Abstract
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.
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References
Manson, S.S.: Behavior of materials under conditions of thermal stress. Heat Transf. Symp., 9–75 (1953)
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)
Suresh, S.: Fatigue of materials. Cambridge University Press, ISBN: 978-0-52-157847-9 (1998)
Mughrabi, H.: Fatigue, an everlasting materials problem—still en vogue. Proc. Eng. 2(1), 3–26 (2010)
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)
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)
Kumar, P.S., Sivaprasad, S., Dhar, S., Tarafder, S.: Key issues in cyclic plastic deformation: experimentation. Mech. Mater. 43(11), 705–720 (2011)
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)
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)
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)
Sinha, A.K.: Ferrous Physical Metallurgy. Butterworths, London (1989)
Zhang, S., Wu, C.: Ferrous Materials. Metallurgical Industry Press, Beijing (1992)
Totten, G.E.: Steel Heat Treatment—Metallurgy and Technologies, 2nd edn. CRC Press, Taylor and Francis Group, Boca Raton (2007)
Ross, R.B.: Metallic Materials Specification Handbook, 4th edn. Chapman & Hall, London (1992)
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)
Bauschinger, J.: Über die Veranderung der Elasticitatsgrenze und elastcitatsmodulverschiedener, Metal Civiling N.F., 27, 289–348 (1881)
Chai, H.-F., Laird, C.: Mechanisms of cyclic softening and cyclic creep in low carbon steel. Mater. Sci. Eng. 93, 159–174 (1987)
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)
Masing, G.: Eigenspannungen und verfestigungbeim messing. In: Proceedings of the Second International Congress for Applied Mechanics, Zurich, pp. 332–335 (1926)
Bauschinger, J.: Mitteilung XV ausdemMechanisch-technischenLaboratorium der KöniglichenTechnischenHochschule in München 13, 1–115 (1886)
Boller, C., Seeger, T.: Materials Data for Cyclic Loading. Elsevier, ISBN: 978-0-44-442875-2 (1987)
Doong, S.H., Socie, D.F., Robertson, I.M.: Dislocation substructures and nonproportional hardening. ASME J. Eng. Mater. Technol. 112(4), 456–465 (1990)
Jiang, Y., Kurath, P.: Nonproportional cyclic deformation: critical experiments and analytical modeling. Int. J. Plast. 13, 743–763 (1997)
Jiang, Y.: An experimental study of inhomogeneous cyclic plastic deformation. J. Eng. Mater. Technol. 123(3), 274–280 (2001)
Zhang, J., Jiang, Y.: A study of inhomogeneous plastic deformation of 1045 steel. J. Eng. Mater. Technol. 126(2), 164–171 (2004)
Jiang, Y., Sehitoglu, H.: Cyclic ratchetting of 1070 steel under multiaxial stress states. Int. J. Plast. 10(5), 579–608 (1994)
Jiang, Y., Sehitoglu, H.: Multiaxial cyclic ratchetting under multiple step loading. Int. J. Plast. 10(8), 849–870 (1994)
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)
Jiang, Y., Zhang, J.: Benchmark experiments and characteristic cyclic plasticity deformation. Int. J. Plast. 24(9), 1481–1515 (2008)
Estrin, Y., Vinogradov, A.: Extreme grain refinement by severe plastic deformation: a wealth of challenging science. Acta Materialla 61(3), 782–817 (2013)
Mughrabi, H.: On the current understanding of strain gradient plasticity. Mater. Sci. Eng. A 387–389, 209–213 (2004)
Lu, K., Lu, L., Suresh, S.: Strengthening materials by engineering coherent internal boundaries at the nanoscale. Science 324(5925), 349–352 (2009)
Pan, Q.S., Lu, L.: Strain-controlled cyclic stability and properties of Cu with highly oriented nanoscale twins. Acta Mater. 81, 248–257 (2014)
Masing, G.: ZurHeyn’schenTheorie der Verfestigung der MetalledurchverborgenelastischeSpannungen. In: Harries, C.D. (ed.) WissenschaftlicheVeröffentlichungenausdem Siemens-Konzern, pp. 231–239. Springer, Heidelberg (1923)
Elline, F., Kujawaski, D.: Plastic strain energy in fatigue failure. J. Eng. Mater. Technol. Trans. 106, 342–347 (1984)
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)
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)
Wang, Z., Laird, C.: Relationship between loading process and Masing behavior in cyclic deformation. Mater. Sci. Eng. A 101, L1–L5 (1988)
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)
Plumtree, A., Abdel-Raouf, H.A.: Cyclic stress-strain response and substructure. Int. J. Fatigue 23, 799–805 (2001)
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)
Orowan, E.Z.: Zur Kristallplastizität. I, Tieftemperaturplastizität und Beckersche Formel, ZeitschriftfürPhysik 89(9–10), 605–613 (1934)
Taylor, G.I.: Plastic strain in metals. J. Inst. Metals 62, 307–324 (1938)
Estrin, Y.: Dislocation theory based constitutive modelling: foundations and applications. J. Mater. Process. Technol. 80–81, 33–39 (1998)
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)
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)
Kocks, U.F.: On the temperature and stress dependence of the dislocation velocity stress exponent. ScriptaMetallurgica 4(1), 29–31 (1970)
Mecking, H., Kocks, U.F.: Physics and phenomenology of strain hardening: the FCC case. In: Progress in Material Science, pp. 171–273 (2003)
Rauch, E.F., Schmitt, J.H.: Dislocation substructures in mild steel deformed in simple shear. Mater. Sci. Eng. A 113, 441–448 (1989)
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)
Wood, W.A.: Bulletin/Institute of Metals 3, 5–6 (1955)
Plumbridge, W.J., Ryder, D.A.: Metall. Rev. 14(136), 119–142 (1969)
Ashby, M.F.: Philos. Mag. 21, 399–424 (1970)
Franciosi, P.: The concepts of latent hardening and strain hardening in metallic single crystals. Acta Metall. 33(9), 1601–1612 (1985)
Read, W.T., Shockley, W.: Dislocation models of grain boundaries. Phys. Rev. 78(3), 275–289 (1950)
Sedláček, R., Blum, W., Kratochvíl, J., Forest, S.: Subgrain formation during deformation—physical origin and consequences. Metall. Mater. Trans. A 33A, 319–327 (2002)
Bragg, W.L.: The structure of a cold worked metal. Proc. Phys. Soc. 52(1), 105–109 (1940)
Burger, J.M.: Geometrical considerations concerning the structural irregularities to be assumed in a crystal. Proc. Phys. Soc. 52(1), 23–33 (1940)
Read and Shockley: Quantitative predictions from dislocation models of crystal grain boundaries. Phys. Rev. 75(4), 692 (1949)
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)
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)
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)
Li, Y., Laird, C.: Masing behavior observed in monocrystalline copper during cyclic deformation. Mater. Sci. Eng. A 161(1), 23–29 (1993)
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)
Prnadtl, L.: Proceedings of the First International Congress of Applied Mechanics, Delft, 43 (1924)
Levy, M.: ComptesRendus. Académie des Sciences 70, 1323 (1870)
Mises, R.: Mechanik der festenKörperimplastisch-deformablenZustand. Nachrichten von der Gesellschaft der WissenschaftenzuGöttingen, Mathematisch-PhysikalischeKlasse 4, 582–593 (1913)
Reuss, A.: Berücksichtigung der elastischenFormänderung in der Plastizitätstheorie, ZeitschriftfürAngewandteMathematik und Mechanik (ZAMM—Journal of Applied Mathematics and Mechanics) 10(3), 266–274 (1930)
Lode, W.: Versucheüber den Einfluss der mittlerenHauptspannung auf das Fliessen der MetalleEisen. Kupfer und Nickel, ZeitschriftfürPhysik 36(11–12), 913–939 (1926)
Taylor, G.I., Quinney, H.: The Plastic Distortion of Metals. Philos. Trans. R. Soc. A 230, 681–693 (1931)
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)
Morrison, J.L.M., Shepherd, W.M.: An experimental investigation of plastic stress-strain relations. Proc. Inst. Mech. Eng. 163(1), 1–17 (1950)
Hill, R.: Mathematical Theory of Plasticity. Oxford University Press, ISBN: 978-0-19-850367-5 (1950)
Prager, W.: The theory of plasticity: a survey of recent achievements. Proc. Inst. Mech. Eng. 169(1), 41–57 (1955)
Drucker, D.C.: Stress-Strain Relations in the Plastic Range—A Survey of Theory and Experiment. O.N.R. Report, NR-041-032 (1950)
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)
Drucker, D.C.: A definition of stable inelastic material. J. Appl. Mech. 26, 101–106 (1959)
Mises, R.: Mechanik der plastischenFormänderung von Kristallen, ZeitschriftfürAngewandteMathematik und Mechanik (ZAMM—Journal of Applied Mathematics and Mechanics) 8(3), 161–185 (1928)
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)
Hill, R.: A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. A 193(1033), 281–297 (1948)
Prager, W.: A new method of analyzing stresses and strains in work hardening plastic solids. ASME J. Appl. Mech. 23, 493–496 (1956)
Mróz, Z.: On the description of anisotropic work hardening. J. Mech. Phys. Solids 15(3), 163–175 (1967)
Besseling, J.F.: A theory of elastic, plastic and creep deformations of an initially isotropic material. J. Appl. Mech. 25, 529–536 (1958)
Li, J.C.M.: Some elastic properties of an edge dislocation wall. Acta Metall. 8(8), 563–574 (1960)
Li, J.C.M.: Petch relation and grain boundary sources. Trans. Metall. Soc. AIME 227, 239–247 (1963)
Armstrong, P.J., Frederick, C.O.: A Mathematical Representation of the Multiaxial Bauschinger Effect, CEGB Report No.-RD/B/N 731 (1967)
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)
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)
Chaboche, J.L.: Time-independent constitutive theories for cyclic plasticity. Int. J. Plast 2(2), 149–188 (1986)
Chaboche, J.L.: On some modifications of kinematic hardening to improve the description of ratchetting effects. Int. J. Plast. 7(7), 661–678 (1991)
Guionnet, C.: Modeling of ratcheting in biaxial experiments. J. Eng. Mater. Technol. 114, 56–62 (1992)
Bari, S., Hassan, T.: Anatomy of coupled constitutive models for ratcheting simulation. Int. J. Plast. 16(3–4), 381–409 (2000)
Dafalias, Y.F., Popov, E.P.: Plastic internal variables formalism of cyclic plasticity. J. Appl. Mech. 43, 645–650 (1976)
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)
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)
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)
Ohno, N., Wang, J.D.: Kinematic hardening rules for simulation of ratchetting behaviour. Eur. J Mech. A Solids 13(4), 519–531 (1994)
McDowell, D.L.: Stress state dependence of cyclic ratcheting behavior of two rail steels. Int. J. Plast. 11, 397–421 (1995)
Jiang, Y., Sehitoglu, H.: Modeling of cyclic ratchetting plasticity—Part I: development of constitutive relations. ASME J. Appl. Mech. 63(3), 720 (1996)
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)
Voyiadjis, G.Z., Sivakumar, S.M.: Cyclic plasticity and ratchetting. Stud. Appl. Mech. 35, 253–295 (1994)
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)
Phillips, A., Lee, C.W.: Yield surfaces and loading surfaces. Exp. Recommendations Int. J. Solids Struct. 15, 715–729 (1979)
Tseng, N.T., Lee, G.C.: Simple plasticity model of two-surface type. ASCE J. Eng. Mech. 109(3), 795–810 (1983)
Voyiadjis, G.Z., Basuroychowdhury, I.N.: A plasticity model for multiaxial cyclic loading and ratcheting. Acta Mech. 126, 19–35 (1998)
Basuroychowdhury, I.N., Voyiadjis, G.Z.: A multiaxial cyclic plasticity model for nonproportional loading cases. Int. J. Plast. 14(9), 855–870 (1998)
Abdel-Karim, M., Ohno, N.: Kinematic hardening model suitable for ratcheting with steady-state. Int. J. Plast. 16(3–4), 225–240 (2000)
Zaverl Jr., F., Lee, D.: Constitutive relations for nuclear reactor core materials. J. Nucl. Mater. 75, 14–19 (1978)
Marquis, D.: Modelisation et Identification de I’Ecrouisage Anisotropic des Metaux, These Paris VI (1979)
Haupt, P., Kamlah, M., TsakmakisCh, : Continuous representation of hardening properties in cyclic plasticity. Int. J. Plast. 8, 803–817 (1992)
Jiang, Y., Kurath, P.: Characteristics of Armstrong-Frederick type plasticity models. Int. J. Plast. 12(3), 387–415 (1996)
SIMULIA ABAQUS CAE v6.8 user manual
Rahaman, S.M.: Finite Element Analysis and Related Numerical Schemes for Ratcheting Simulation, Ph.D. thesis submitted in North Carolina State University, USA (2006)
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)
Simo, J.C., Taylor, R.L.: Consistent tangent operators for rate-independent elastoplasticity. Comput. Methods Appl. Mech. Eng. 48, 101–118 (1985)
Simo, J.C., Taylor, R.L.: A return mapping algorithm for plane stress elastoplasticity. Int. J. Numer. Meth. Eng. 22, 649–670 (1986)
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)
Dunne, F., Patrinik, N.: Introduction to Computational Plasticity. Oxford University Press, ISBN: 978-0-19-856826-1 (2005)
Ypma, T.J.: Historical development of Newton-Raphson method. Soc. Ind. Appl. Math. 37(4), 531–551 (1995)
Acknowledgements
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.
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Bhattacharjee, S., Dhar, S., Acharyya, S.K., Gupta, S.K. (2019). Comparative Study of Cyclic Softening Modelling and Proposition of a Modification to ‘MARQUIS’ Approach. In: Sahoo, P., Davim, J. (eds) Advances in Materials, Mechanical and Industrial Engineering. INCOM 2018. Lecture Notes on Multidisciplinary Industrial Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-96968-8_7
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