Rubric: "1.1.8. Mechanics of a Deformable Solid Body (physical and mathematical sciences)"
Dimitrienko Y. I. (Bauman Moscow State Technical University), Sborschikov S. V. (Bauman Moscow State Technical University), Dimitrienko A. Y. (Lomonosov Moscow State University), Yurin Y. V. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2022-1-1541
Within the framework of the deformation theory of plasticity under active loading, a model of constitutive relations for elastic-plastic composites belonging to the class of transversally isotropic materials is proposed. The theory of spectral expansions of stress and strain tensors and the spectral representation of nonlinear tensor functions for transversely isotropic media are used to develop a nonlinear constitutive relations. Specific models of plasticity functions are proposed, depending on the spectral invariants of the strain tensor. To determine the model constants, a method is proposed in which these constants are calculated based on the approximation of deformation curves obtained by direct numerical solution of three-dimensional problems on the periodicity cell of elastic-plastic composites. Problems on the periodicity cell are formulated using the method of asymptotic averaging of periodic media. The numerical solution of problems on the periodicity cell is carried out using the finite element method within the framework of software developed at the Scientific and Educational Center "Supercomputer Engineering Modeling and Development of Software Systems" of Bauman Moscow State Technical University. An example of numerical calculation of the constants of a composite model using the proposed method for a unidirectionally reinforced composite based on carbon fibers and an aluminum alloy matrix is given. Examples of verification of the proposed model for different loading trajectories of the composite in a 6-dimensional stress space are given. It is shown that the proposed microstructural model and the algorithm for determining its constants provide a sufficiently high accuracy in predicting the elastic-plastic deformation of transversely isotropic composites
Димитриенко Ю.И., Сборщиков С.В., Димитриенко А.Ю., Юрин Ю.В. Микроструктурная модель деформационной теории пластичности трансверсально-изотропных композитов. Математическое моделирование и численные методы, 2022, № 1, с. 15–41.
Dimitrienko Y. I. (Bauman Moscow State Technical University), Sborschikov S. V. (Bauman Moscow State Technical University), Dimitrienko A. Y. (Lomonosov Moscow State University), Yurin Y. V. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2021-4-1744
A model of constitutive relations for elastic-plastic composites with cubic symmetry of properties is proposed. This class includes a significant number of composite materials: dispersed-reinforced composites, which have an ordered rather than a chaotic reinforcement system, as well as some types of spatially reinforced composites. To build a model of nonlinear constitutive relations, a tensor-symmetry approach was used, based on the spectral expansions of stress and strain tensors, as well as the spectral representation of nonlinear tensor relations between these tensors. The deformation theory of plasticity is considered, for which the tensor-symmetric approach is used, and specific models are proposed for plasticity functions that depend on the spectral invariants of the strain tensor. To determine the model constants, a method is proposed in which these constants are calculated based on the approximation of deformation curves obtained by direct numerical solution of three-dimensional problems on the periodicity cell of elastic-plastic composites. These problems arise in the method of asymptotic averaging of periodic media. To solve problems on a periodicity cell, a finite element method and special software was used that implements solutions to problems on periodicity cells, developed at the Scientific and Educational Center for Supercomputer Engineering Modeling and Development of Software Packages of Bauman Moscow State Technical University. An example of calculating the constants of a composite model using the proposed method for a dispersed-reinforced composite based on a metal matrix is considered. Also, the verification of the proposed model for various ways of multiaxial loading of the composite was carried out with direct numerical simulation. It is shown that the proposed microstructural model and the algorithm for determining its constants provide a sufficiently high accuracy in predicting the elastic-plastic deformation of the composite.
Димитриенко Ю.И., Сборщиков С.В., Димитриенко А.Ю., Юрин Ю.В. Микроструктурная модель деформационной теории пластичности квази-изотропных композиционных материалов. Математическое моделирование и численные методы, 2021, № 4, с. 17–44.
539.36 Finite element modeling of natural vibrations of shell structures
Dimitrienko Y. I. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University), Bogdanov I. O. (Bauman Moscow State Technical University), Maremshaova A. A. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2023-3-317
A numerical algorithm for solving the problem of natural vibrations for thin-walled shell structures based on the finite element method is proposed. A software module has been developed as part of the SMCM software package, which implements the proposed numerical algorithm. A test problem was solved for natural vibrations of a cylindrical shell structural element. A comparative analysis of eigenfrequencies and eigenmodes was carried out with similar results obtained using a two-dimensional shell solution in the ANSYS software package, as well as with the results of solving a three-dimensional problem for natural vibrations in the ANSYS software package.
Димитриенко Ю.И., Юрин Ю.В., Богданов И.О., Маремшаова А.А. Конечно-элементное моделирование собственных колебаний оболочечных конструкций. Математическое моделирование и численные методы, 2023, № 3, с. 3–17.
Golovina N. Y. (Industrial University of Tyumen), Belov P. A. (Institute of Applied Mechanics of RAS)
doi: 10.18698/2309-3684-2022-2-1427
This article is a continuation of the review of works devoted to the study of the properties of elastic-plastic materials. In the first part, universal laws of deformation containing less than four formal parameters considered. As result of the review, requirements for the formulation of empirical laws of deformation of elastic-plastic materials formulated.In particular, it concluded that the deformation law must be at least four-parameter. In the second part of this paper, empirical laws of deformation containing four or more parameters considered and analyzed. Comparison of the considered empirical curves with a sample of experimental points carried out according to the standard procedure of minimization of the total quadratic deviation and using the method of gradient descent to determine the minimum of a function of many variables. A representative sample of 158 experimental points of the deformation curve of the Russian titanium alloy VT6 used to evaluate the predictive ability of the models for experimental agreement. Universal empirical strain laws containing four formal parameters allow describing the strain curve with specified stresses and tangential moduli at the ends of the curve. This fact allows us to state that the elastic-plastic properties of materials can expressed through the geometric parameters of the strain curve. In turn, the relationship between the elastic-plastic properties of the material and the geometry of the strain curve can interpreted as the principle of "geometrization" of the elastic-plastic properties of materials.
Головина Н.Я., Белов П.А. Анализ эмпирических моделей кривых деформирования упругопластических материалов (обзор). Часть 2. Математическое моделирование и численные методы, 2022, № 2, с. 16–29
Belov P. A. (Institute of Applied Mechanics of RAS), Golovina N. Y. (Industrial University of Tyumen)
doi: 10.18698/2309-3684-2022-1-6396
The article presents the result the review of works devoted to the research the properties of elastoplastic materials. The article consists of two parts. In the first part, universal single, two- and three-parametric laws describing the nonlinear dependence between the stress and deformation up to the destruction. The review includes: power laws, parabolic laws, exponential laws, harmonic law. A comparison the considered empirical curves with a sample experimental points is carried out by the standard procedure for minimizing the total quadratic deviation and using the method the gradient descent to determine the minimum function of many variables. To assess the predictive force for models on the compliance with the experiment, a representative sample used from 158 experimental points in the deformation curve of the Russian titanium alloy WT6. The analysis showed that the empirical laws of deformation containing less than four formal parameters cannot describe the universal deformation curve with the stress specified at the ends and the tangent module. Analysis of the advantages and disadvantages of existing empirical laws of deformation, made it possible to formulate certain requirements for their wording.
Головина Н.Я., Белов П.А. Анализ эмпирических моделей кривых деформирования упругопластических материалов (обзор). Часть 1. Математическое моделирование и численные методы, 2022, № 1, с. 63–96
624.04 A new method for calculating the torsional stiffness of a naturally twisted bar
Temis Y. M. (Центральный институт авиационного моторостроения им. П.И. Баранова), Ziyatdinov I. Z. (Центральный институт авиационного моторостроения им. П.И. Баранова)
doi: 10.18698/2309-3684-2023-1-6480
At the initial stages of designing compressor blades, screws, cutting tools, it is advisable to use a finite element model based on a model of a naturally twisted beam. This model takes into account the influence of the angle of natural twist on the rigidity of the part. The torsional stiffness of a bar significantly affects the stiffness parameters of the finite element model. It is shown that the torsional stiffness correction obtained on the basis of the relations of the technical theory of naturally twisted beams makes it possible to obtain results at small angles of natural twist that are in good agreement with the three-dimensional calculation of a twisted FEM beam. At large specific angles of initial twist, the technical theory gives overestimated values of the torsional stiffness. The article proposes a modification of the relations of the technical theory to determine the torsional rigidity, taking into account large angles of initial twist.
Темис Ю.М., Зиятдинов И.З. Новый метод вычисления жесткости на кручение в модели естественно-закрученного стержня. Математическое моделирование и численные методы, 2023, No 1, с. 64–80
Odinokov V. I. (Komsomolsk-na-Amure State University), Dmitriyev E. A. (Komsomolsk-na-Amure State University), Evstigneev A. I. (Komsomolsk-na-Amure State University), Potianikhin D. A. (Komsomolsk-na-Amure State University), Kvashnin A. E. (Komsomolsk-na-Amure State University)
doi: 10.18698/2309-3684-2021-3-323
This paper presents the mathematical formulation and the results of calculations of the problem of metal deformation on a casting-forging module with modified side strikers’ drive. A complex spatial problem of determination the stress-strain state of the flow region under loading with an external load that changes over time is considered. The fundamental equations are based on flow theory. At solving the problem, a proven numerical method is used, as well as numerical schemes and the software package used earlier at solving similar problems. The software package implements a step-by-step loading algorithm considering the history of the process and the changing geometry of the flow region. A small time step is associated with a 10° rotation of the eccentric shaft. The deformation area is divided into elements by an orthogonal system of surfaces (elements have an orthogonal shape). For each element, the formulated system of equations is written in a difference form, which is solved according to the developed numerical schemes and algorithms, that consider the initial and boundary conditions. The result of the solution is the fields of stresses and velocities of displacements in the spatial area. The analysis of the obtained results is given. A comparison with the results of the current structure module solving has been made. Lead is taken as a deformed material, the physical properties of which are approximated by an analytical dependence according to the available experimental data. The physical nonlinearity of the system of equations is realized during solving by the iterative method. Local calculations of the solution of the problem were carried out on three variants of division of the area into elements. The choice of the mesh density imposed on the considered deformation region is substantiated. The solution results are presented graphically. The efficiency of the deformation process according to the improved method on a new design of the casting and forging module is shown.
Одиноков В.И., Дмитриев Э.А., Евстигнеев А.И., Потянихин Д.А., Квашнин А.Е. Математическое моделирование процесса деформации металла на литейно-ковочном модуле с измененным приводом боковых бойков. Математическое моделирование и численные методы, 2021, № 3, с. 3–23.
539.36 Microstructural model anisotropic flow theory for elastic-plastic layered composites
Dimitrienko Y. I. (Bauman Moscow State Technical University), Черкасова М. С. (Bauman Moscow State Technical University), Dimitrienko A. Y. (Lomonosov Moscow State University)
doi: 10.18698/2309-3684-2022-3-4770
A microstructural model of layered elastic-plastic composites based on the anisotropic flow theory is proposed. The model represents the effective constitutive relations of the transversally isotropic theory of plastic flow, in which the model constants are determined not experimentally, but on the basis of approximations of the deformation curves of composites obtained by direct numerical solution of problems on the periodicity cell for basic loading trajectories, which arise in the method of asymptotic averaging. The problem of identifying the constants of this composite model is formulated; for the numerical solution of this problem, methods of optimizing the error functional are used. The results of numerical simulation by the proposed method for layered elastic-plastic composites are presented, which showed good accuracy of approximation of numerical strain diagrams.
Димитриенко Ю.И., Черкасова М.С., Димитриенко А.Ю. Микроструктурная модель анизотропной теории течения для упруго-пластических слоистых композитов. Математическое моделирование и численные методы, 2022, № 3, с. 47–70.
539.3 Modeling the bending of beams made of rubber-like materials
Firsanov V. V. (Moscow Aviation Institute (National Research University))
doi: 10.18698/2309-3684-2021-4-316
Since the classical hypotheses of Bernoulli for beams and Kirchhoff for thin plates contradict the additional condition of incompressibility for rubber-like (incompressible) materials (invariability of the volume during deformation), a calculation model for a bending beam is proposed, which does not lead to a serious complication of the problem in comparison with the classical solution. The invariability of the volume is manifested under the action of a power load; in the case of a temperature load, the deformation of the volume change is not zero. The absence of volumetric deformations for rubber-like materials is a consequence of Hooke's law for such materials. Summing the linear deformations expressed in terms of stresses and taking Poisson's ratio 0.5, we obtain the equality of the indicated sum to zero. Many rubber-like materials are incompressible and low-modulus, which means their weak resistance to tension and shear, but the resistance of the material to change in volume tends to infinity, therefore the physical relations of the generalized Hooke's law are transformed into the so-called "neo- Hooke " equations of the relationship between stresses and strains. Of the two independent physical characteristics (modules) for incompressible materials, only one module remains, which characterizes the resistance of the medium to change in shape. In physical relations for an incompressible material, the product of an infinitely large volumetric modulus by the deformation of a change in volume equal to zero is an uncertainty that is replaced by some force function that has the dimension of stresses and is an additional unknown. At the same time, the system of governing equations of the mechanics of incompressible media is supplemented by the equation of invariability of volume. The scheme for solving the problem in displacements for traditional structural materials turns into a mixed scheme for rubber-like materials, since for them not only displacements but also the mentioned force function act as the main unknown sought function.
Фирсанов В.В. Моделирование изгиба балок из резиноподобных материалов. Математическое моделирование и численные методы, 2021, № 4, с. 3–16.