and Computational Methods

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: 10.18698/2309-3684-2023-1-331

This article is the third part of a review of works devoted to the study of the properties of elastic-plastic materials. The first and second parts were devoted to the analysis of universal empirical laws of deformation, which model the material properties over the entire range of deformation, up to fracture. It was concluded that in order to create a model of the material response to stress growth, the deformation law must be at least four-parametric. The empirical Ramberg-Osgood law was found to be the most qualitative, at least for the titanium alloy VT6 considered. However, despite its accuracy, it does not reflect the material properties in the zone of large plastic strains, including in the vicinity of the point of ultimate strength. This paper presents an analysis of multilink models describing the relationship between strain and stress by different laws in the elastic zone and in the plastic zone. The review includes two-link models by Nadai, Mirambell-Real, Rasmussen, Abdella, formulated for materials whose strain curve has no positive curvature section. Also considered in the review are the three-link models of Quach; Hertele; Belov-Golovina, which allow modeling of deformation curves with a positive curvature region. The evaluation of the quality of empirical laws and their correspondence to the sample of experimental points was carried out by minimizing the standard quadratic deviation and using the method of gradient descent to determine the minimum of the function of many variables. The material for the comparative analysis of empirical models is titanium alloy VT6; for the Hertele and Belov-Golovina models — steel St3sp. It is shown that the models built on the basis of multi-line splines determine the properties of elastic-plastic materials more accurately than the models built on the basis of universal laws.

Головина Н.Я. Анализ эмпирических моделей кривых деформирования упруго-пластических материалов (обзор). Часть 3. Математическое моделирование и численные методы, 2023, No 1, с. 3–31.