Historical aspects of mathematical analysis of metal material deformation diagrams

The article presents a retrospective of the formation and development of one of the methods to study the materials strain hardening – mathematical analysis of strain diagrams. Discussed in detail the most important mathematically reasonable direction of the analysis of the diagrams of deformation based on the use of their rebuild in the coordinates, due to the defined model representations, describe the process of strain hardening. The paper illustrates changing of analysis methods of stretching curves (load-elongation) and their mathematical description starting from representations of the 17${}^{th}$–18${}^{th}$ centuries in the works by L. da Vinci, R. Hooke, I. Newton up to the present day. A phased development of the mathematical description of strain hardening using strain diagrams from phenomenological approaches within the theories of Ludwik, Hollomon, Jaul-Krussard to modern physical theories of strain hardening by J. Taylor, N.F. Mott, E. Orovana, Ya.I. Frenkel, Ya.B. Friedman, A. Zeger, A. Cottrell, J. Reed, based on the analysis of the evolution of complexes of structural defects, the results of metallographic grounded thin metalphysical experiments. A critical analysis of the shortcomings of modern mathematical methods for estimating the parameters of strain diagrams in terms of the degradation and destruction (damage) development in the testing process is given. For example, mild steel experimental testing of the use of the listed model representations in comparison with the approaches Ludvika, Hollomon, Joule-Crosara to assess their conformity with modern views on the contribution of degradation and destruction processes (damage) to work hardening. The quantitative analysis of the coefficients of hardening, q-factor and destruction, which allowed to link the stages of deformation hardening with the parallel development of two main processes: the transformation of dislocation substructure and the development of deformation damage such as microcracks and pores. The critical values of deformation determining the boundaries of the ranges of sharp changes in the parameters of deformation hardening and destruction are revealed.