calculations which show its high efficiency are discussed.
{"title":"Free Oscillations of a Conical Shell","authors":"S. D. Algazin, A. A. Sinitsyn","doi":"10.3103/S0027133023020024","DOIUrl":"10.3103/S0027133023020024","url":null,"abstract":"<p>Free oscillations of a conical shell of finite length are considered. This problem was formulated in the 1960s. A modern algorithm without saturation is given in this paper, and specific calculations which show its high efficiency are discussed.</p>","PeriodicalId":710,"journal":{"name":"Moscow University Mechanics Bulletin","volume":"78 2","pages":"29 - 35"},"PeriodicalIF":0.3,"publicationDate":"2023-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4863232","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}