Abstract. Hollow axisymmetric parts made from round bar stock are widely represented among the products of general and special mechanical engineering. In most cases, simple bulk forming schemes are used in their manufacture, in particular reverse extrusion. The main disadvantage of a simple scheme is the extremely high working pressure on the punch, which severely limits the process capabilities of the operation. At the same time, the mean strain of the workpiece material is relatively small, which does not allow the workpiece structure to be adequately worked. One approach to solving this problem may be a transition to combined extrusion schemes, such as longitudinal-transverse extrusion. When introducing new metal forming schemes, it is important not only to maintain the integrity of the workpieces, but also to determine the maximum forming force in advance and maintain the strength of the tooling. Data are presented on the influence of characteristic geometric parameters of the combined longitudinal-transverse extrusion scheme on the magnitude of the forming force and the working pressure on the punch and counterpunch. A comparison of the results obtained using the previously proposed equation and the power balance method (PBM) revealed large errors in the existing recommendations for selecting tool and workpiece dimensions. The need to calculate the forming force for the counterpunch is also noted. Based on computer simulation results, correction factors are proposed for calculating the forming force on the punch and counterpunch using the derived equation and the power balance method, which allow results to be obtained with acceptable accuracy within the previously formulated recommendations for selecting tool and workpiece geometry.

Keywords: cold forging, combined extrusion, forming force, working pressure, longitudinal-transverse extrusion, design recommendations

For citation: Filin D. S., Tikhonova E. S., Kunitsyn K. A., Khorevsky K. K. Recommendations for calculating the maximum forming force in longitudinal-transverse extrusion. Aerospace Engineering and Technology. 2026. Vol. 4, no. 1, pp. 198–214. (In Russ.) https://doi.org/10.52467/2949-401X-2026-4-1-198-214; https://elibrary.ru/CKYEAX

References
1. Lyasnikov A. V., Ageev N. P., Kuznetsov D. P. [et al.]. Resistance of Materials to Plastic Deformation in Applications to Pressure Working Processes. St. Petersburg: Vneshtorgizdat-Petersburg, 1995. 527 p. (In Russ.) https://elibrary.ru/TQOIPH
2. Vorontsov A. L. Theory and calculations of metal forming processes: A textbook for students of higher educational institutions. in 2 volumes. Volume 1. Moscow: Bauman Moscow State Technical University (National Research University), 2014. 400 p. (In Russ.)
3. Vorontsov A. L. Theory and calculations of metal forming processes: A textbook for students of higher educational institutions. in 2 volumes. Volume 2. Moscow: Bauman Moscow State Technical University (National Research University), 2014. 448 p. (In Russ.)
4. Gronostajski Z., Pater Z., Madej L., Gontarz A., Lisiecki L., Lukaszek-Solek A., Luksza J., Mróz S., Muskalski Z., Muzykiewicz W., Pietrzyk M., Sliwa R. E., Tomczak J., Wiewiórowska S., Winiarski G., Zasadzinski J., Ziólkiewicz S. Recent development trends in metal forming. Archives of Civil and Mechanical Engineering. 2019. Vol. 19, no. 3, рр. 898–941. https://doi.org/10.1016/j.acme.2019.04.005; https://elibrary.ru/CSKNAJ
5. Pasynkov A. A. Evaluation of the possibility of manufacturing the cylindrical case-shaped products of hard-to-deform non-ferrous alloys by reverse extrusion. Non-FerrousMetals. 2020. Vol. 48, no. 1, рр. 49–51. https://doi.org/10.17580/nfm.2020.01.08; https://elibrary.ru/LHPFLH
6. Khomyak B. S. Tool for cold-heading machines. Moscow: Research Institute of Information on Mechanical Engineering, 1972. 119 p. (In Russ.)
7. Gurulev D., Levin Yu., Rogachev S., Vodovatov V. Investigation of stresses and deformation forces in energy-intensive pressing processes with backflow of metal. E3S Web of Conferences. 2024. Vol. 515. pp. 20–23. (In Russ.) https://doi.org/10.1051/e3sconf/202451502023; https://elibrary.ru/KSZBJO
8. Kapustin A. I. Stamping of glass-type forgings. Tutorial for the course “Forging and Stamping Technology”. Moscow: Bauman Moscow State Technical University, 1991. 239 p. (In Russ.)
9. Danilin G. A., Filin D. S., Volzhin S. G. Longitudinal-transverse extrusion of hollow components from bar. Russian Engineering Research. 2016. Vol. 36, no. 10, pp. 836–840. (In Russ.) https://doi.org/10.3103/S1068798X16100075; https://elibrary.ru/XFUOEN
10. Danilin G. A., Filin D. S., Lukin S. A. Feasibility of cold extrusion methods in the manufacture of critical-purpose hollow parts from bar stock. Metalworking. 2017. Vol. 102, no. 6, pp. 17–23. (In Russ.) https://elibrary.ru/YNMPTR
11. Filin D. S., Fedorov D. V. Justification of a rational tool shape for the longitudinal-transverse extrusion process of hollow semi-finished products. Bulletin of Tula State University. Engineering Sciences. 2018. No. 8. pp. 294–301. (In Russ.) https://elibrary.ru/YLHSZF
12. Filin D. S., Sobolev I. A., Abu Fadda T. M. Refinements of a rational tool shape for the longitudinal-transverse extrusion operation. Bulletin of Tula State University. Engineering Sciences. 2023. No. 3, pp. 141–147. (In Russ.) https://doi.org/10.24412/2071-6168-2023-3-141-148; https://elibrary.ru/UMTWLB
13. Glezer A. M. On the nature of ultra-high plastic (megaplastic) deformation. Bulletin of the Russian Academy of Sciences. Physical Series. 2007. Vol. 71, no. 12, pp. 1764–1772. https://elibrary.ru/IBMRNX (In Russ.) https://elibrary.ru/IBMRNX
14. Beyerlein I. J., Tóth L. S. Texture evolution in equal-channel angular extrusion. Bulk Nanostructured Materials. 2009. pp. 387–421. https://doi.org/10.1002/9783527626892.ch18
15. Newbery, A. P., Han, B. Q., Lavernia, E. J. [et al.]. Mechanical alloying and severe plasticdeformation. Materials Processing Handbook. 2007. pp. 273–300. https://elibrary.ru/QPYSKM
16. Ageyev N. P., Tereshchenko V. G. Study of the Technological possibilitiesof using preliminary, prior to heat treatment, severe cold plastic deformation by widening extrusion for strengthening rod parts. Metalworking. 2012. Vol. 71–72, No. 5–6, pp. 40–44. (In Russ.) https://elibrary.ru/RKWMBH
17. Ivanov A. M., Petrova N. D., Lepov V. V. The effect of extrusion and screw pressing on the structure and mechanical properties of low-alloy steel. Science and Education. 2015. Vol. 80, no. 4, pp. 87–90. (In Russ.) https://elibrary.ru/UZJTQP
18. Yin X., Deng W., Zou Y., Zhang J. Ultrafine grained Al 7075 alloy fabricated by cryogenic temperature large strain extrusion machining combined with aging treatment. Materials Science and Engineering: A. 2019. Vol. 762. pp. 138106. https://doi.org/10.1016/j.msea.2019.138106 6; https://elibrary.ru/XOFEUY
19. Nagasekhar A. V., Yoon S. C., Yoo J. H., Kang S. Y. Plastic flow and strain homogeneity of an equal channel angular pressing process enhanced through forward extrusion. Materials Transactions. 2010. Vol. 51, no. 5. pp. 977–981. https://doi.org/10.2320/matertrans.M2009407; https://elibrary.ru/MYYYOV
20. Matsubara K., Miyahara Y., Horita Z., Makii K. Using extrusion and ECAP processing to achieve low temperature and high strain rate superplasticity. Materials Science Forum. 2003. Vol. 419–422, No. I, pp. 497–502. https://doi.org/10.4028/www.scientific.net/msf.419-422.497; https://elibrary.ru/LYJWAN
21. Naizabekov A. B., Lezhnev S. N., Volokitina I. E. Change in copper microstructure and mechanical properties with deformation in an equal channel stepped die. Metal Science and Heat Treatment. 2015. Vol. 57, no. 5–6. pp. 254–260. https://doi.org/10.1007/s11041-015-9870-x; https://elibrary.ru/XLGOXR
22. Stepansky L. G. Calculations of metal forming processes. Moscow: Mashinostroenie, 1979. 215 p. (In Russ.)
23. Danilin G. A., Ogorodnikov V. P. Theory and calculations of combined plastic forming processes. BSTU “VOENMEH”. St. Petersburg, 2004. 304 p. (In Russ.)
24. Danilin G. A., Zaterukha E. V., Filin D. S., Tereshchenko V. G. Design and technological support for the reliability of small arms cartridges. BSTU “VOENMEH”. St. Petersburg, 2017. 493 p. (In Russ.)
25. Smirnov-Alyaev G. A. Resistance of materials to plastic deformation. 3rd ed. L.: Mechanical Engineering, 1978. 368 p. (In Russ.)
26. Kuznetsov D. P., Nesterov N. I., Ivanov K. M. Theory of plasticity and theory of metal forming: laboratory practical training [for universities]. BSTU “VOENMEH”. St. Petersburg, 2007. 77 p. (In Russ.)