Comparative Study of Cyclic Softening Modelling and Proposition of a Modification to ‘MARQUIS’ Approach

  • Snehasish BhattacharjeeEmail author
  • Sankar Dhar
  • Sanjib Kumar Acharyya
  • Suneel Kumar Gupta
Conference paper
Part of the Lecture Notes on Multidisciplinary Industrial Engineering book series (LNMUINEN)


Cyclic hardening and softening of materials can be modelled by a single exponential decay function. Marquis proposed that similar function can be used to modify the dynamic recovery contribution of kinematic hardening rule to simulate cyclic hardening or softening by changing only the sign of a function parameter. According to Marquis, only kinematic hardening rule, then, can be able to simulate cyclic hardening and softening with reasonable physical justification. Though it is observed that, adoption of the function in multi-segmented kinematic hardening rule is not very capable, and a separate softening approach is proposed using the same Marquis function. The cyclic plastic response of SA333 steel subjected to uniaxial tension–compression cyclic loading is experimented, and predominant cyclic softening is observed with initially non-Masing plastic curvature. Three different softening models approached with multi-segmented Ohno–Wang kinematic hardening rule in commercial FE platform. The simulations are discussed in a comparative manner, and the modification proposed is found to be showing promising agreement with experimental results.


SA333 Cyclic softening Modified dynamic recovery Isotropic softening 



The authors acknowledge Bhabha Atomic Research Centre, Mumbai, for financial assistance through collaborative project and National Metallurgical Laboratory, Jamshedpur, for experimental support. The authors also acknowledge Dr. Surajit Kumar Paul, National Metallurgical Laboratory, Jamshedpur, for TEM micrographs.


  1. 1.
    Manson, S.S.: Behavior of materials under conditions of thermal stress. Heat Transf. Symp., 9–75 (1953)Google Scholar
  2. 2.
    Coffin, L.F.: A study of the effects of cyclic thermal stresses on a ductile metal. Trans. Am. Soc. Test. Mater. 76, 931–950 (1954)Google Scholar
  3. 3.
    Suresh, S.: Fatigue of materials. Cambridge University Press, ISBN: 978-0-52-157847-9 (1998)Google Scholar
  4. 4.
    Mughrabi, H.: Fatigue, an everlasting materials problem—still en vogue. Proc. Eng. 2(1), 3–26 (2010)CrossRefGoogle Scholar
  5. 5.
    Kumar, P.S., Sivaprasad, S., Dhar, S., Tarafder, S.: Ratcheting and low cycle fatigue behavior of SA333 steel and their life prediction. J. Nucl. Mater. 401(1–3), 17–24 (2010)Google Scholar
  6. 6.
    Kumar, P.S., Sivaprasad. S., Dhar. S., Tarafder. S.: Cyclic plastic deformation behavior in SA333 Gr. 6 C–Mn steel. Mater. Sci. Eng. A 528(24), 7341–7349 (2011)Google Scholar
  7. 7.
    Kumar, P.S., Sivaprasad, S., Dhar, S., Tarafder, S.: Key issues in cyclic plastic deformation: experimentation. Mech. Mater. 43(11), 705–720 (2011)CrossRefGoogle Scholar
  8. 8.
    Khutia, N., Dey, P.P., Kumar, P.S., Tarafder, S.: Development of non masing characteristic model for LCF and ratcheting fatigue simulation of SA333 C-Mn steel. Mech. Mater. 65, 88–102 (2013)CrossRefGoogle Scholar
  9. 9.
    Sivaprasad, S., Kumar, P.S., Arpan, D., Narasaiah, N., Tarafder, S.: Cyclic plastic behaviour of primary heat transport piping materials: influence of loading schemes on hysteresis loop. Mater. Sci. Eng. A 527(26), 6858–6869 (2010)CrossRefGoogle Scholar
  10. 10.
    Sivaprasad, S., Bar, H.N., Kumar, G.S., Punit, A., Bhasin, V., Tarafder, S.: A comparative assessment of cyclic deformation behaviour in SA333 Gr.6 steel using solid, hollow specimens under axial and shear strain paths. Int. J. Fatigue 61, 76–86 (2014)CrossRefGoogle Scholar
  11. 11.
    Sinha, A.K.: Ferrous Physical Metallurgy. Butterworths, London (1989)Google Scholar
  12. 12.
    Zhang, S., Wu, C.: Ferrous Materials. Metallurgical Industry Press, Beijing (1992)Google Scholar
  13. 13.
    Totten, G.E.: Steel Heat Treatment—Metallurgy and Technologies, 2nd edn. CRC Press, Taylor and Francis Group, Boca Raton (2007)Google Scholar
  14. 14.
    Ross, R.B.: Metallic Materials Specification Handbook, 4th edn. Chapman & Hall, London (1992)CrossRefGoogle Scholar
  15. 15.
    Mehran, M.: The Effects of Alloying Elements on Steels (I), Christian Doppler Laboratory for Early Stages of Precipitation, InstitutfürWerkstoffkunde, Schweißtechnik und SpanloseFormgebungsverfahren, TechnischeUniversität Graz (2007)Google Scholar
  16. 16.
    Bauschinger, J.: Über die Veranderung der Elasticitatsgrenze und elastcitatsmodulverschiedener, Metal Civiling N.F., 27, 289–348 (1881)Google Scholar
  17. 17.
    Chai, H.-F., Laird, C.: Mechanisms of cyclic softening and cyclic creep in low carbon steel. Mater. Sci. Eng. 93, 159–174 (1987)CrossRefGoogle Scholar
  18. 18.
    Sivaprasad, S., Swaminathan, J., Tiwary, Y.N., Roy, P.K., Singh, R.: Remaining life assessment of service exposed reactor and distillation column materials of a petrochemical plant. Eng. Fail. Anal. 10(3), 275–289 (2003)CrossRefGoogle Scholar
  19. 19.
    Masing, G.: Eigenspannungen und verfestigungbeim messing. In: Proceedings of the Second International Congress for Applied Mechanics, Zurich, pp. 332–335 (1926)Google Scholar
  20. 20.
    Bauschinger, J.: Mitteilung XV ausdemMechanisch-technischenLaboratorium der KöniglichenTechnischenHochschule in München 13, 1–115 (1886)Google Scholar
  21. 21.
    Boller, C., Seeger, T.: Materials Data for Cyclic Loading. Elsevier, ISBN: 978-0-44-442875-2 (1987)Google Scholar
  22. 22.
    Doong, S.H., Socie, D.F., Robertson, I.M.: Dislocation substructures and nonproportional hardening. ASME J. Eng. Mater. Technol. 112(4), 456–465 (1990)CrossRefGoogle Scholar
  23. 23.
    Jiang, Y., Kurath, P.: Nonproportional cyclic deformation: critical experiments and analytical modeling. Int. J. Plast. 13, 743–763 (1997)zbMATHCrossRefGoogle Scholar
  24. 24.
    Jiang, Y.: An experimental study of inhomogeneous cyclic plastic deformation. J. Eng. Mater. Technol. 123(3), 274–280 (2001)CrossRefGoogle Scholar
  25. 25.
    Zhang, J., Jiang, Y.: A study of inhomogeneous plastic deformation of 1045 steel. J. Eng. Mater. Technol. 126(2), 164–171 (2004)CrossRefGoogle Scholar
  26. 26.
    Jiang, Y., Sehitoglu, H.: Cyclic ratchetting of 1070 steel under multiaxial stress states. Int. J. Plast. 10(5), 579–608 (1994)CrossRefGoogle Scholar
  27. 27.
    Jiang, Y., Sehitoglu, H.: Multiaxial cyclic ratchetting under multiple step loading. Int. J. Plast. 10(8), 849–870 (1994)CrossRefGoogle Scholar
  28. 28.
    Véronique, A., Philippe, Q., Suzanne, D.: Cyclic plasticity of a duplex stainless steel under non-proportional loading. Mater. Sci. Eng. A 346(1–2), 208–215 (2003)Google Scholar
  29. 29.
    Jiang, Y., Zhang, J.: Benchmark experiments and characteristic cyclic plasticity deformation. Int. J. Plast. 24(9), 1481–1515 (2008)zbMATHCrossRefGoogle Scholar
  30. 30.
    Estrin, Y., Vinogradov, A.: Extreme grain refinement by severe plastic deformation: a wealth of challenging science. Acta Materialla 61(3), 782–817 (2013)CrossRefGoogle Scholar
  31. 31.
    Mughrabi, H.: On the current understanding of strain gradient plasticity. Mater. Sci. Eng. A 387–389, 209–213 (2004)CrossRefGoogle Scholar
  32. 32.
    Lu, K., Lu, L., Suresh, S.: Strengthening materials by engineering coherent internal boundaries at the nanoscale. Science 324(5925), 349–352 (2009)CrossRefGoogle Scholar
  33. 33.
    Pan, Q.S., Lu, L.: Strain-controlled cyclic stability and properties of Cu with highly oriented nanoscale twins. Acta Mater. 81, 248–257 (2014)CrossRefGoogle Scholar
  34. 34.
    Masing, G.: ZurHeyn’schenTheorie der Verfestigung der MetalledurchverborgenelastischeSpannungen. In: Harries, C.D. (ed.) WissenschaftlicheVeröffentlichungenausdem Siemens-Konzern, pp. 231–239. Springer, Heidelberg (1923)CrossRefGoogle Scholar
  35. 35.
    Elline, F., Kujawaski, D.: Plastic strain energy in fatigue failure. J. Eng. Mater. Technol. Trans. 106, 342–347 (1984)Google Scholar
  36. 36.
    Fan, Z., Jiang, J.: Investigation of low cycle fatigue behavior of 16MnR steel at elevated temperature, Zhejiang DaxueXuebao (Gongxue Ban)/Journal of Zhejiang University (Engineering Science) 38, 1190–1195 (2004)Google Scholar
  37. 37.
    Maier, H.J., Gabor, P., Gupta, N., Karaman, I., Haouaoui, M.: Cyclic stress-strain response of ultrafine grained copper. Int. J. Fatigue 28, 243–250 (2006)CrossRefGoogle Scholar
  38. 38.
    Wang, Z., Laird, C.: Relationship between loading process and Masing behavior in cyclic deformation. Mater. Sci. Eng. A 101, L1–L5 (1988)CrossRefGoogle Scholar
  39. 39.
    Raman, S.G.S., Padmanabhan, K.A.: Effect of prior cold work on the room temperature low-cycle fatigue behavior of AISI 304LN stainless steel. Int. J. Fatigue 18, 71–79 (1996)CrossRefGoogle Scholar
  40. 40.
    Plumtree, A., Abdel-Raouf, H.A.: Cyclic stress-strain response and substructure. Int. J. Fatigue 23, 799–805 (2001)CrossRefGoogle Scholar
  41. 41.
    Gough, H.J.: Crystalline structure in relation to failure of metals—especially by fatigue, Edgar Marburg Lecture. In: Proceedings of the American Society for Testing and Materials, vol. 33, II, pp. 3–114 (1933)Google Scholar
  42. 42.
    Orowan, E.Z.: Zur Kristallplastizität. I, Tieftemperaturplastizität und Beckersche Formel, ZeitschriftfürPhysik 89(9–10), 605–613 (1934)Google Scholar
  43. 43.
    Taylor, G.I.: Plastic strain in metals. J. Inst. Metals 62, 307–324 (1938)Google Scholar
  44. 44.
    Estrin, Y.: Dislocation theory based constitutive modelling: foundations and applications. J. Mater. Process. Technol. 80–81, 33–39 (1998)CrossRefGoogle Scholar
  45. 45.
    Kassner, M.E., Kyle, K.: Taylor hardening in five power law creep of metals and class M alloys. Nano Microstructural Des. Adv. Mater. 255–271 (2003)Google Scholar
  46. 46.
    Mughrabi, H.: The α-factor in the Taylor flow-stress law in monotonic, cyclic and quasi-stationary deformations: dependence on slip mode, dislocation arrangement and density. Curr. Opin. Solid State Mater. Sci. 26(6), 411–420 (2016)CrossRefGoogle Scholar
  47. 47.
    Kocks, U.F.: On the temperature and stress dependence of the dislocation velocity stress exponent. ScriptaMetallurgica 4(1), 29–31 (1970)CrossRefGoogle Scholar
  48. 48.
    Mecking, H., Kocks, U.F.: Physics and phenomenology of strain hardening: the FCC case. In: Progress in Material Science, pp. 171–273 (2003)Google Scholar
  49. 49.
    Rauch, E.F., Schmitt, J.H.: Dislocation substructures in mild steel deformed in simple shear. Mater. Sci. Eng. A 113, 441–448 (1989)CrossRefGoogle Scholar
  50. 50.
    Kubin, L., Devincre, B., Hoc, T.: Modeling dislocation storage rates and mean free paths in face-centered cubic crystals. Acta Mater. 56(20), 6040–6049 (2008)CrossRefGoogle Scholar
  51. 51.
    Wood, W.A.: Bulletin/Institute of Metals 3, 5–6 (1955)Google Scholar
  52. 52.
    Plumbridge, W.J., Ryder, D.A.: Metall. Rev. 14(136), 119–142 (1969)Google Scholar
  53. 53.
    Ashby, M.F.: Philos. Mag. 21, 399–424 (1970)CrossRefGoogle Scholar
  54. 54.
    Franciosi, P.: The concepts of latent hardening and strain hardening in metallic single crystals. Acta Metall. 33(9), 1601–1612 (1985)CrossRefGoogle Scholar
  55. 55.
    Read, W.T., Shockley, W.: Dislocation models of grain boundaries. Phys. Rev. 78(3), 275–289 (1950)zbMATHCrossRefGoogle Scholar
  56. 56.
    Sedláček, R., Blum, W., Kratochvíl, J., Forest, S.: Subgrain formation during deformation—physical origin and consequences. Metall. Mater. Trans. A 33A, 319–327 (2002)CrossRefGoogle Scholar
  57. 57.
    Bragg, W.L.: The structure of a cold worked metal. Proc. Phys. Soc. 52(1), 105–109 (1940)CrossRefGoogle Scholar
  58. 58.
    Burger, J.M.: Geometrical considerations concerning the structural irregularities to be assumed in a crystal. Proc. Phys. Soc. 52(1), 23–33 (1940)CrossRefGoogle Scholar
  59. 59.
    Read and Shockley: Quantitative predictions from dislocation models of crystal grain boundaries. Phys. Rev. 75(4), 692 (1949)CrossRefGoogle Scholar
  60. 60.
    Sauzay, M., Brillet, H., Monnet, I., Mottot, M., Barcelo, F., Fournier, B., Pineau, A.: Cyclically induced softening due to low-angle boundary annihilation in a martensitic steel. Mater. Sci. Eng. A 400–401, 241–244 (2005)CrossRefGoogle Scholar
  61. 61.
    Fournier, B., Sauzay, M., Pineau, A.: Micromechanical model of the high temperature cyclic behavior of 9-12%Cr martensitic steels. Int. J. Plast. 27(11), 1803–1816 (2011)zbMATHCrossRefGoogle Scholar
  62. 62.
    Mughrabi, H.: The long-range internal stress field in the dislocation wall structure of persistent slip bands. Physica Status Solidi (A) 104(1), 107–120 (1987)CrossRefGoogle Scholar
  63. 63.
    Li, Y., Laird, C.: Masing behavior observed in monocrystalline copper during cyclic deformation. Mater. Sci. Eng. A 161(1), 23–29 (1993)CrossRefGoogle Scholar
  64. 64.
    Watanabe, E., Asao, T., Toda, M., Yoshida, M., Horibe, S.: Relationship between Masing behavior and dislocation structure of AISI 1025 under different stress ratios in cyclic deformation. Mater. Sci. Eng. A 582, 55–62 (2013)CrossRefGoogle Scholar
  65. 65.
    Prnadtl, L.: Proceedings of the First International Congress of Applied Mechanics, Delft, 43 (1924)Google Scholar
  66. 66.
    Levy, M.: ComptesRendus. Académie des Sciences 70, 1323 (1870)Google Scholar
  67. 67.
    Mises, R.: Mechanik der festenKörperimplastisch-deformablenZustand. Nachrichten von der Gesellschaft der WissenschaftenzuGöttingen, Mathematisch-PhysikalischeKlasse 4, 582–593 (1913)Google Scholar
  68. 68.
    Reuss, A.: Berücksichtigung der elastischenFormänderung in der Plastizitätstheorie, ZeitschriftfürAngewandteMathematik und Mechanik (ZAMM—Journal of Applied Mathematics and Mechanics) 10(3), 266–274 (1930)Google Scholar
  69. 69.
    Lode, W.: Versucheüber den Einfluss der mittlerenHauptspannung auf das Fliessen der MetalleEisen. Kupfer und Nickel, ZeitschriftfürPhysik 36(11–12), 913–939 (1926)Google Scholar
  70. 70.
    Taylor, G.I., Quinney, H.: The Plastic Distortion of Metals. Philos. Trans. R. Soc. A 230, 681–693 (1931)zbMATHGoogle Scholar
  71. 71.
    Hohenemser, K.: Fließversuche an Rohrenaus Stahl beikombinierter Zug- und Torsionsbeanspruchung, ZeitschriftfürAngewandteMathematik und Mechanik (ZAMM—Journal of Applied Mathematics and Mechanics) 11(1), 15–19 (1931)Google Scholar
  72. 72.
    Morrison, J.L.M., Shepherd, W.M.: An experimental investigation of plastic stress-strain relations. Proc. Inst. Mech. Eng. 163(1), 1–17 (1950)CrossRefGoogle Scholar
  73. 73.
    Hill, R.: Mathematical Theory of Plasticity. Oxford University Press, ISBN: 978-0-19-850367-5 (1950)Google Scholar
  74. 74.
    Prager, W.: The theory of plasticity: a survey of recent achievements. Proc. Inst. Mech. Eng. 169(1), 41–57 (1955)MathSciNetCrossRefGoogle Scholar
  75. 75.
    Drucker, D.C.: Stress-Strain Relations in the Plastic Range—A Survey of Theory and Experiment. O.N.R. Report, NR-041-032 (1950)Google Scholar
  76. 76.
    Drucker, D.C.: A more fundamental approach to plastic stress-strain relations. In: Proceedings of the First US National Congress of Applied Mechanics, ASME, pp. 487–491 (1951)Google Scholar
  77. 77.
    Drucker, D.C.: A definition of stable inelastic material. J. Appl. Mech. 26, 101–106 (1959)MathSciNetzbMATHGoogle Scholar
  78. 78.
    Mises, R.: Mechanik der plastischenFormänderung von Kristallen, ZeitschriftfürAngewandteMathematik und Mechanik (ZAMM—Journal of Applied Mathematics and Mechanics) 8(3), 161–185 (1928)Google Scholar
  79. 79.
    Taylor, G.I.: A connexion between the criteria of yield and the strain ratio relationship in plastic solids. Proc. R. Soc. A 191(1027), 441–446 (1947)Google Scholar
  80. 80.
    Hill, R.: A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. A 193(1033), 281–297 (1948)MathSciNetzbMATHGoogle Scholar
  81. 81.
    Prager, W.: A new method of analyzing stresses and strains in work hardening plastic solids. ASME J. Appl. Mech. 23, 493–496 (1956)MathSciNetzbMATHGoogle Scholar
  82. 82.
    Mróz, Z.: On the description of anisotropic work hardening. J. Mech. Phys. Solids 15(3), 163–175 (1967)CrossRefGoogle Scholar
  83. 83.
    Besseling, J.F.: A theory of elastic, plastic and creep deformations of an initially isotropic material. J. Appl. Mech. 25, 529–536 (1958)zbMATHGoogle Scholar
  84. 84.
    Li, J.C.M.: Some elastic properties of an edge dislocation wall. Acta Metall. 8(8), 563–574 (1960)CrossRefGoogle Scholar
  85. 85.
    Li, J.C.M.: Petch relation and grain boundary sources. Trans. Metall. Soc. AIME 227, 239–247 (1963)Google Scholar
  86. 86.
    Armstrong, P.J., Frederick, C.O.: A Mathematical Representation of the Multiaxial Bauschinger Effect, CEGB Report No.-RD/B/N 731 (1967)Google Scholar
  87. 87.
    Essmann, U., Mughrabi, H.: Annihilation of dislocations during tensile and cyclic deformation and limits of dislocation densities. Philos. Mag. A 40(6), 731–756 (1979)CrossRefGoogle Scholar
  88. 88.
    Chaboche, J.L., DangVan, K., Cordier, G.: Modelization of the Strain Memory Effect on the Cyclic Hardening of 316 Stainless Steel, SMiRT-5, Berlin (1979)Google Scholar
  89. 89.
    Chaboche, J.L.: Time-independent constitutive theories for cyclic plasticity. Int. J. Plast 2(2), 149–188 (1986)zbMATHCrossRefGoogle Scholar
  90. 90.
    Chaboche, J.L.: On some modifications of kinematic hardening to improve the description of ratchetting effects. Int. J. Plast. 7(7), 661–678 (1991)CrossRefGoogle Scholar
  91. 91.
    Guionnet, C.: Modeling of ratcheting in biaxial experiments. J. Eng. Mater. Technol. 114, 56–62 (1992)CrossRefGoogle Scholar
  92. 92.
    Bari, S., Hassan, T.: Anatomy of coupled constitutive models for ratcheting simulation. Int. J. Plast. 16(3–4), 381–409 (2000)zbMATHCrossRefGoogle Scholar
  93. 93.
    Dafalias, Y.F., Popov, E.P.: Plastic internal variables formalism of cyclic plasticity. J. Appl. Mech. 43, 645–650 (1976)zbMATHCrossRefGoogle Scholar
  94. 94.
    Corona, E., Hassan, T., Kyriakides, S.: On the performance of kinematic hardening rules in predicting a class of biaxial ratcheting histories. Int. J. Plast. 12(1), 117–145 (1996)CrossRefGoogle Scholar
  95. 95.
    Ohno, N., Wang, J.D.: Kinematic hardening rules with critical state of dynamic recovery, part I: formulation. Int. J. Plast. 9(3), 375–391 (1993)zbMATHCrossRefGoogle Scholar
  96. 96.
    Ohno, N., Wang, J.D.: Kinematic hardening rules with critical state of dynamic recovery, part II: application. Int. J. Plast. 9(3), 391–403 (1993)CrossRefGoogle Scholar
  97. 97.
    Ohno, N., Wang, J.D.: Kinematic hardening rules for simulation of ratchetting behaviour. Eur. J Mech. A Solids 13(4), 519–531 (1994)zbMATHGoogle Scholar
  98. 98.
    McDowell, D.L.: Stress state dependence of cyclic ratcheting behavior of two rail steels. Int. J. Plast. 11, 397–421 (1995)CrossRefGoogle Scholar
  99. 99.
    Jiang, Y., Sehitoglu, H.: Modeling of cyclic ratchetting plasticity—Part I: development of constitutive relations. ASME J. Appl. Mech. 63(3), 720 (1996)zbMATHCrossRefGoogle Scholar
  100. 100.
    Voyiadjis, G.Z., Sivakumar, S.M.: A robust kinematic hardening rule for cyclic plasticity with ratcheting effects, part I: theoretical formulation. Acta Mech. 90, 105–123 (1991)zbMATHCrossRefGoogle Scholar
  101. 101.
    Voyiadjis, G.Z., Sivakumar, S.M.: Cyclic plasticity and ratchetting. Stud. Appl. Mech. 35, 253–295 (1994)zbMATHCrossRefGoogle Scholar
  102. 102.
    Phillips, A., Tang, J.L.: The effect of loading paths on the yield surface at elevated temperatures. Int. J. Solids Struct. 8(4), 463–474 (1972)CrossRefGoogle Scholar
  103. 103.
    Phillips, A., Lee, C.W.: Yield surfaces and loading surfaces. Exp. Recommendations Int. J. Solids Struct. 15, 715–729 (1979)CrossRefGoogle Scholar
  104. 104.
    Tseng, N.T., Lee, G.C.: Simple plasticity model of two-surface type. ASCE J. Eng. Mech. 109(3), 795–810 (1983)CrossRefGoogle Scholar
  105. 105.
    Voyiadjis, G.Z., Basuroychowdhury, I.N.: A plasticity model for multiaxial cyclic loading and ratcheting. Acta Mech. 126, 19–35 (1998)zbMATHCrossRefGoogle Scholar
  106. 106.
    Basuroychowdhury, I.N., Voyiadjis, G.Z.: A multiaxial cyclic plasticity model for nonproportional loading cases. Int. J. Plast. 14(9), 855–870 (1998)zbMATHCrossRefGoogle Scholar
  107. 107.
    Abdel-Karim, M., Ohno, N.: Kinematic hardening model suitable for ratcheting with steady-state. Int. J. Plast. 16(3–4), 225–240 (2000)zbMATHCrossRefGoogle Scholar
  108. 108.
    Zaverl Jr., F., Lee, D.: Constitutive relations for nuclear reactor core materials. J. Nucl. Mater. 75, 14–19 (1978)CrossRefGoogle Scholar
  109. 109.
    Marquis, D.: Modelisation et Identification de I’Ecrouisage Anisotropic des Metaux, These Paris VI (1979)Google Scholar
  110. 110.
    Haupt, P., Kamlah, M., TsakmakisCh, : Continuous representation of hardening properties in cyclic plasticity. Int. J. Plast. 8, 803–817 (1992)zbMATHCrossRefGoogle Scholar
  111. 111.
    Jiang, Y., Kurath, P.: Characteristics of Armstrong-Frederick type plasticity models. Int. J. Plast. 12(3), 387–415 (1996)zbMATHCrossRefGoogle Scholar
  112. 112.
    SIMULIA ABAQUS CAE v6.8 user manualGoogle Scholar
  113. 113.
    Rahaman, S.M.: Finite Element Analysis and Related Numerical Schemes for Ratcheting Simulation, Ph.D. thesis submitted in North Carolina State University, USA (2006)Google Scholar
  114. 114.
    Nagtegaal, J.C.: On the implementation of inelastic constitutive equations with special reference to large deformation problems. Comput. Methods Appl. Mech. Eng. 33, 469–484 (1982)zbMATHCrossRefGoogle Scholar
  115. 115.
    Simo, J.C., Taylor, R.L.: Consistent tangent operators for rate-independent elastoplasticity. Comput. Methods Appl. Mech. Eng. 48, 101–118 (1985)zbMATHCrossRefGoogle Scholar
  116. 116.
    Simo, J.C., Taylor, R.L.: A return mapping algorithm for plane stress elastoplasticity. Int. J. Numer. Meth. Eng. 22, 649–670 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  117. 117.
    Kobayashi, M., Ohno, N.: Implementation of cyclic plasticity models based on a general form of kinematic hardening. Int. J. Numer. Meth. Eng. 53, 2217–2238 (2002)zbMATHCrossRefGoogle Scholar
  118. 118.
    Dunne, F., Patrinik, N.: Introduction to Computational Plasticity. Oxford University Press, ISBN: 978-0-19-856826-1 (2005)Google Scholar
  119. 119.
    Ypma, T.J.: Historical development of Newton-Raphson method. Soc. Ind. Appl. Math. 37(4), 531–551 (1995)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Snehasish Bhattacharjee
    • 1
    Email author
  • Sankar Dhar
    • 1
  • Sanjib Kumar Acharyya
    • 1
  • Suneel Kumar Gupta
    • 2
  1. 1.Department of Mechanical EngineeringJadavpur UniversityKolkataIndia
  2. 2.Reactor Safety DivisionBhabha Atomic Research CentreMumbaiIndia

Personalised recommendations