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Foundry Processes and Metal Science
ArticleName Direct and inverse non-linear approximation of hardening zone of steel
ArticleAuthor V. N. Shinkin

National University of Science and Technology “MISIS” (Moscow, Russia):

V. N. Shinkin, Dr. Phys.-Math., Prof., e-mail:


The mechanical properties of steels and alloys play an important role in the production of the steel products from the steel sheet and straight beam (with the round and rectangular crosssections). Experimentally, the mechanical properties are studied on the universal tensile machines, which can stretch, compress or twist the steel specimens of standard shape. For example, under a tension of the round or flat steel specimen, made from a low-carbon steel, in an electromechanical universal testing machine, we get the classic diagram “stress - relative elongation” (the diagram of the dependence of the normal stress on the relative elongation of the specimen). We define the two most important mechanical characteristics of steels on the diagram - the yield strength and the ultimate strength. At the beginning of the diagram, there is a linear zone of the stress-strain dependence obeying the linear Hooke’s law for the elastic deformations (up to the proportionality limit). Then there is a small zone from the proportionality limit through the elastic limit to the yield point (usually this zone is neglected in the analytical and numerical calculations). Then on the diagram we clearly see the non-linear hardening zone (the zone of the irreversible plastic deformations), which begins at the end of the yield area (the yield point) and ends at the time of the neck’s formation (at the time of the reaching of the ultimate strength). Behind the hardening zone, there is the local fluidity zone in the neck’s area. At the end of the local fluidity zone, the specimen’s rupture takes place. In metallurgy and mechanical engineering, the elastic-plastic deformation of steel billets is carried out in the zone of the steel’s plastic hardening (usually in the first half of the hardening zone). It is not recommended to come close to the second rear half of the steel’s hardening zone during the billet’s deformation, as in this case there is a high risk of an appearance of the surface and internal hidden defects of the steel billet, which are almost impossible to eliminate during the subsequent stages of metal product. Under the numerical and mathematical modeling of processes of hot and cold deformation of steel products, it is essential to know the analytical dependence (approximation) of stresses on strains in the steel’s hardening zone. The most well-known classical approximations of the steel’s hardening zone are the simplest Prandtl’s approximation, the linear approximation, and the non-linear approximations by Nadai and Ludwik. However, all of them are not without significant drawbacks, as they offer to approximate the non-linear hardening curve by only one term (a power function). Below we propose other methods of the direct and inverse non-linear approximations using the finite or infinite power series with the displacement of the argument (relative deformation). The series members can have both integer and fractional power, and the coefficients of the series members can have both positive and negative values. It is shown that this method is much more accurate than the classical approximation methods.

keywords Elastic and plastic deformation, non-linear hardening zone of low-carbon steel, yield strength, ultimate strength, direct and inverse approximation

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