In standard linear finite-element formulations, volumetric locking because of the incompressibility constraint that may occur in computational plasticity is often encountered. This study uses crossed patch arrangements of triangles to form quadrilateral elements in order to overcome the locking in the upper bound finite-element analysis of plane strain deformation problems. The velocity field is described in terms of linear triangular elements, while the incompressibility constraint is imposed by quadrilateral elements. Rigid, perfectly plastic materials, and strain hardening materials that form the von Mises model have been considered. The velocity formulation is presented and has been implemented in a finite-element code. Several examples, some benchmarks problems, are presented to illustrate the applicability of the approach for predicting the load, strain, and velocity field during the plastic deformation. Numerical results show that the crossed patch arrangements of linear triangular elements are free of volumetric locking and achieve well-defined limit loads. This study shows that the presented method can be used to simulate large plastic deformation under plane strain conditions.
HillR., The mathematical theory of plasticity (USA: Oxford University Press1998) First published in 1950.
3.
JohnsonW., ‘Over-estimates of load for some two-dimensional forging operations’ -estimates of load for some two-dimensional forging operationsProceedings of the Third US Congress on Applied mechanics, Providence, Rhode Island, USA, 1958, pp. 571–579.
KudoH., ‘An upper-bound approach to plane-strain forging and extrusion - I’Int. J. Mech. Sci.1 (1961): 57–83.
6.
KudoH., ‘Some analytical and experimental studies of axisymmetric cold forging and extrusion - I’Int. J. Mech. Sci21 (1960): 102–127.
7.
KobayashiS., ‘Upper-bound solution of axisymmetric forming problems - I’Trans. ASME, B, J. Engng Ind.86 (1961): 122–126.
8.
KobayashiS., ‘Upper-bound solution of axisymmetric forming problems - II’Trans. ASME, B, J. Engng Ind.86 (1961): 326–332.
9.
McDermottR. P.BramleyA. N., ‘Forging analysis - a new approach’Metall. Metal Forming415 (1974): 127–129.
10.
LugoraC. F.BramleyA. N., ‘Three-dimensional analysis of closed-die forging processes - I. Formulation’Proc. IMechE, Part B: J. Engineering Manufacture203 (1981): 33–37 DOI: 10.1243/PIME_PROC_203_043_02.
11.
LeeR. S.KwanC. T., ‘Upper-bound elemental technique (UBET) with convex circular parallelepiped and convex spherical elements for three-dimensional forging analysis’Int. J. Mach. Tools Manuf.378 (1997): 1053–1067.
12.
LeeR. S.KwanC. T., ‘A three-dimensional upper bound elemental technique for forging analysis’Proc. IMechE, Part B: J. Engineering Manufacture2104 (1996): 353–359. DOI: 10.1243/PIME_PROC_210_128_02.
13.
YangD. Y.KimJ. H.LiraC. K., ‘An arbitrarily inclined triangular UBET element and its application to combined forging’ ASME, B, J. Engng Ind. (1985): 107, 134.
14.
KiuchiM.KaratoA., ‘Application of UBET to nonaxisymmetric forging’Adv. Technol. Plast.2 (1981): 967–972.
15.
SloanS. W., ‘Upper bound limit analysis using finite elements and linear programming’Int. J. Numer. Anal. Methods Geomech.13 (1981): 263–282.
16.
SloanS. W.KleemanP. W., ‘Upper bound limit analysis using discontinuous velocity fields’Comput. Methods Appl. Mech. Eng.1271–4 (1995): 293–314.
17.
CapsoniA.CorradiL., ‘A finite element formulation of the rigid-plastic limit analysis problem’Int. J. Numer. Methods Eng.4011 (1997):2063–2086.
18.
HuhH.LeeC. H., ‘Eulerian finite-element modeling of the extrusion process for work-hardening materials with the extended concept of limit analysis’J. Mater. Process. Technol.381 (1993): 51–62.
19.
YangW. H., ‘Large deformation of structures by sequential limit analysis’Int. J. Solids Struct.307 (1993): 1001–1013.
20.
Pitt-FrancisJ. M.BowyerA.BramleyA. N., ‘TEUBA: a simple 3D formulation for modelling forging using the upper bound method’CIRP Ann.-College Int. Rech. Prod.451 (1996): 245–248.
21.
HwanC. L., ‘An upper bound finite element procedure for solving large plane strain deformation’Int. J. Numer. Methods Eng., 1997 40 (10): 1909–1922.
22.
HwanC. L., ‘Plane strain extrusion by sequential limit analysis’Int. J. Mech. Sci.397 (1997): 807–817.
23.
ChangC. C.BramleyA. N., ‘Forging preform design using a reverse simulation approach with the upper bound finite element procedure’Proc. IMechE, Part C: J. Mechanical Engineering Science2141 (2000): 127–1362. DOI: 10.1243/0954406001522868.
24.
BrezziF.FortinM., Mixed and hybrid finite element methods (New York: Springer-Verlag1991) Springer Series in Computational Mathematics, 15.
25.
ZienkiewiczO. C.TaylorR. L.ZhuJ. Z., The finite element method: its basis and fundamentals (Oxford: Butterworth-Heinemann2005).
26.
WagonerR. H.ChenotJ. L., Metal forming analysis (Cambridge: Cambridge University Press2001).
27.
KobayashiS.OhS. I.AltanT., Metal forming and the finite-element method (New York: Oxford University Press1989).
28.
ArnoldD. N.BrezziF.FortinM., ‘A stable finite element for the Stokes equations’Calcolo214 (1984): 337–344.
29.
TaylorR. L., ‘A mixed-enhanced formulation tetrahedral finite elements’Int. J. Numer. Methods Eng.471–3 (2000): 205–227.
30.
HughesT. J. R., ‘Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods’Comput. Methods Appl. Mech. Eng.1271–4 (1995): 387–401.
31.
ChiumentiM.ValverdeQ.de SaracibarC. AgeletCerveraM., ‘A stabilized formulation for incompressible plasticity using linear triangles and tetrahedra’Int. J. Plast.208–9 (2004): 1487–1504.
32.
OnateE.RojekJ.TaylorR. L.ZienkiewiczO. C., ‘Finite calculus formulation for incompressible solids using linear triangles and tetrahedra’Int. J. Numer. Methods Eng.5911 (2004): 1473–1500.
33.
HerrmannL., ‘Finite element bending analysis for plates’J. Eng. Mech. Div. ASCE EM593 (1961): 13–26.
34.
NagtegaalJ. C.ParksD. M.RiceJ. R., ‘On numerically accurate finite element solutions in the fully plastic range’Comput. Methods Appl. Mech. Eng.42 (1974): 153–177.
35.
BohatierC.ChenotJ. L., ‘Finite element formulations for non-steady-state large viscoplastic deformation’Int. J. Numer. Methods Eng.219 (1985): 1697–1708.
36.
Le TallecP., ‘Numerical solution of viscoplastic flow problems by augmented Lagrangians’IMA J. Numer. Anal.62 (1986): 185–219.
37.
SquillacoteA., The ParaView guide (New York: Kitware2008).
38.
HosfordW. F.CaddellR. M., Metal forming: mechanics and metallurgy (USA: PTR Prentice-Hall1993).
