and Computational Methods

Rubric: "1.1.8. Mechanics of a Deformable Solid Body (Physical and Mathematical Sciences)"

doi: 10.18698/2309-3684-2022-2-2862

The problem of development a theory for calculating the stress-strain state of thin multilayer elastic plates, for which linearized slip conditions are specified at the interface between the layers, is considered. The solution of this problem is constructed using an asymptotic analysis of the general equations of the 3-dimensional theory of elasticity with the conditions of non-ideal contact of the layers. The asymptotic analysis is carried out with respect to a small geometric parameter representing the ratio of the plate thickness to its characteristic length. Recurrent formulations of local quasi-one-dimensional problems of the theory of elasticity with slip are obtained. Explicit analytical solutions are obtained for these problems. The derivation of the averaged equations of elastic equilibrium of multilayer plates is presented, taking into account the slippage of the layers. It is shown that due to the effect of slippage of layers, the system of averaged equations of the theory of multilayer plates has an increased - 5th order of derivatives, in contrast to the classical 4th order, which takes place in the theory of Kirchhoff-Love plates. It is shown that the asymptotic theory makes it possible to obtain an explicit analytical expression for all 6 components of the stress tensor in the layers of the plate. As a special case, the problem of calculating the stress-strain state of a 4-layer plate under uniform pressure bending with one slip coefficient is considered. A complete analytical solution of this problem is obtained, including explicit expressions for all non-zero components of the stress tensor. A numerical analysis of the solution of the averaged problem for a composite plate is carried out, in which the layers are unidirectional reinforced fibrous materials oriented at different angles. A comparative analysis of the influence of the fiber reinforcement angles and the slip coefficient of the layers on the displacement of the plate and the distribution of stresses in the layers was carried out. It is shown that the problem of bending a plate with slip admits the existence of a spectrum of critical values of the slip coefficient, when passing through which the displacements and stresses in the layers of the plate change significantly, and these critical values depend on the angle of reinforcement of the composite layers.

Димитриенко Ю.И., Губарева Е.А. Асимптотическая теория многослойных тонких упругих пластин с проскальзыванием слоев. Математическое моделирование и численные методы, 2022, № 2, с. 30–64

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.

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

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

doi: 10.18698/2309-3684-2022-1-4262

When designing products made of composite materials intended for use in difficult conditions of inhomogeneous deformations and temperature, it is important to take into account viscoelastic, including spectral and dynamic, properties of the binder and fillers. The article considers dynamic characteristics (complex modulus, complex malleability,their real and imaginary parts, loss angle tangent) and spectral characteristics of relaxation and creep and their dependence on each other. The characteristics mentioned above were found for all known types of creep kernel and relaxation kernel. To find the spectral characteristics, one of the numerical methods of inverting the Laplace transform was used — the method of quadrature formulas with equal coefficients. Algorithms and computer programs for the implementation of this method have been compiled. The obtained graphs are quite accurate (the maximum error of calculations in the average does not exceed 5%), despite the fact that the error is very noticeable in the initial time segments.

Валишин А.А., Тиняев М.А. Моделирование динамических и спектральных вязкоупругих характеристик материалов на основе численного обращения преобразования Лапласа. Математическое моделирование и численные методы, 2022, № 1, с. 42–62.

doi: 10.18698/2309-3684-2023-2-3366

The problem of development of a theory for calculating the stress-strain state of thin multilayer elastic plates in the moment (micropolar) theory, is considered. The solution of this problem is built using an asymptotic analysis of the general equations for a 3-dimensional quasi-static problem of the moment theory of elasticity. The asymptotic analysis is carried out with respect to a small parameter representing the ratio of the plate thickness to its characteristic length. Recurrent formulations of local problems of the moment theory of elasticity are obtained. Explicit analytical solutions are obtained for these problems. The derivation of the averaged system of equations for multilayer plates is presented. It is shown that the asymptotic theory makes it possible to obtain an explicit analytical expression for all 9 components of the stress tensor and the moment stress tensor (in general) in the plate. As a special case, the problem of calculating the stress-strain state of a centrally symmetrical hingedly fixed plate when bending under the action of a uniformly distributed pressure. A complete analytical solution of this problem for all non-zero components of the stress tensor and the moment stress tensor is obtained. A numerical analysis of the solution of the problem for a single layer plate for the stress tensor is carried out, basing on the obtained expressions. A comparative analysis of the obtained results with similar calculations for the classical theory of elasticity is carried out, with revealing of similarities and differences for all components of the stress tensor.

Димитриенко Ю.И., Бойко С.В. Асимптотическая теория многослойных тонких микрополярных упругих пластин. Математическое моделирование и численные методы, 2023, № 2, с. 33–66.

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.

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.

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.