An Improved Finite Element Model for Numerical Simulation of Phase Changes of Iron Under Extreme Conditions

  • Ladislav ÉcsiEmail author
  • Pavel Élesztős
  • Kinga Balázsová
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 35)


In this chapter an improved finite element model for numerical simulation of phase changes of iron is presented, which is capable of simulating iron construction behaviour under extreme conditions that include large strains/large deformations at high strain rates and temperatures. The model is based on an improved variational formulation of the conservation of energy with convective heat transfer. It employs the updated Lagrangian formulation and uses the extended NoIHKH material model for cyclic plasticity of metals. It also uses the Kelvin–Voigt model for internal damping, the Jaumann objective rate in the Cauchy’s stress update calculation and simplified rate forms of the Mehl-Avrami and Koisten-Marburger equations for ferrite, pearlite, bainite, martensite and the retaining austenite phase calculation. A numerical experiment using a cooled bar in cyclic bending is presented and briefly discussed.


FEM Thermal-structural analysis Phase changes Cyclic plasticity of metals Updated Lagrangian formulation 



Funding using the VEGA grant 1/0488/09 and 1/0051/10 resources is greatly appreciated.


  1. 1.
    Skrzypek, J.J., Ganczarski, A.W., Rustichelli, F., Egner, H.: Advanced Materials and Structures for Extreme Operating Conditions. Springer, Berlin (2008)Google Scholar
  2. 2.
    Maugin, G.A.: The Thermomechanics of Plasticity and Fracture. Cambridge University Press, Cambridge (1992)CrossRefGoogle Scholar
  3. 3.
    Šilhavý, M.: The Mechanics and Thermodynamics of Continuous Media. Springer, Berlin (1997)Google Scholar
  4. 4.
    Müller, I.: Thermodynamics. Pitman Publishing LTD, London (1985)Google Scholar
  5. 5.
    Haupt, P.: Continuum Mechanics and Theory of Materials, 2nd edn. Springer, Berlin (2002)CrossRefGoogle Scholar
  6. 6.
    Holzapfel, G.A.: Nonlinear Solid Mechanics. A Continuum Approach for Engineering. Wiley, Chichester (2001)Google Scholar
  7. 7.
    Liu, W.K.: Development of mixed time partition procedures for thermal analysis of structures. Int. J. Num. Meth. Eng. 19, 125–140 (1983)CrossRefGoogle Scholar
  8. 8.
    Ray, S.K., Utku, S.: A numerical model for the thermo-elasto-plastic behaviour of a material. Int. J. Num. Meth. Eng. 28, 1103–1114 (1989)CrossRefGoogle Scholar
  9. 9.
    Kleiber, M.: Computational coupled non-associative thermo-plasticity. Comput. Meth. Appl. Mech. Eng. 90, 943–967 (1991)CrossRefGoogle Scholar
  10. 10.
    Simo, J.C., Miehe, C.: Associative coupled thermoplasticity at finite strains: Formulation, numerical analysis and implementation. Comput. Meth. Appl. Mech. Eng. 98, 41–104 (1992)CrossRefGoogle Scholar
  11. 11.
    Služalec, A.: Temperature rise in elastic-plastic metal. Comput. Meth. Appl. Mech. Eng. 96, 293–302 (1992)CrossRefGoogle Scholar
  12. 12.
    Huang, Z.P.: Arate independent thermoplastic theory at finite deformations. Arch. Mech. 46(6), 855–879 (1994)Google Scholar
  13. 13.
    Pantuso, D., Bathe, K.J., Bouzinov, P.A.: A finite element procedure for the analysis of thermo-mechanical solids in contact. Comput. Struct. 75, 551–573 (2000)CrossRefGoogle Scholar
  14. 14.
    Batra, R.C., Love, B.M.: Mesoscale analysis of shear bands in high strain rate deformations of tungsten/nickel-iron composites. J. Therm. Stresses 28, 747–782 (2005)CrossRefGoogle Scholar
  15. 15.
    Maugin, A.G., Berezovski, A.: On the propagation of singular surfaces in thermoelasticity. J. Therm. Stresses 32, 557–592 (2009)CrossRefGoogle Scholar
  16. 16.
    Služalec, A.: An evaluation of the internal dissipation factor in coupled thermoplasticity. Int. J. Non-Lin. Mech. 25(4), 395–403 (1990)CrossRefGoogle Scholar
  17. 17.
    Dillon Jr, O.W.: Coupled thermoplasticity. J. Mech. Phys. Solids 11, 21–33 (1963)CrossRefGoogle Scholar
  18. 18.
    Saracibar, C.A., Cervena, M., Chiumenti, M.: On the constitutive modeling of coupled thermomechanical phase-change problems. Int. J. Plast. 17, 1565–1622 (2001)CrossRefGoogle Scholar
  19. 19.
    Júnior, M.V: Computational approaches to simulation of metal cutting processes. Thesis, University of Wales. Swansea (1998)Google Scholar
  20. 20.
    Schönauer, M.: Unified numerical analysis of cold and hot metal forming processes. Thesis, University of Wales, Swansea (1993)Google Scholar
  21. 21.
    Oden, T.J.: Finite Elements of Nonlinear Continua. McGraw-Hill, New York (1972)Google Scholar
  22. 22.
    Lemaitre, J., Chaboche, J.L.: Mechanics of Solid Materials. Cambridge University Press, Cambridge (1994)Google Scholar
  23. 23.
    Ibrahimbegovic, A.: Nonlinear Solid Mechanics. Theoretical Formulations and Finite Element Solution Methods. Springer, Dordrecht (2009)Google Scholar
  24. 24.
    Écsi, L., Élesztős, P.: Constitutive equation with internal damping for materials under cyclic and dynamic loadings using a fully coupled thermal-structural finite element analysis. Int. J. Multiphys. 3(2), 155–165 (2009)CrossRefGoogle Scholar
  25. 25.
    Porter, D.A., Easterling, K.E.: Phase Transformations in Metals and Alloys, 2nd edn. Chapman & Hall, London (1992)CrossRefGoogle Scholar
  26. 26.
    Simo, J.C., Hughes, T.J.R.: Computational Inelasticity. Springer, New York (1998)Google Scholar
  27. 27.
    Nemat-Nasser, S.: Plasticity. A Treatsie on Finite Deformation of Heterogenous Inelastic Materials. Cambridge University Press, Cambridge (2009)Google Scholar
  28. 28.
    Lemaitre, J.: Handbook of Material Behavior Models. Deformations of Materials, vol. 1. Academic Press, London (2001)Google Scholar
  29. 29.
    Lemaitre, J.: Handbook of Material Behavior Models. Failures of Materials, vol. 2. Academic Press, London (2001)Google Scholar
  30. 30.
    Lemaitre, J.: Handbook of Material Behavior Models. Multiphysics Behaviors, vol. 3. Academic Press, London (2001)Google Scholar
  31. 31.
    Suresh, S.: Fatigue of Materials, 2nd edn. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  32. 32.
    Lemaitre, J.: A Course on Damage Mechanics. Springer, Berlin (1992)CrossRefGoogle Scholar
  33. 33.
    Needleman, A., Tvergaard, V.: Analysis of a brittle-ductile transition under dynamic shear loading. Int. J. Solids Struct. 32(17–18), 2571–2590 (1995)CrossRefGoogle Scholar
  34. 34.
    Zhou, M., Ravichandran, G., Rosakis, A.J.: Dynamically propagating shear bands in impact-loaded prenotched plates—II. Numerical simulations. J. Mech. Phys. Solids 44(6), 1007–1032 (1996)CrossRefGoogle Scholar
  35. 35.
    Li, S., Liu, W.K., Rosakis, A.J., Belytschko, T., Hao, V.: Mesh-Free Galerkin simulations of dynamic shear band propagation and failure mode transition. Int. J. Solids Struct. 39, 1213–1240 (2002)CrossRefGoogle Scholar
  36. 36.
    Liang, R., Khan, A.S.: A critical review of experimental results and constitutive models for BCC and FCC metals over wide range of strain rates and temperatures. Int. J. Plast. 15, 963–980 (1999)CrossRefGoogle Scholar
  37. 37.
    Caboche, J.L.: A review of some plasticity and viscoplasticity constitutive theories. Int. J. Plast. 24, 1642–1693 (2008)CrossRefGoogle Scholar
  38. 38.
    Rusinek, A., Rodríguez-Martínez, J.A., Arias, A.: A thermo-viscoplastic model for FCC metals with application to OFHC copper. Int. J. Mech. Sci. 52, 120–135 (2010)CrossRefGoogle Scholar
  39. 39.
    Holmquist, T.J., Templeton, D.W., Bishnoi, K.D.: Constitutive modelling of aluminium nitride for large strain high-strain rate, and high-pressure applications. Int. J. Impact Eng. 25, 211–231 (2001)CrossRefGoogle Scholar
  40. 40.
    Rodríguez-Martínez, J.A., Rusinek, A., Klepaczko, J.R.: Constitutive relation for steels approximating quasi-static and intermediate strain rates at large deformations. Mech. Res. Commun. 36, 419–427 (2009)CrossRefGoogle Scholar
  41. 41.
    Yu, H., Guo, Y., Zhang, K., Lai, X.: Constitutive model on the description of plastic behavior of DP600 steel at strain rate from 10−4 to 103 s−1. Comput. Mater. Sci. 46, 36–41 (2009)CrossRefGoogle Scholar
  42. 42.
    Rusinek, A., Zaera, R., Klepaczko, J.R.: Constitutive relation in 3-D for wide range of strain rates and temperatures-application to mild steels. Int. J. Solid. Struct. 44, 5611–5634 (2007)CrossRefGoogle Scholar
  43. 43.
    Yu, H., Guo, Y., Lai, X.: Rate-dependent behavior and constitutive model of DP600 steel at strain rate from 10−4 to 103 s−1. Mater. Des. 30, 2501–2505 (2009)CrossRefGoogle Scholar
  44. 44.
    Yin, Z.N., Wang, T.J.: Deformation of PC/ABS alloys at elevated temperatures and high strain rates. Mat. Sci. Eng. A 494, 304–313 (2008)CrossRefGoogle Scholar
  45. 45.
    Deseri, L., Mares, R.: A class of viscoelastoplastic constitutive models based on the maximum dissipation principle. Mech. Mater. 32, 389–403 (2000)CrossRefGoogle Scholar
  46. 46.
    Ramrakhyani, D.S., Lesieutre, G.A., Smith, E.C.: Modeling of elastomeric materials using nonlinear fractional derivative and continuously yielding friction elements. Int. J. Solids Struct. 41, 3929–3948 (2004)CrossRefGoogle Scholar
  47. 47.
    Naumenko, K., Altenbach, H.: Modeling of Creep for Structural Analysis. Springer, Berlin (2007)CrossRefGoogle Scholar
  48. 48.
    Hyde, T.H., Sun, W., Wiliams, J.A.: Prediction of creep failure life of initially pressurized thick walled CrMoV pipes. Int. J. Press Vessels Pip. 76, 925–933 (1999)CrossRefGoogle Scholar
  49. 49.
    Staroselsky, A., Cassenti, B.N.: Combined rate-independent plasticity and creep model for single crystal. Mech Mater. 42, 945–959 (2010)CrossRefGoogle Scholar
  50. 50.
    Aktaa, J., Petersen, C.: Challenges in the constitutive modelling of the thermo-mechanical deformation and damage behaviour of EUROFER 97. Eng. Fract. Mech. 76, 1474–1484 (2009)CrossRefGoogle Scholar
  51. 51.
    Saleeb, A.F., Padula II, S.A., Kumar, A.: A multi-axial, multimechanism based constitutive model for the comprehensive representation of evolutionary response of SMAs under general thermomechanical loading conditions. Int. J. Plast. 27, 655–687 (2011)CrossRefGoogle Scholar
  52. 52.
    Auricchio, F., Taylor, R.L., Lubliner, J.: Shape-memory alloys: macromodelling and numerical simulations of the superelastic behaviour. Comput. Methods Appl. Mech. Eng. 146, 281–312 (1997)CrossRefGoogle Scholar
  53. 53.
    Dan, W.J., Zhang, W.G., Li, S.H., Lin, Z.Q.: A model for strain-induced martensitic transformation of TRIP steel with strain rate. Comput. Mater. Sci. 40, 101–107 (2007)CrossRefGoogle Scholar
  54. 54.
    Lee, M.G., Kim, S.J., Han, H.N., Jeong, WCh.: Implicit finite element formulations for multi-phase transformation in high carbon steel. Int. J. Plast. 25, 1720–1758 (2009)Google Scholar
  55. 55.
    Lee, M.G., Kim, S.J., Han, H.N., Jeong, WCh.: Implicit finite element formulations for multi-phase transformation in high carbon steel. Int. J. Plast. 25, 1720–1758 (2009)Google Scholar
  56. 56.
    Field, J.E., Walley, S.M., Proud, W.G., Goldrein, H.T., Siviour, C.R.: Review of experimental techniques for high rate deformation and shock studies. Int. J. Impact. Eng. 30, 725–775 (2004)CrossRefGoogle Scholar
  57. 57.
    Das, A., Tarafder, S.: Experimental investigation on martensitic transformation and fracture morphologies of austenitic stainless steel. Int. J. Plast. 25, 2222–2247 (2009)CrossRefGoogle Scholar
  58. 58.
    Sedighi, M., Khandaei, M., Shokrollahi, H.: An approach in parametric identification of high strain rate constitutive model using Hopkinson pressure bar test results. Mater. Sci. Eng. A 527, 3521–3528 (2010)CrossRefGoogle Scholar
  59. 59.
    Crisfield, M.A.: Non-linear Finite Element Analysis of Solids and Structures. Advanced Topics, vol. 2. Wiley, Chichester (2001)Google Scholar
  60. 60.
    Bathe, K.J., Kojić, M.: Inelastic Analysis of Solids and Structures. Springer, Berlin (2005)Google Scholar
  61. 61.
    Wriggers, P.: Nonlinear Finite Element Methods. Springer, Berlin (2008)Google Scholar
  62. 62.
    Evans, L.C.: Partial Differential Equations, American Mathematical Society. Providence. Rhode Island (1998)Google Scholar
  63. 63.
    Rektorys, K.: Variační metody v inženýrských problémech a v problémech matematické fyziky. Academia, Praha (1999)Google Scholar
  64. 64.
    Bathe, K.J.: Finite Element Procedures in Engineering Analysis. Prentice-Hall Inc., Englewood Cliffs (1982)Google Scholar
  65. 65.
    Howell, J.R.: A catalog of radiation configuration factors. McGraw-Hill, New York (1976)Google Scholar
  66. 66.
    Écsi, L.: Numerical behaviour of a solid body under various mechanical loads using finite element method with new energy balance equation for fully coupled thermal-structural analysis. In: Proceedings of the Sixth International Congress on Thermal Stresses 2005, vol. 2, pp. 543–546. Technische Universität Wien, Wien (2005). ISBN 3-901167-12Google Scholar
  67. 67.
    Écsi, L, Élesztős, P.: Hysteretic heating of a solid bar using a universal constitutive equation with internal damping for fully coupled thermal-structural finite element analysis. In: Proceedings of the 8th International Congress Thermal Stresses 2009, vol. 1, pp. 233–236. University of Illinois Press, Illinois (2009). ISBN 978-0-615-28233-6Google Scholar
  68. 68.
    Écsi, L, Élesztős, P.: Trying to model the thermo-mechanical behaviour of a solid body under cyclic loading using a material model with internal damping. Acta Mechanica Slovaca Roč 12(3-B), 143–150 (2008)Google Scholar
  69. 69.
    Écsi, L, Élesztős, P.: One of the possible variational formulations of fully coupled thermal-structural analysis. J. Mech. Eng. Roč 60(3), 135–144 (2009)Google Scholar
  70. 70.
    Oden, T.J., Carey, F.G.: Finite Elements. A Second Course, vol. 2. Prentice-Hall Inc., Englewood Cliffs (1983)Google Scholar
  71. 71.
    Oden, T.J., Carey, F.G.: Finite Elements, Mathematical Aspects, vol. 4. Prentice-Hall Inc., Englewood Cliffs (1983)Google Scholar
  72. 72.
    Grutin, M.E.: The linear theory of elasticity. In: Flügge, S., Thruesdell, C. (eds.) Handbuch der Physic. VIa/2, pp. 1–295. Springer, New York (1972)Google Scholar
  73. 73.
    Oden, T.J.: Applied Functional Analysis. Prentice Hall, Englewood Cliffs (1979)Google Scholar
  74. 74.
    Reddy, B.D.: Introductory Functional Analysis. Springer, Berlin (1998)CrossRefGoogle Scholar
  75. 75.
    Sun, C.T., Lu, J.P.: Vibration Damping of Structural Elements. Prentice Hall PTR, Englewood Clifs (1995)Google Scholar
  76. 76.
    Asszonyi, Csl, et al.: Izotrop kontinuumok anyagtulajdonságai. Műegyetemi kiadó, Budapest (2008)Google Scholar
  77. 77.
    Mesquita, A.D., Coda, H.B.: Alternative Kelvin viscoelastic procedure for finite elements. Appl. Math. Model. 26, 501–516 (2002)CrossRefGoogle Scholar
  78. 78.
    Argiris, J., Mlejnek, H.P.: Dynamics of Structures. Elsevier Science Publishers, Amsterdam (1991)Google Scholar
  79. 79.
    Bathe, K.J.: Finite Element Procedures. Prentice-Hall Inc., Englewood Cliffs (1995)Google Scholar
  80. 80.
    Lee, U.: Spectral Element Method in Structural Dynamics. John Wiley & Sons Pte Ltd., Singapore (2009)CrossRefGoogle Scholar
  81. 81.
    Liu, M., Gorman, D.G.: Formulation of Rayleigh damping and its extensions. Comput. Struct. 57(2), 277–285 (1995)CrossRefGoogle Scholar
  82. 82.
    Belytschko, T., Liu, W.K., Moran, B.: Nonlinear finite elements for continua and structures. Wiley, Chichester (2000)Google Scholar
  83. 83.
    Écsi, L., Élesztős, P., Kosnáč, J.: Constitutive equation with internal damping for materials under cyclic and dynamic loadings using large strain/large deformation formulation. In: Proceedings of the International Conference on Computational Modelling and Advanced Simulations 2009, Bratislava, Slovak Republic (2009)Google Scholar
  84. 84.
    de Souza Neto, E.A., Perić, D., Owen, D.R.J.: Computational Methods for Plasticity. Theory and Applications. John Wiley & Sons Ltd., Singapore (2008)CrossRefGoogle Scholar
  85. 85.
    Bonet, J., Wood, R.D.: Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, Cambridge (1997)Google Scholar
  86. 86.
    Écsi, L.: Extended NOIHKH model usage for cyclic plasticity of metals. Eng. Mech. Roč 13(2), 83–92Google Scholar
  87. 87.
    Crisfield, M.A.: Non-linear Finite Element Analysis of Solids and Structures. Essentials, vol. 1. Wiley, Chichester (2000)Google Scholar
  88. 88.
    Trebuňa, F., Šimčák, F.: Príručka experimentálnej mechaniky. Edícia vedeckej a odbornej literatúry. TypoPress, Košice (2007)Google Scholar
  89. 89.
    Budó, Á.: Kísérleti fizika I. Nemzeti tankönyvkiadó, Budapest (1997)Google Scholar
  90. 90.
    Klepazko, J.R., Rusinek, A.: Experiments on heat generation during plastic deformation and stored energy for TRIP steels. Mater. Des. 30, 35–48 (2009)CrossRefGoogle Scholar
  91. 91.
    Johnson, W.A., Mehl, R.F.: Reaction kinetics in processes of nucleation and growth. Trans. AIME 135, 416–458 (1939)Google Scholar
  92. 92.
    Avrami, M.: Kinetics of phase change I. J. Chem. Phys. 7, 1103–1112 (1939)CrossRefGoogle Scholar
  93. 93.
    Koistinen, D.P., Marburger, R.E.: A general equation prescribing the extent of the austenite-martensite transformation in pure iron-carbon alloys and carbon steels. Acta Metall. 7, 59–60 (1959)CrossRefGoogle Scholar
  94. 94.
    Lee, M.G., Kim, S.J., Han, H.N., Jeong, WCh.: Implicit finite element formulations for multi-phase transformation in high carbon steel. Int. J. Plast. 25, 1726–1758 (2009)CrossRefGoogle Scholar
  95. 95.
    Kang, S.H., Im, Y.T.: Three-dimensional thermo-elastic-plastic finite element modelling of quenching process of plain-carbon steel in couple with phase transformation. Int. J. Mech. Sci. 49, 423–439 (2007)CrossRefGoogle Scholar
  96. 96.
    Ronda, J., Oliver, G.J.: Consistent thermo-mechano-metallurgical model of welded steel with unified approach to derivation of phase evolution laws and transformation induced plasticity. Comput. Methods. Appl. Mech. Eng. 189, 361–417 (2000)CrossRefGoogle Scholar
  97. 97.
    Hömberg, D.: A numerical simulation of the Jominy end-quench test. Acta Matter. 44(11), 4375–4385 (1996)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Ladislav Écsi
    • 1
    Email author
  • Pavel Élesztős
    • 1
  • Kinga Balázsová
    • 1
  1. 1.Faculty of Mechanical EngineeringSlovak University of Technology in BratislavaBratislavaSlovakia

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