Quasi-Deterministic Processes with Monotonic Trajectories and Unsupervised Machine Learning

This paper aims to consider approximation-estimation tests for decision-making by machine-learning methods, and integral-estimation tests are defined, which is a generalization for the continuous case. Approximation-estimation tests are measurable sampling functions (statistics) that estimate the approximation error of monotonically increasing number sequences in different classes of functions. These tests make it possible to determine the Markov moments of a qualitative change in the increase in such sequences, from linear to nonlinear type. If these sequences are trajectories of discrete quasi-deterministic random processes, then moments of change in the nature of their growth and qualitative change in the process match up. For example, in cluster analysis, approximation-estimation tests are a formal generalization of the “elbow method” heuristic. In solid mechanics, they can be used to determine the proportionality limit for the stress strain curve (boundaries of application of Hooke’s law). In molecular biology methods, approximation-estimation tests make it possible to determine the beginning of the exponential phase and the transition to the plateau phase for the curves of fluorescence accumulation of the real-time polymerase chain reaction, etc.

MOR and MOE of Serbian Spruce (Picea omorika Pančić/Purkyně) Wood from Natural Stands

The paper presents the results of testing the bending stress of Serbian spruce wood from natural stands. In testing the samples, in addition to the modulus of rupture, the bending stress at the proportionality limit, the ratio between the stress at the proportionality limit and the modulus of rupture as well as the modulus of elasticity of wood were determined. The study included nine trees from natural stands, and a total of 261 samples were tested. Regression analysis determined the dependences of these mechanical properties on the annual ring width, the proportion of late wood and wood density, as well as the dependence of the modulus of elasticity on the modulus of rupture.

METHOD FOR DETERMINING THE LIMITS OF ELASTICITY AND PROPORTIONALITY OF MATERIALS FOR SUBSEQUENT DRILLING OF WORKPIECES IN THE STRESS–STRAIN STATE

Drilling holes is one of the most common operations in the manufacture of parts. As a result, improving the efficiency of this process is an urgent task. To improve the efficiency of the drilling process, a method was developed at the Southwestern University for drilling with pre–stressed–deformed material of the workpiece, in which the sample is subjected to elastic deformation at a load that does not exceed the proportionality limit of the workpiece material, that is, when the load is removed, the dimensions of the workpiece remain the same. As part of this work, an experimental device was developed and designed to determine the limits of elasticity and proportionality of materials for subsequent drilling of workpieces in a stress–strain state. This invention will improve the automation and accuracy of measurement. An example of measurement and calculation is considered. Calculation formulas for determining the measurement error are given.

MODEL OF A CURVE OF NONLINEAR DEFORMATION OF STEEL 20HGR AND STEEL 35

In most cases the deformation curves that determine the properties of elastoplastic materials are empirical dependencies that interpolate a certain sample of experimental points. At the same time, empirical curves do not provide an idea of the physics of deformation processes in such materials. As an alternative, the article proposes to construct deformation curves as a solution to a differential equation, each term of which has a physical meaning and determines a certain deformation mechanism specific to each material. The offered differential equation is a fourth-order equation with variable coefficients, whose eigenfunctions are polynomials and power functions, which ensures the continuity of this model with the empirical Ramberg-Osgood model. It is shown that the operator of this equation is the Euler operator and is the product of two second-order operators. The first operator defines the mechanism of linear deformation, and the second - the mechanism of nonlinear deformation of the material. The parameters of the material for processing experimental data are determined by the methods of applied mathematics. To reduce the amount of calculations, a method of compressing a two-dimensional (in the coordinates of deformation-stress) region is proposed to find the characteristic point of the deformation curve, which is the proportionality limit. The proposed method for determining the proportionality limit is formal, mathematical and devoid of subjectivity, in contrast to the method prescribed by the current GOST. A consistent physically substantiated system of four boundary conditions on the interval of nonlinear elasticity of the deformation curve is formulated. The mathematical model of building of deformation curves of various physically nonlinear materials allows to create mathematical models of the resource of such materials.

Experimental studies of hole drilling operations in stress-strain workpiecematerial

Drilling holes is one of the most common operations in the part production. Consequently, increasing the efficiency of this process is an urgent task. To improve the efficiency of the drilling process, the South- West University developed a method of drilling for pre-stressed and prestrained workpiece material, which requires a sample to be subjected to elastic strain under load not exceeding the proportionality limit of the workpiece material. That is, when the load is removed, the dimensions of the workpiece remain unchanged. The paper presents the experimental device designed to determine the axial force and torque when drilling holes in the stress-strain workpiece material. Multi-factor experiments were carried out to obtain empirical dependences of the axial force and torque arising in drilling holes in the stress-strain workpiecematerial on the process parameters.

Limit Dependences in Stability Calculations With Account for Physical Nonlinearity

Stability of bars, plates, shells, and other thin-walled structures in conditions of small physical nonlinearity is considered, when stresses exceed the proportionality limit, the amount of deformations being limited. Shanley's concept is used. The critical state is determined by means of some limit dependences. In a large number of cases, when creating efficient highly-stressed constructions, limited plastic deformations are allowed in them. When analysing stability in the critical state, the calculated stresses turn out to exceed the proportionality limit and the Young's modulus of elasticity turns out to be greater than the tangent modulus corresponding to the calculated stress on the diagram “deformation-stress”. The objective of this work is to show that stability calculation beyond the proportionality limit is reduced to the analysis of some limit dependences as well as to develop a general solution algorithm for similar problems.