On the Relations between the ‘Flow’ and ‘Solid’ Approaches in the Metal Forming Process

  • Wen-chao Zhang
Conference paper


In this paper we attempt to clarify the relation between the ‘Flow’ and ‘Solid’ approaches to the numerical analysis of the metal-forming process in order to set up a physically meaningful foundation for the ‘flow’ approach. Such clarification provides a better understanding of the viscoplastic flow formulation as applied to metal forming rather than relying on the analogy between Newtonian fluid flow and the behaviour of an incompressible elastic solid. In addition, some limitations of the ‘Flow’ approach are clearly exposed.


Finite Element Analysis Constitutive Relation Deformation Theory Flow Formulation Incremental Theory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Goon, G.Y., Poluchin, P.I., Poluchin, W.P. and Prudcowsky, B.A. ‘The plastic deformation of metals’, Metallurgica Moscow (in Russian) (1968).Google Scholar
  2. [2]
    Cornfield, G.C. and Johnson, R.H. ‘Theoretical prediction of flow in hot rolling including the effect of various temperature distributions’, J. Iron and Steel Inst. Vol. 211, PP. 567–573 (1973).Google Scholar
  3. [3]
    Lee, C.H. and Kobayashi, S. ‘New solution to rigid plastic deformation problems using a matrix method’, J. Eng. for Ind., Trans. ASME, Vol.95, PP. 865–873 (1973).CrossRefGoogle Scholar
  4. [4]
    Zienkiewicz, O.C. and Godbole, P.N. ‘Flow of plastic and viscoplastic solids with special reference to extrusion and forming processes’, Int. J. Num. Meth. in Eng., Vol.8, PP. 3–16 (1974).MATHGoogle Scholar
  5. [5]
    McMeeking, R.M. and Rice, R.J. ‘Finite element formulations for problems of large elastic-plastic deformations’, Int. J. Solids and Structures, Vol. 11, P.601 (1975).MATHCrossRefGoogle Scholar
  6. [6]
    Lee, E.H. and Mallett, R.L. ‘Stress and deformation analysis of the metal extrusion process’, Com. Meth. in Appl. Mech. Engng., Vol. 10, PP. 339–353 (1977).CrossRefGoogle Scholar
  7. [7]
    Nagtegaal, J.C. and De Jong, J.E. ‘Some computational aspects of elastic-plastic large strain analysis’, Int. J. Num. Meth. in Eng., Vol. 17, P.15 (1981).MATHCrossRefGoogle Scholar
  8. [8]
    Nagtegaal, J.C. ‘On the implementation of inelastic constitutive equations with special reference to large deformation problems’, presented at Fenomech II, Stuttgart, (1981).Google Scholar
  9. [9]
    Dawson, P.R. and Thompson, E.G. ‘Finite element analysis of steady state elasto-viscoplastic flow by the initial stress rate method’, Int. J. Num. Meth. in Eng., Vol.12, PP.47–57, 382–383 (1978).MATHCrossRefGoogle Scholar
  10. [10a]
    Shimizaki, K. and Thompson, E.G. ‘Elasto-visco-plastic flow with special attention to boundary conditions’, Int., J. Num. Meth. in Eng., Vol. 17, PP. 97–112 (1981).CrossRefGoogle Scholar
  11. [10b]
    Thompson, E.G. ‘Finite element analysis of steady state elasto- viscoplastic flows in metal forming’, Pre-Conference Course, Swansea, 21st Int. Machine Tool Design and Res. Conf. (1980).Google Scholar
  12. [11]
    Zienkiewicz, O.C. ‘Flow formulation for numerical solution of forming processes’ PP.1–44, of Numerical Analysis of Forming Processes, ed. Pittman et al. Wiley and Sons (1984).Google Scholar
  13. [12]
    Chandra, A. and Mukherjee, S. ‘A finite element analysis of metal- forming problems with an elastic-viscoplastic materials model’, Int. J. Num. Method in Eng., Vol. 20, PP. 1613–1628 (1984).MATHCrossRefGoogle Scholar
  14. [13]
    Zienkiewicz, O.C. and Godbole, P.N. ‘Viscous incompressible flow with special reference to Non-Newtonian (plastic) fluids’, F.E. in Fluids, Vol.1, Wiley & Sons (1975).Google Scholar
  15. [14]
    Prager, W. ‘An elementary discussion of definition of stres rate’, Quart. Appl. Math., Vol. 18, p. 403 (1961).MathSciNetMATHGoogle Scholar
  16. [15]
    Hill, R. ‘The Mathematical theory of plasticity’, Oxford (1950).MATHGoogle Scholar
  17. [16]
    Kachanov, L.M. ‘Fundamentals of the theory of plasticity’, (English Translation), Mir Publishers (1974).Google Scholar
  18. [17]
    Prandtl, L. ‘On the penetration hardness of plastic materials and hardness of indentors’, Z. Angew, Math. Meth., Vol. 1, No. 15 (1921).Google Scholar
  19. [18]
    Reuss, A. A. ang Math. Mech. Vol.1 P.266 (1930).CrossRefGoogle Scholar
  20. [19]
    Mises R. Von, ‘Mechanik der festen korper imm plastisch deformablen Zustand’, Machr. Akad. Wiss. Gottingen., Math. Phys. Kl., H.4., PP.582–592 (1913).Google Scholar
  21. [20]
    Levy, M. ‘Memoire sur les equations generales des monvement interieurs des corps soliods ductitles ou deta des limites on 1 elasticite pourrait les ramener a leur premier etat’, J. Math. Pures et Appl. ser. II, Vol. 16, PP. 369–372 (1871).Google Scholar
  22. [21]
    Hencky, H. ‘Zur Theorie Plastischer deformationen und der Hierdurch im material herforgerufenen nashspannungen’, Z. angew. Math. und Mech., Vol.4, No.4, PP. 323–334 (1924).CrossRefGoogle Scholar
  23. [22]
    Zhang, W.C., ‘Finite element analysis of metal forming process’, Ph.D. Thesis, University College of Swansea, Swansea, U.K. (1986).Google Scholar
  24. [23]
    Walters, K. ‘Rheometry’, London: Chapman Hall (1975).Google Scholar
  25. [24]
    Caddell, R.M., ‘Deformation and fracture of solids’, Prentice-Hall, Inc. (1980).Google Scholar
  26. [25]
    Davies, A.R., Lee, S.J. and Webster, M.F., ‘Numerical simulations of viscoelastic flow: The effect of mesh size’, J. of Non-Newtonian Fluid Mechanics, Vol. 16, PP. 117–139, (1984).MATHCrossRefGoogle Scholar

Copyright information

© Springer Japan 1986

Authors and Affiliations

  • Wen-chao Zhang
    • 1
  1. 1.Department of MechanicsTianjin UniversityTianjinChina

Personalised recommendations