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Rolling and Tubemaking
ArticleName Moment at elastic-plastic bending of steel sheet. Part 2. Cubic approximation of steel's hardening zone
DOI 10.17580/chm.2022.02.03
ArticleAuthor V. N. Shinkin

National University of Science and Technology “MISiS”, Moscow, Russia:

V. N. Shinkin, Dr. Phys.-Math., Professor, Dept. of Physics, e- mail:


The classical approximations by P. Ludwik and A. Nadai of the steel hardening zone do not accurately describe the plastic deformation of steel. That often leads to the significant errors (defects) in the final form of metal products made from a steel sheet during its forming (based on the preliminary analytical and numerical calculations of the given model of metallurgical production) on the presses and dies. For example, the Nadai’s curve does not pass through the ultimate strength’s point of the experimental curve, signifi cantly exceeding the ultimate strength at this point, which is not permissible at calculating stresses and deformations. The Ludwik’s curve does not take into account the elastic deformation zone of the steel (immediately has a stress equal to the yield strength at zero relative deformation) and does not have the maximum (observed on the experimental curve) at the point of the ultimate strength. To eliminate the above-mentioned disadvantages, in the first published part of the paper, the author considered the parabolic approximations of the steel hardening zone in the form of the second-order polynomials, which satisfy three boundary conditions. The constructed parabolic approximations of the steel hardening zone turned out to be an order of magnitude more accurate than the classical approximations by Ludwik and Nadai. In this paper, the author considers even more exact cubic approximations of the steel hardening zone with the same supporting parameters of the experimental curve of the steel hardening zone. In this case, the cubic approximations already satisfy four boundary conditions. The constructed cubic approximations of the steel hardening zone are two orders of magnitude more accurate than the classical approximations by Ludwik and Nadai of the steel hardening zone.

keywords Steel sheet, bending, steel hardening, elasticity, plasticity, normal stress, bending moment, boundary conditions, cubic approximation

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