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Metal Finishing
ArticleName Influence of non-linearity of hardening curve on elasticoplastic bend of rectangular rod
DOI 10.17580/cisisr.2019.01.07
ArticleAuthor V. N. Shinkin

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

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


For the description of the hardening curve, they use the various classical approximations, for instance, the Prandtl’s approximation (Prandtl L.), the linear approximation (Smirnov-Alyaev G. A., Birger I. A., Moshnin E. N.), the Nadai’s approximation (Nadai A. L.) and the Ludwik’s approximation (Ludwik P.). However, all these approximations have a large relative error (up to 35–45%) with respect to the experimental hardening curves obtained on the modern universal tensile machines. In addition, the significant drawback of the Nadai’s and Ludwik’s approximations is the infinite derivative (the tangent of inclination angle) of the stress curve at the beginning of the stretching of the rod (ε = 0). It contradicts the classical Hooke’s law for the small elastic deformations and is not observed in none of the metals in practice. Therefore, below we will use a more accurate approximation by Shinkin V. N. in the form of a power series (a relative error 1–3 %). When metalware’s bend of a by the external forces (the concentrated forces, distributed forces or pairs of forces), there are both positive and negative stresses in the metalware. In this case, one part of the metalware is stretched, and the second part is compressed. However, always in the center of the metalware there is a non-stresses surface (an axis), which bends, but has no stresses. The stresses, that arise during the bend of metalwares, create a bend’s moment in its perpendicular plane, which tends to straighten the metalware after removing the external forces. The degree of straightening of the non-stresses surface after the bend is characterized by the spring-back factor. With the help of the spring-back factor we can find out the final form of the metalware after forces. In addition, after the elasticoplastic bend in the metalware the harmful stresses remain. Often the harmful stresses manifest themselves during the heating, cutting or secondary forming of the metalware. As a practical example of using of the proposed non-linear approximation method, the bend of the steel rectangular rod is considered, for which the bend moment, the spring-back factor and the residual stresses are calculated.

keywords Elasticoplastic deformation, non-linear hardening zone of low-carbon steel, yield strength, ultimate strength, direct approximation

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