Abstract
An efficient method for describing plane strain bending of wide rigid viscoplastic sheets at large strains is developed. Both pure bending and bending under tension are considered. The approach is based on the transformation equations between Eulerian and Lagrangian coordinates. In addition, it is shown that it is advantageous to use the equivalent strain rate as an independent variable instead of the space coordinate. Due to this change in the independent variable, a uniform treatment of an arbitrary dependence of the yield stress on the equivalent strain rate is possible. The solution reduces to an ordinary differential equation that should be solved numerically. However, one term of this equation reduces to the expression 0/0 at the initial instant. Therefore, analytic treatment of the differential equation is required to describe an initial stage of the process. An illustrative example is provided. This example illustrates the effect of the parameter involved in the Bingham model on the through-thickness distribution of stresses and the bending moment. It is shown that the thickness of the sheet decreases as the deformation proceeds.
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References
Alexandrov S, Kim J-H, Chung K, Kang T-J (2006) An alternative approach to analysis of plane-strain pure bending at large strains. J Strain Anal Eng Des. https://doi.org/10.1243/03093247JSA154
Alexandrov S, Manabe K, Furushima T (2011) A general general analytic solution for plane strain bending under tension for strain-hardening material at large strains. Arch Appl Mech. https://doi.org/10.1007/s00419-011-0529-9
Alexandrov S, Lyamina E, Hwang Y-M (2021) Finite pure plane strain bending of inhomogeneous anisotropic sheets. Symmetry. https://doi.org/10.3390/sym13010145
Bruhns OT, Gupta NK, Meyers ATM, Xiao H (2003) Bending of an elastoplastic strip with isotropic and kinematic hardening. Arch Appl Mech. https://doi.org/10.1007/s00419-002-0273-2
Dadras P, Majless SA (1982) Plastic bending of work hardening materials. Trans ASME J Eng Ind. https://doi.org/10.1115/1.3185823
Hill R (1950) The mathematical theory of plasticity. Clarendon Press, Oxford
Ikumapayi OM, Akinlabi ET, Madushele N, Fatoba SO (2020) A brief overview of bending operation in sheet metal forming. In: Emamian S, Awang M, Yusof F (eds) Advances in manufacturing engineering; Lecture notes in mechanical engineering. Springer, Singapore, pp. 149–159. https://doi.org/10.1007/978-981-15-5753-8_14
Kato H, Tottori Y, Sasaki K (2014) Four-point bending test of determining stress-strain curves asymmetric between tension and compression. Exp Mech. https://doi.org/10.1007/s11340-013-9791-9
Laws V (1981) Derivation of the tensile stress-strain curve from bending data. J Mater Sci. https://doi.org/10.1007/BF01033845
Lee Y, Dawson PR (1989) Obtaining residual stresses in metal forming after neglecting elasticity on loading. ASME J Appl Mech. https://doi.org/10.1115/1.3176086
Maeda T, Noma N, Kuwabara T, Barlat F, Korkolis YP (2018) Measurement of the strength differential effect of DP980 steel sheet and experimental validation using pure bending test. J Mater Process Technol. https://doi.org/10.1016/j.jmatprotec.2018.02.009
Oldroyd PWJ (1971) Tension-compression stress-strain curves from bending tests. J Strain Anal. https://doi.org/10.1243/03093247V064286
Tan Z, Persson B, Magnusson C (1995) Plastic bending of anisotropic sheet metals. Int J Mech Sci. https://doi.org/10.1016/0020-7403(94)00069-V
Urriolagoitia-Sosa G, Durodola JF, Lopez-Castro A, Fellows NA (2006) A method for the simultaneous derivation of tensile and compressive behaviour of materials under Bauschinger effect using bend tests. Proc Inst Mech Eng Part C. https://doi.org/10.1243/09544062JMES180
Verguts H, Sowerby R (1975) The pure plastic bending of laminated sheet metals. Int J Mech Sci. https://doi.org/10.1016/0020-7403(75)90061-2
Acknowledgements
This research was made possible by the grants RFBR-19-51-52003 (Russia) and AAAA-A20- 120011690136-2 (Russia).
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Alexandrov, S., Lyamina, E., Pirumov, A. (2022). An Efficient Method for Describing Plane Strain Bending of Viscoplastic Sheets at Large Strains. In: Altenbach, H., Eremeyev, V.A., Galybin, A., Vasiliev, A. (eds) Advanced Materials Modelling for Mechanical, Medical and Biological Applications. Advanced Structured Materials, vol 155. Springer, Cham. https://doi.org/10.1007/978-3-030-81705-3_2
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