Abstract
In Chapter 5, we discussed the details of finite element method as well as the finite element modeling of metal forming processes using Eulerian formulation. In this chapter, we extend the finite element technique to the updated Lagrangian formulation. In Eulerian formulation, the domain is a fixed region in space (called control volume). However, in a Lagrangian formulation, the domain consists of a set of material particles that changes its shape continuously with the deformation. The updated Lagrangian formulation is an incremental method in which the domain is updated incrementally. Further, the measure of deformation used in Eulerian formulation is the rate of deformation tensor and the constitutive equation is expressed in terms of the stress and rate of deformation tensors. On the other hand, in updated Lagrangian formulation, the measure of deformation is an incremental strain tensor and the constitutive equation is expressed in terms of the incremental stress and incremental strain tensors. We shall discuss how finite element modeling needs to be modified in the light of these changes in the governing equations. Like that of the Eulerian formulation, the governing equations of the updated Lagrangian formulation also are non-linear and need an iterative scheme to obtain a solution. But the iterative scheme we adopt here is different from that of the previous chapter. However, like in the previous chapter, here also we adopt the Galerkin formulation for developing finite element equations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
6.6 References
Malvern, L.E. (1969), Introduction to the Mechanics of a Continuous Medium, Prentice-Hall Inc., Englewood Cliffs
Crisfield, M.A. (1991), Non-linear Finite Element Analysis of Solids and Structures, Vol. 1: Essentials, John Wiley and Sons, Chichester.
Bathe, K.J. (1996), Finite Element Procedures, Prentice-Hall of India, New Delhi.
Altan, T. (1971), Computer simulation to predict load, stress and metal flow in an axisymmetric closed die forging, in A.L. Hoffmanner (Ed.), Metal Forming, Plenum Press, New York, pp. 249–274.
Hill, R., Lee, E.H. and Tupper, S.J. (1923), A method of numerical analysis of plastic flow in plane strain and its application to the compression of a ductile material between rough platens, Transaction of ASME, Journal of Applied Mechanics, Vol. 73, pp. 46–52.
Green, A.P. (1951), A theoretical investigation of the compression of a ductile material between smooth flat dies, Philosphical Magazine, Vol. 42, pp. 900–918.
Kudo, H. (1960), Some analytical and experimental studies of axisymmetric cold forging and extrusion, I and II, International Journal of Mechanical Sciences, Vol. 2, pp. 102–127.
Avitzur, B. (1968), Metal Forming: Processes and Analysis, McGraw-Hill, New York.
Lee, C.H. and Kobayashi, (1971), Analysis of axisymmetric upsetting and plane strain side-pressing of solid cylinders by finite element method, Transaction of ASME Journal of Engineering for Industry, Vol. 93, pp. 445–454.
Shah, S.N., Lee, C.H. and Kobayashi, S. (1974), Compression of tall, circular solid cylinders between parallel flat dies, in: Proceedings of the International Conference on Production Engineering, Tokyo, pp. 295–300.
Chen, C.C. and Kobayashi, S. (1978), Rigid-plastic finite-element analysis of ring compression, in: H. Armen and R.F. Jones, Jr. (Eds.), Application of Numerical Methods to Forming Processes, Proceedings of the Winter Annual Meeting of ASME, AMD, Vol. 28, pp. 163–174.
Oh, S.I. and Kobayashi, S. (1976), Workability of aluminum alloy 7075-t6 in upsetting and rolling, Transaction of ASME, Journal of Engineering for Industry, Vol. 98, pp. 800–806.
Hartley, P., Sturgess, C.E.N. and Rowe, G.W. (1979), Friction in finite element analysis of metal forming process, International Journal of Mechanical Sciences, Vol. 21, pp. 301–311.
Hartley, P., Sturgess, C.E.N. and Rowe, G.W. (1980), Influence of friction on the prediction of forces, pressure distributions and properties in upset forging, International Journal of Mechanical Sciences, Vol. 22, pp. 743–753.
Shima, S., Mori, K. and Osakada, K. (1978), Analysis of metal forming by the rigid-plastic finite element method based on plasticity theory for porous metals, in H. Lippmann (Ed.), Metal Forming Plasticity, Springer, Berlin, pp. 305–317.
Kobayashi, S. Oh, S.I. and Altan, T. (1989), Metal Forming and the Finite Element Method, Oxford University Press, New York.
Rowe, G.W., Sturgess, C.E.N., Hartley, P. and Pillinger, I. (1991), Finite Element Plasticity and Metal Forming Analysis, Cambridge University Press, Cambridge.
Hartley, P. and Pillinger, I. and Sturgess, C.E.N. (Eds.) (1992), Numerical Modeling of Material Deformation Processes: Research, Development and Application, Springer, London.
Bathe, K.J., Ramm, E. and Wilson, E.L. (1975), Finite element formulations for large deformation dynamic analysis, International Journal of Numerical Methods for Engineering, Vol. 9, pp. 353–386.
Dadras, P. and Thomas, J.F. (1983), Analysis of axisymmetric upsetting based on flow pattern observation, International Journal of Mechanical Sciences, Vol. 25, pp. 421–427.
Carter, Jr., W.T. and Lee, D. (1986), Further analysis of axisymmetric upsetting, Transaction of ASME, Journal of Engineering for Industry, Vol. 108, pp. 198–204.
Michel, B. and Boyer, J.C. (1995), Elasto-visco-plastic finite-element analysis of a cold upsetting test and stress-state validation by residual-stress measurements, Journal of Materials Processing Technology, Vol. 54, pp. 120–128.
Zhao, G., Wright, E. and Grandhi, R.V. (1996), Computer aided perform design in forging using the inverse die contact tracking method, International Journal of Machine Tools & Manufacture, Vol. 36, pp. 755–769.
Joun, M.S., Lee, S.W. and Chung, J.H. (1998), Finite element analysis of a multistage axisymmetric forging process, International Journal of Machine Tools & Manufacture, Vol. 38, pp. 843–854.
Kim, H.S., Im, Y.T. and Geiger, M. (1999), Prediction of ductile fracture in cold forging of aluminum alloy, Transaction of ASME, Journal of Manufacturing Science and Engineering, Vol. 121(3), pp. 336–344.
Yang, D.Y., Im, Y.T., Yoo, Y.C., Park, J.J., Kim, J.H., Chun, M.S., Lee, C.H., Lee, Y.K., Park, C.H., Song, J.H., Kim, D.Y., Hong, K.K., Lee, M.C. and Kim, S.I. (2000), Development of integrated and intelligent design and analysis system for forging processes, Annals of CIRP, Vol. 49, pp. 177–180.
Gupta, S., Reddy, N.V. and Dixit, P.M. (2003), Ductile fracture prediction in axisymmetric upsetting using continuum damage mechanics, Journal of Materials Processing Technology, Vol. 141, pp. 256–265.
Mungi, M.P., Rasane, S.D. and Dixit, P.M. (2003), Residual stresses in cold axisymmetric forging, Journal of Materials Processing Technology, Vol. 142, pp. 256–266.
Riks, E. (1972), The application of Newton’s method to the problem of elastic stability, Transaction of ASME, Journal of Applied Mechanics, Vol. 39, pp. 1060–1066.
Batoz, J.L. and Dhatt, G. (1979), Incremental displacement algorithms for non-linear problems, International Journal of Numerical Methods for Engineering, Vol. 14, pp. 1262–1267.
Park, J.J. and Kobayashi, S. (1984), Three-dimensional finite element analysis of block compression, International Journal of Mechanical Sciences, Vol. 26(3), pp. 165–176.
Lemaitre, J. and Chaboche, J.L. (1990), Mechanics of Solid Materials, Cambridge University Press, Cambridge.
Dhar, S., Sethuraman, R. and Dixit, P.M. (1996), A continuum damage mechanics model for void growth and micro-crack initiation, Engineering Fracture Mechanics, Vol. 53, pp. 917–928.
Reddy, N.V., Dixit, P.M. and Lal, G.K. (1996), Central bursting and optimal die profile for axisymmetric extrusion, Transaction of ASME, Journal of Manufacturing Science and Engineering, Vol. 118, pp. 579–584.
Le Roy, G., Embury, J.D., Edward, G. and Ashby, M.F. (1981), A model of ductile fracture based on the nucleation and growth of voids, Acta Metallurgica, Vol. 29, pp. 1509–1522.
Predeleanu, M., Cordebois, J.P. and Belkhiri, L. (1986), Failure analysis of cold upsetting by computer and experimental simulation, in Proceedings of the NUMIFORM’86 Conference, pp. 277–282.
Kuhn, H.A. and Lee, P.W. (1971), strain instability and fracture at the surface of upset cylinders, Metallurgical Transactions, Vol. 2, pp. 3197–3202.
Kobayashi, S. (1970), Deformation characteristics and ductile fracture of 1040 steel in simple upsetting of solid cylinders and rings, Transaction of ASME, Journal of Engineering for Industry, Vol. 92, pp. 391–399.
Semiatin, S.L., Goetz, T.L., Shell, E.B., Seetharaman, V. and Ghosh, A.K. (1999), Cavitation and failure during hot forging of Ti-6Al-4V, Metallurgical and Materials Transaction A, Vol. 30, pp. 1411–1424.
Johnson, W. and Mellor, P.B. (1972), Engineering Plasticity, von Nostrand Co. Ltd.
Chakrabarty, J. (1987), Theory of Plasticity, McGraw-Hill Book Company, New York.
Toh, C.H. and Kobayashi, S. (1985), Deformation analysis and blank design in square cup drawing, International Journal of Machine Tool Design and Research, Vol. 25, pp. 15–32.
Saran, M.J. and Samuelsson, A. (1990), Elastic-viscoplastic implicit formulation for finite element simulation of complex sheet forming processes, International Journal for Numerical Methods in Engineering, Vol. 30, pp. 1675–1697.
Majlessi, S.A. and Lee, D. (1993), Deep drawing of square-shaped sheet metal parts, part 1: Finite element analysis, Transaction of ASME, Journal of Engineering for Industry, Vol. 115, pp. 102–109.
Onate, E. and Saracibar, C.A.D. (1990), Finite element analysis of sheet metal forming problems using a selective viscous bending/membrane formulation, International Journal for Numerical Methods in Engineering, Vol. 30, pp. 1577–1593.
Chou, C.H., Pan, J. and Tang, S.C. (1994), Analysis of sheet metal forming operations by a stress resultant constitutive law, International Journal for Numerical Methods in Engineering, Vol. 37, pp. 717–735.
Shi, X., Wei, Y. and Ruan, X. (2001), Simulation of sheet metal forming by a one-step approach: Choice of element, Journal of Materials Processing Technology, Vol. 108, pp. 300–306.
Chou, C.H., Pan, J. and Tang, S.C. (1994), An anisotropic stress resultant constitutive law for sheet metal forming, International Journal for Numerical Methods in Engineering, Vol. 39, pp. 435–449.
Menezes, L.F. and Teodosiu, C. (2000), Three dimensional numerical simulation of deep drawing process using solid finite elements, Journal of Materials Processing Technology, Vol. 97, pp. 100–106.
Colgan, M. and Monaghan, J. (2003), Deep drawing process: Analysis and experiment, Journal of Materials Processing Technology, Vol. 132, pp. 35–41.
Osakada, K., Wang, C.C. and Mori, K.I. (1995), Controlled FEM simulation for determining history of blank holding force in deep drawing, Annals of CIRP, Vol. 44, pp. 243–246.
Lorenzo, R.D., Fratini, L. and Micari, F. (1999), Optimal blank holder force path in sheet metal forming processes: an AI based procedure, Annals of CIRP, Vol. 48, pp. 231–234.
Guo, Y.Q., Batoz, J.L., Detraux, J.M. and Duroux, P. (1990), Finite element procedures for strain estimation of sheet metal forming parts, Annals of CIRP, Vol. 30, pp. 1385–1401.
Kim, S.H. and Huh, H. (2001), Finite element inverse analysis for the design of intermediate dies in multi-stage deep drawing with large aspect ratio, Journal of Materials Processing Technology, Vol. 113, pp. 779–785.
Ku, T.W., Lim, H.J., Choi, H.H., Hwang, S.M. and Kang, B.S. (2001), Implementation of backward tracing scheme of the FEM to blank design in sheet metal forming, Journal of Materials Processing Technology, Vol. 111, pp. 90–97.
Pegada, V.P., Chun, Y. and Santhanam, S. (2002), An algorithm for determining the optimum blank shape for deep drawing of aluminum cups, Journal of Materials Processing Technology, Vol. 125/126, pp. 743–750.
Chakka, V.M. (2006), Optimum Blank Shape Design for Cylindrical Cup Drawing Using Finite Element Method, M.Tech. Thesis, Department of Mechanical Engineering, Indian Institute of Technology Kanpur.
Raja, S. (2007), Optimum Blank Shape Design for Cylindrical Cup Drawing Using Various Anisotropic Criteria, M.Tech. Thesis, Department of Mechanical Engineering, Indian Institute of Technology Kanpur.
Yoon, J.W., Barlat, F., Dick, R.E. and Karabin, M.E. (2006), Prediction of six or eight ears in a drawn cup based on a new anisotropic yield function, International Journal of Plasticity, Vol. 22, pp. 174–193.
Barlat, F., Aretz, H., Yoon, J.W., Karabin, M.E., Brem, J.C. and Dick, R.E. (2005), Linear transformation-based anisotropic yield functions, International Journal of Plasticity, Vol. 21, pp. 1009–1039.
Yoon, J.W., Barlat, F., Gracio, J.J. and Rauch, E. (2005), Anisotropic strain hardening behavior in simple shear for cube textured aluminum alloy sheets, International Journal of Plasticity, Vol. 21, pp. 2426–2447.
Rights and permissions
Copyright information
© 2008 Springer-Verlag London Limited
About this chapter
Cite this chapter
(2008). Finite Element Modeling of Metal Forming Processes Using Updated Lagrangian Formulation. In: Modeling of Metal Forming and Machining Processes. Engineering Materials and Processes. Springer, London. https://doi.org/10.1007/978-1-84800-189-3_6
Download citation
DOI: https://doi.org/10.1007/978-1-84800-189-3_6
Publisher Name: Springer, London
Print ISBN: 978-1-84800-188-6
Online ISBN: 978-1-84800-189-3
eBook Packages: EngineeringEngineering (R0)