A mathematical model of transition of the material from elastic to elasto-plastic state

The goal of this study is to develop a mathematical model of the transition of the structural material from elastic state to elastoplastic state upon tension. The model is based on a modified three-parameter transition operator from one mathematical function to another. A procedure of the mathematical approximation of the transition and corresponding algorithm, which provides a generalized canonical description of the transition, regardless of the form of the functions that characterize the system behavior before and after the transition are presented. The technique is used to describe the transition of two structural materials from the elastic to elastoplastic state upon tension of the samples. Initial sections of the tension diagram are described using three empirical parameters. The role of each of them — maximum permissible relative deformation, transition rate and asymmetry of the transition — are determined. Statistical interpretation of the elastoplastic transition is developed and substantiated. Mathematical expressions for the integral probability function and probability density functions that provide numerical statistical estimation of the degree of change in the state of the structural elements of the material during loading are derived. Analytical description of the initial part of the tension diagram of the material can be used to rearrange the diagrams when modeling deformation processes in conditions of reversible elastoplastic loading.

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