REVIEW
Selected issues in the upper bound theorem of plasticity
Sergei Alexandrov,
Elena Lyamina,
Yeau-Ren Jeng,
Sergei Alexandrov
Laboratory for Technological Processes, Ishlinsky Institute for Problems in Mechanics RAS, Moscow, Russia
Search for more papers by this authorElena Lyamina
Laboratory for Technological Processes, Ishlinsky Institute for Problems in Mechanics RAS, Moscow, Russia
Search for more papers by this authorCorresponding Author
Yeau-Ren Jeng
Academy of Innovative Semiconductor and Sustainable Manufacturing, National Cheng Kung University, Tainan, Taiwan
Department of Biomedical Engineering, National Cheng Kung University, Tainan, Taiwan
Correspondence
Yeau-Ren Jeng, Department of Biomedical Engineering, National Cheng Kung University, No. 1, University Road, East District, Tainan 70101, Taiwan.
Email: [email protected]
Search for more papers by this authorSergei Alexandrov,
Elena Lyamina,
Yeau-Ren Jeng,
Sergei Alexandrov
Laboratory for Technological Processes, Ishlinsky Institute for Problems in Mechanics RAS, Moscow, Russia
Search for more papers by this authorElena Lyamina
Laboratory for Technological Processes, Ishlinsky Institute for Problems in Mechanics RAS, Moscow, Russia
Search for more papers by this authorCorresponding Author
Yeau-Ren Jeng
Academy of Innovative Semiconductor and Sustainable Manufacturing, National Cheng Kung University, Tainan, Taiwan
Department of Biomedical Engineering, National Cheng Kung University, Tainan, Taiwan
Correspondence
Yeau-Ren Jeng, Department of Biomedical Engineering, National Cheng Kung University, No. 1, University Road, East District, Tainan 70101, Taiwan.
Email: [email protected]
Search for more papers by this authorFirst published: 24 February 2025
Abstract
The paper reviews the upper bound theorem in rigid plasticity, emphasizing its application to analyzing and designing metal forming processes. Two formulations of the theorem are considered. The peculiarities in applying the theorem to non-stationary and stationary processes are discussed. Special attention is devoted to friction boundary conditions. General methods of constructing trial velocity fields are presented. On the other hand, routine upper bound solutions based on the von Mises yield criterion are not reviewed, though their applied significance is not questionable. The article ends with a simple numerical example that illustrates all qualitative features of the theorem considered.
REFERENCES
- 1Avitzur, B.: Metal forming: the application of limit analysis. Annu. Rev. Mater. Sci. 7, 261–300 (1977). http://doi.org/10.1146/annurev.ms.07.080177.001401
- 2Zhou, W., Shi, Z., Lin, J.: Upper bound analysis of differential velocity sideways extrusion process for curved profiles using a fan-shaped flow line model. Int. J. Lightweight Mater. 1, 21–32 (2018). http://doi.org/10.1080/15397734.2023.2165100
10.1080/15397734.2023.2165100 Google Scholar
- 3Hamidpour, S., Parvizi, A., Nosrati, A.S.: Upper bound analysis of wire flat rolling with experimental and FEM verifications. Meccanica 54, 2247–2261 (2019). http://doi.org/10.1007/s11012-019-01066-4
10.1007/s11012-019-01066-4 Google Scholar
- 4Sheikhpour, M., Hosseinipour, S., Mirnia, M.: Prediction of exit profile distortion in forward extrusion process using Riemann mapping theorem and upper bound method. Meccanica 55, 1099–1118 (2020). http://doi.org/10.1007/s11012-020-01141-1
10.1007/s11012-020-01141-1 Google Scholar
- 5Agarwal, M., Singh, S.: Upper bound analysis of closed-die forging of eccentrically-located SiCp AMC preforms. J. Appl. Sci. Eng. 25, 275–285 (2022). https://doi.org/10.6180/jase.202204_25(2).0004
10.6180/jase.202204_25(2).0004 Google Scholar
- 6Jo, D., Moon, Y., Kim, J.: Upper bound analysis of friction stir spot welding of 6061-T6 aluminum alloys. Int. J. Adv. Manuf. Technol. 120, 8311–8320 (2022). http://doi.org/10.1007/s00170-022-09294-x
10.1007/s00170-022-09294-x Google Scholar
- 7Zhang, X., Li, F., Wang, Y., Chen, Z.: An analysis for magnesium alloy curvature products formed by staggered extrusion (SE) based on the upper bound method. Int. J. Adv. Manuf. Technol. 119, 303–313 (2022). http://doi.org/10.1007/s00170-021-08167-z
10.1007/s00170-021-08167-z Google Scholar
- 8Farokhpey, A., Parsa, M.: Analyzing the accumulative roll bonding deformation zone behavior by FEM, upper bound, and experimental methods. J. Manuf. Process. 81, 328–345 (2022). http://doi.org/10.1016/j.jmapro.2022.07.003
10.1016/j.jmapro.2022.07.003 Google Scholar
- 9Bermudo, C., Sevilla, L., Martín, F., Trujillo, F.: Study of the tool geometry influence in indentation for the analysis and validation of the new modular upper bound technique. Appl. Sci. 6, 203 (2016). http://doi.org/10.3390/app6070203
10.3390/app6070203 Google Scholar
- 10Kamenjarzh, J.: Limit Analysis of Solids and Structures. CRC Press, Boca Ration (1996)
- 11Füssl, J., Lackner, R., Eberhardsteiner, J., Mang, H.: Failure modes and effective strength of two-phase materials determined by means of numerical limit analysis. Acta Mech. 195, 185–2002 (2008). http://doi.org/10.1007/s00707-007-0550-9
10.1007/s00707-007-0550-9 Google Scholar
- 12Zerbst, U., Ainsworth, R., Schwalbe, K.-H.: Basic principles of analytical flaw assessment methods. Int. J. Press. Vessels Pip. 77(14), 855–867 (2000). http://doi.org/10.1016/S0308-0161(01)00008-4
10.1016/S0308-0161(01)00008-4 Google Scholar
- 13Chen, W.-F.: Limit Analysis and Soil Plasticity. Elsevier, Amsterdam (1975)
- 14Kobayashi, S., Oh, S.-I., Altan, T.: Metal Forming and the Finite-Element Method. Oxford University Press, New York (1989)
10.1093/oso/9780195044027.001.0001 Google Scholar
- 15Pastor, F., Loute, E., Pastor, J., Trillat, M.: Mixed method and convex optimization for limit analysis of homogeneous Gurson materials: a kinematical approach. Eur. J. Mech. A-Solid 28(1), 25–35 (2009). http://doi.org/10.1016/j.euromechsol.2008.02.008
- 16Pastor, J., Thoré, P., Pastor, F.: Limit analysis and numerical modeling of spherically porous solids with Coulomb and DruckerPrager matrices. J. Comput. Appl. Math. 234(7), 2162–2174 (2010). http://doi.org/10.1016/j.cam.2009.08.079
10.1016/j.cam.2009.08.079 Google Scholar
- 17Hill, R.: The Mathematical Theory of Plasticity. Clarendon Press, Oxford (1950)
- 18Hill, R.: New horizons in the mechanics of solids. J. Mech. Phys. Solids 5, 66–74 (1956). http://doi.org/10.1016/0022-5096(56)90009-6
- 19Collins, I.: The upper bound theorem for rigid/plastic solids generalized to include Coulomb friction. J. Mech. Phys. Solids 17, 323–338 (1969). http://doi.org/10.1016/0022-5096(69)90021-0
- 20Das, N., Joardar, H., Rout, M., Behera, B., Maity, K.: On generalized upper bound method and its application to some plane strain compression and extrusion problems. Adv. Mater. Technol. 46, 7–18 (2021). http://doi.org/10.24867/ATM-2021-2-002
10.24867/ATM?2021?2?002 Google Scholar
- 21Tsai, S., Alexandrov, S., Strashnov, S.: Application of Collins upper bound theorem to upsetting processes. Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci. 238(9), 4101–4111 (2024). http://doi.org/10.1177/09544062231207211
10.1177/09544062231207211 Google Scholar
- 22Drucker, D.C., Prager, W., Greenberg, H.J.: Extended limit design theorems for continuous media. Q. Appl. Math. 9(4), 381–389 (1952)
- 23Tamai, Y., Inazumi, T., Manabe, K.: Fe forming analysis with nonlinear friction coefficient model considering contact pressure, sliding velocity and sliding length. J. Mater. Process. Technol. 227, 161–168 (2016). http://doi.org/10.1016/j.jmatprotec.2015.08.023
- 24Gil, I., Mendiguren, J., Galdos, L., Mugarra, E., Sáenz de Argando na, E.: Influence of the pressure dependent coefficient of friction on deep drawing springback predictions. Trib. Int. 103, 266–273 (2016). http://doi.org/10.1016/j.triboint.2016.07.004
- 25Zabala, A., Sáenz de Argando na, E., Ca nizares, D., Llavori, I., Otegi, N., Mendiguren, J.: Numerical study of advanced friction modelling for sheet metal forming: Influence of the die local roughness. Trib. Int. 165, 107259 (2022). http://doi.org/10.1016/j.triboint.2021.107259
10.1016/j.triboint.2021.107259 Google Scholar
- 26Druyanov, B., Nepershin, R.: Problems of Technological Plasticity. Elsevier Science, Amsterdam (1994)
- 27Puzrin, A., Randolph, M.: New planar velocity fields for upper bound limit analysis. Int. J. Solids Struct. 40, 3603–3619 (2003). http://doi.org/10.1016/S0020-7683(03)00152-5
- 28Wang, G., Du, H., Zhao, D., Liu, X., W. G.: Application of geometric midline yield criterion for strip drawing. Iron Steel Res. Int. 16, 13–16 (2009). http://doi.org/10.1016/S1006-706X(10)60020-9
- 29Hosford, W.F.: A generalized isotropic yield criterion. J. Appl. Mech. 39, 607–609 (1972). http://doi.org/10.1115/1.3422732
- 30Altenbach, H., Kolupaev, V.A.: Reviewing yield criteria in plasticity theory. In: H. Altenbach, J. Hohe, C. Mittelstedt (eds.) Progress in Structural Mechanics. Advanced Structured Materials, vol 199, pp. 19–106. Springer International Publishing, Cham (2024)
10.1007/978-3-031-45554-4_2 Google Scholar
- 31Alexandrov, S., Lyamina, E., Jeng, Y.-R.: Application of the upper bound theorem for metal forming processes considering an arbitrary isotropic pressure-independent yield criterion with no strength differential effect. Int. J. Adv. Manufact. Technol. 126, 3311–3321 (2023). http://doi.org/10.1007/s00170-023-11312-5
10.1007/s00170-023-11312-5 Google Scholar
- 32Alexandrov, S., Vilotic, M., Dacevic, N., Li, Y.: Identification of Hosfords yield criterion using compression tests. Metals 13, 471 (2023). http://doi.org/10.3390/met13030471
- 33Alexandrov, S., Strashnov, S., Li, Y.: An upper bound solution for axisymmetric extrusion and drawing considering a generalized yield criterion. Metals 13(3), 602 (2023). http://doi.org/10.3390/met13030602
10.3390/met13030602 Google Scholar
- 34Druyanov, B.: Technological Mechanics of Porous Bodies. Oxford University Press, New York (1993)
- 35Green, R.: A plasticity theory for porous solids. Int. J. Mech. Sci. 14, 215–224 (1972). http://doi.org/10.1016/0020-7403(72)90063-X
- 36Shima, S., Oyane, M.: Plasticity theory for porous metals. Int. J. Mech. Sci. 18, 285–291 (1976). http://doi.org/10.1016/0020-7403(76)90030-8
- 37Doraivelu, S., Gegel, H., Gunasekera, J., Malas, J., Morgan, J., Thomas, J.: A new yield function for compressible PM materials. Int. J. Mech. Sci. 26, 527–535 (1984). http://doi.org/10.1016/0020-7403(84)90006-7
- 38Hwang, Y., Yin, S., Yu, H., Tsai, Y.: Finite element analysis of rotating compression forming of powder materials. Int. J. Adv. Manuf. Technol. 123, 793–807 (2022). http://doi.org/10.1007/s00170-022-10218-y
10.1007/s00170-022-10218-y Google Scholar
- 39Chandra, N.: Constitutive behavior of superplastic materials. Int. J. Non Linear Mech. 37(3), 461–484 (2002). http://doi.org/10.1016/S0020-7462(01)00021-X
10.1016/S0020-7462(01)00021-X Google Scholar
- 40Tully, K., Monaghan, J.: Bulk forming of superplastic alloys. J. Mater. Process. Technol. 26(2), 159–171 (1991). http://doi.org/10.1016/0924-0136(91)90130-7
10.1016/0924-0136(91)90130-7 Google Scholar
- 41Dialami, N., Cervera, M., Chiumenti, M., Agelet de Saracibar, C.: A fast and accurate two-stage strategy to evaluate the effect of the pin tool profile on metal flow, torque and forces in friction stir welding. Int. J. Mech. Sci. 122, 215–227 (2017). http://doi.org/10.1016/j.ijmecsci.2016.12.016
- 42Milesi, M., Lecoq, J., Pradille, C., Vitu, L., Boudeau, N., Bouchard, P.-O.: Impact of strain rate sensitivity on the identification of the material parameters scattering and on the formability of zinc sheet. Int. J. Mater. Form. 13, 203–218 (2020). http://doi.org/10.1007/s12289-019-01479-2
10.1007/s12289-019-01479-2 Google Scholar
- 43Zheng, Z., Chen, Y., Kong, F., Wang, X., Yu, Y.: Hot deformation behavior and hot rolling properties of a nano- addition near- titanium alloy. Metals 11(5), 837 (2021). http://doi.org/10.3390/met11050837
- 44Vakhrushev, A., Kharicha, A., Wu, M., Ludwig, A., Nitzl, G., Tang, Y., Hackl, G., Watzinger, J., Rodrigues, C.M.G.: Norton-Hoff model for deformation of growing solid shell of thin slab casting in funnel-shape mold. J. Iron Steel Res. Int. 29, 88–102 (2022). http://doi.org/10.1007/s42243-021-00734-8
- 45Oldroyd, J.G.: A rational formulation of the equations of plastic flow for a Bingham solid. Math. Proc. Cambridge Philos. Soc. 43, 100–105 (1947). http://doi.org/10.1017/S0305004100023239
- 46Taylor-West, J., Hogg, A.: Viscoplastic flow between hinged plates. J. Fluid Mech. 958, A38 (2023).
10.1017/jfm.2023.106 Google Scholar
- 47Alexandrov, S.: An analysis of the axisymmetric compression of viscous materials. J. Mater. Process. Technol. 105(3), 278–283 (2000). http://doi.org/10.1016/S0924-0136(00)00652-X
10.1016/S0924-0136(00)00652-X Google Scholar
- 48Alexandrov, S., Alexandrova, N.: On the maximum friction law in viscoplasticity. Mech. Time-Depend. Mat. 4, 99–104 (2000). http://doi.org/10.1023/A:1009851621518
- 49Cristescu, N.: Plastic flow through conical converging dies, using a viscoplastic constitutive equation. Int. J. Mech. Sci. 17(6), 425–433 (1975). http://doi.org/10.1016/0020-7403(75)90040-5
10.1016/0020-7403(75)90040-5 Google Scholar
- 50Forcellese, A., Gabrielli, F., Micari, F., Zurla, O.: Finite element modelling of isothermal forging of a high strength aluminium alloy. J. Mater. Process. Technol. 34(1), 77–84 (1992). http://doi.org/10.1016/0924-0136(92)90092-7
10.1016/0924-0136(92)90092-7 Google Scholar
- 51Sahi, M., Rahouadj, R., Herbach, R., Choulier, D.: The influence of viscoplasticity in the interpretation of the ring test. J. Mater. Process. Technol. 58(2), 286–292 (1996). http://doi.org/10.1016/0924-0136(95)02151-5
10.1016/0924-0136(95)02151-5 Google Scholar
- 52Abrinia, K., Fazlirad, A.: Investigation of single pass shape rolling using an upper bound method. J. Mater. Eng. Perform. 19, 541–552 (2010). http://doi.org/10.1007/s11665-009-9511-x
- 53Richmond, O.: Plane strain necking of V-notched and un-notched tensile bars. J. Mech. Phys. Solids 17, 83–90 (1969). http://doi.org/10.1016/0022-5096(69)90036-2
10.1016/0022-5096(69)90036-2 Google Scholar
- 54Yang, W.: Large deformation of structures by sequential limit analysis. Int. J. Solids Struct. 30(7), 1001–1013 (1993). http://doi.org/10.1016/0020-7683(93)90023-Z
- 55Leblond, J.-B., Kondo, D., Morin, L., Remmal, A.: Classical and sequential limit analysis revisited. Comptes Rendus. Mécanique 346(4), 336–349 (2018). http://doi.org/10.1016/j.crme.2017.12.015
10.1016/j.crme.2017.12.015 Google Scholar
- 56Johnson, R.W., Rowe, G.W.: Bulge formation in strip drawing with light reductions in area. Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci. 182, 521–530 (1967). http://doi.org/10.1243/PIME_PROC_1967_182_040_02
10.1243/PIME_PROC_1967_182_040_02 Google Scholar
- 57Zhang, J., Merkwae, M., Baker, G.: Bulge formation in strip drawing with light reduction. J. Appl. Metalworking 17, 342–350 (1987). http://doi.org/10.1007/BF02833945
10.1007/BF02833945 Google Scholar
- 58Azarkhin, A., Richmond, O.: A model of ploughing by a pyramidal indenter Upper bound method for stress-free surfaces. Wear 157, 409–418 (1992). http://doi.org/10.1016/0043-1648(92)90077-L
- 59Hartley, C.: Upper bound analysis of extrusion of axisymmetric, piecewise homogeneous tubes. Int. J. Mech. Sci. 15, 651–663 (1973). http://doi.org/10.1016/0020-7403(73)90097-0
- 60Marques, M.B., Martins, P.: A solution to plane strain rolling by the weighted residuals method. Int. J. Mech. Sci. 32, 817–827 (1990). http://doi.org/10.1016/0020-7403(90)90161-B
10.1016/0020-7403(90)90161-B Google Scholar
- 61Oxley, P.: Note: Allowing for friction in estimating upper bound loads. Int. J. Mech. Sci. 5(2), 183–184 (1963). http://doi.org/10.1016/0020-7403(63)90021-3
10.1016/0020-7403(63)90021-3 Google Scholar
- 62Bramley, A., Davies, B.: Computer aided forging design. CIRP Annals 36, 135–138 (1987). http://doi.org/10.1016/S0007-8506(07)62571-2
10.1016/S0007-8506(07)62571-2 Google Scholar
- 63Lin, Y., Wang, J.: A new upper-bound elemental technique approach. Comput. Struct. 65(4), 601–611 (1997). http://doi.org/10.1016/S0045-7949(94)E0287-C
10.1016/S0045-7949(94)E0287-C Google Scholar
- 64Wilson, W.: A simple upper-bound method for axisymmetric metal forming problems. Int. J. Mech. Sci. 19(2), 103–112 (1977). http://doi.org/10.1016/0020-7403(77)90004-2
10.1016/0020-7403(77)90004-2 Google Scholar
- 65Nagpal, V.: General kinematically admissible velocity fields for some axisymmetric metal forming problems. J. Eng. Ind. 96(4), 1197–1201 (1974). http://doi.org/10.1115/1.3438495
10.1115/1.3438495 Google Scholar
- 66Osakada, K., Niimi, Y.: A study on radial flow field for extrusion through conical dies. Int. J. Mech. Sci. 17(4), 241–254 (1975). http://doi.org/10.1016/0020-7403(75)90006-5
10.1016/0020-7403(75)90006-5 Google Scholar
- 67Asgharzadeh, A., Serajzadeh, S.: Development of a stream function-upper bound analysis applicable to the process of plate rolling. Multidiscip. Model. Mater. 12, 254–274 (2016). http://doi.org/10.1108/MMMS-06-2015-0029
- 68Alexandrov, S., Richmond, O.: Singular plastic flow fields near surfaces of maximum friction stress. Int. J. Non-Linear Mech. 36, 1–11 (2001). http://doi.org/10.1016/S0020-7462(99)00075-X
- 69Alexandrov, S., Mishuris, G., Mishuris, W., Sliwa, R.: On the dead zone formation and limit analysis in axially symmetric extrusion. Int. J. Mech. Sci. 43, 367–379 (2001). http://doi.org/10.1016/S0020-7403(00)00016-3
10.1016/S0020-7403(00)00016-3 Google Scholar
- 70Alexandrov, S., Jeng, Y.-R.: Singular rigid/plastic solutions in anisotropic plasticity under plane strain conditions. Cont. Mech. Therm. 25, 685–689 (2013). http://doi.org/10.1007/s00161-013-0304-y
- 71Alexandrov, S., Mishuris, G.: Qualitative behaviour of viscoplastic solutions in the vicinity of maximum-friction surfaces. J. Eng. Math. 65, 143–156 (2009). http://doi.org/10.1007/s10665-009-9277-z
- 72Sińczak, J., Kusiak, J., Łapkowski, W., Okoń, R.: The influence of deformation conditions on the flow of strain rate sensitive materials. J. Mater. Process. Technol. 34(1), 219–224 (1992). http://doi.org/10.1016/0924-0136(92)90110-E
10.1016/0924-0136(92)90110-E Google Scholar
- 73Collins, I.F.: Compression of a rigid-perfectly plastic strip between parallel rotating smooth dies. Q. J. Mech. Appl. Math. 23(3), 329–348 (1970). http://doi.org/10.1093/qjmam/23.3.329
10.1093/qjmam/23.3.329 Google Scholar
- 74Oh, H., Mun, J.: An analysis of the ring upsetting of sintered material. J. Mech. Work. Technol. 9(3), 279–290 (1984). http://doi.org/10.1016/0378-3804(84)90109-8
10.1016/0378-3804(84)90109-8 Google Scholar
- 75Pindra, N., Leblond, J., Kondo, D.: Limit-analysis of a circular cylinder obeying the Green plasticity criterion and loaded in combined tension and torsion. Meccanica 53, 2437–2446 (2018). http://doi.org/10.1007/s11012-018-0833-3
10.1007/s11012-018-0833-3 Google Scholar
- 76Hao, S., Cornec, A., Schwalbe, K.-H.: Plastic stress-strain fields and limit loads of a plane strain cracked tensile panel with a mismatched welded joint. Int. J. Solids Struct. 34(3), 297–326 (1997). http://doi.org/10.1016/S0020-7683(96)00021-2
10.1016/S0020-7683(96)00021-2 Google Scholar
- 77Alexandrov, S., Lyamina, E., Jeng, Y.-R.: Effect of weld geometry on the limit load of a cracked specimen under tension. Mech. Based Des. Struct. Mach. 52(4), 1998–2016 (2024). http://doi.org/10.1080/15397734.2023.2165100
10.1080/15397734.2023.2165100 Google Scholar
