Advances in Constitutive Models of Metals and Alloys During Hot Deformation
LIU Shaofei1, QU Yinhu2, WANG Chonglou1, WANG Yanlong2, CHENG Xiaole2, WANG Ke3
1 Engineering Training Center, Xi’an Polytechnic University, Xi’an 710048; 2 Materials Science and Engineering College,Xi’an Polytechnic University, Xi’an 710048; 3 College of Materials Science and Engineering, Chongqing University, Chongqing 400030
Abstract: Constitutive model is an important way to predict the hot deformation behavior of metals and alloys, which plays a significant role in selecting suitable deformation parameters and preventing defects for various metals and alloys. In the recent investigations of hot deformation behavior of metals and alloys, the original data for constitutive model is generally obtained by various kinds of hot deformation tests under different parameters, and then obtained constitutive model is imported to the corresponding part of simulation software like Deform and Ansys to predict the distribution law of strain, strain rate and temperature of materials during forging process, and further optimize the actual processing parameters, avoid defects, and reduce the waste of materials and resource. Considering the importance of constitutive models in optimizing the working parameters and preventing the defects, tremendous strudies on construction and selection of constitutive models were conducted. The research focuses of constitutive model of metals and alloys in hot deformation can be concluded in the following aspects: the test method to obtain the original data for constitutive model construction, the mathematical or physical method for establishing the constitutive model, the selection of constitutive models for specified object, the advantage and disadvantage of various models and their revision. In past years, the original data for constitutive model are usually acquired by hot compression, hot extension, hot torsion and Split Hopkinson Pressure Bar tests under different hot deformation parameters. The common constitutive models can be generally classified as phenomenological method, physical-based method and the ones using artificial neural network. Each kind of model has its unique applicability, advantages and disadvantages. The disadvantages finally lead to the apparent fitting deviation of several deformation parameters. Aiming at reducing the deviation, scholars over the world have been devoted to revise and improve the models. Apart to the intrinsic reasons of the models, the lack of consideration of the grand factors like friction and deformation heat would also lead to the deviation. Presently, the common phenomenological constitutive models include Arrhenius model, Johnson-Cook model and so forth, the physical-based model includes Zerilli-Armstrong model, while the artificial neural network model is always conducted by three layers. These models have various advantages and disadvantages in dealing with data or physical meaning. This article summarizes the researches and developing direction of constitutive models from the aspects of the experimental methods to achieve the original data, the kinds and revision, and applications of the constitutive models. Also, the advantages and disadvantages of the typical constitutive models are discussed. The apparent deviation phenomenon of predicted and experimental data under several deformation parameters and the modified methods are pointed out. Finally, the future investigation direction of constitutive models of metals and alloys during hot deformation is predicted.
刘少飞, 屈银虎, 王崇楼, 王彦龙, 成小乐, 王柯. 金属和合金高温变形过程本构模型的研究进展[J]. 《材料导报》期刊社, 2018, 32(13): 2241-2251.
LIU Shaofei, QU Yinhu, WANG Chonglou, WANG Yanlong, CHENG Xiaole, WANG Ke. Advances in Constitutive Models of Metals and Alloys During Hot Deformation. Materials Reports, 2018, 32(13): 2241-2251.
1 Xiao J, Li D S, Li X Q, et al. Constitutive modeling and microstructure change of Ti-6Al-4V during the hot tensile deformation[J].Journal of Alloys and Compounds,2012,541(15):346. 2 Ashtiani H R R, Parsa M H, Bisadi H. Constitutive equations for elevated temperature flow behavior of commercial purity aluminum[J].Materials Science and Engineering A,2012,545(5):61. 3 Sun Z C, Yang H, Han G J, et al. A numerical model based on internal-state-variable method for the microstructure evolution during hot-working process of TA15 titanium alloy[J].Materials Science and Engineering A,2010,527(15):3464. 4 Tang J P, Wu W T, Walters J. DEFORM system structure and description[M].SFTC Paper,1995:5. 5 Duan X J, Sheppard T. The influence of the constitutive equation on the simulation of a hot rolling process[J].Journal of Materials Processing Technology,2004,150(1-2):100. 6 Zhou M, Clode M P. Constitutive equations for modeling flow softening due to dynamic recovery and heat generation during plastic deformation[J].Mechanics of Materials,1998,27(2):63. 7 Li M Q, Chen S H, Xiong A M, et al. Acquiring a novel constitutive equation of a TC6 alloy at high-temperature deformation[J].Journal of Materials Engineering and Performance,2005,42(2):263. 8 Feng D, Zhang X M, Liu S D, et al. Constitutive equation and hot deformation behavior of homogenized Al-7.68Zn-2.12Mg-1.98Cu-0.12Zr alloy during compression at elevated temperature[J].Mate-rials Science and Engineering A,2014,608(7):63. 9 Zhang P, Li F G, Wan Q. Constitutive equation and processing map for hotdeformation of SiC particles reinforced metal matrix compo-sites[J].Journal of Materials Engineering and Performance,2010,19(9):1290. 10 Poursina M, Ebrahimi H, Parvizian J. Flow stress behavior of two stainless steels: An experimental-numerical investigation[J].Journal of Materials Processing Technology,2008,199(1):287. 11 Avramovic-Cingara G, McQueen H J, Perovic D D. Comparison of torsion and compression constitutive analyses for elevated temperature deformation of Al-Li-Cu-Mn alloy[J].Materials Science and Technology,2003,19(1):11. 12 Huang Y C, Lin Y C, Deng J, et al. Hot tensile deformation beha-viors and constitutive model of 42CrMo steel[J].Materials and Design,2014,53(2):349. 13 Lou Y, Chen H, Ke C X, et al. Hot tensile deformation characteristics and processing map of extruded AZ80 Mg alloys[J].Journal of Materials Engineering and Performance,2014,23(5):1904. 14 Spigarelli S, Mehtedi M E. A new constitutive model for the plastic flow of metals at elevated temperatures[J].Journal of Materials Engineering and Performance,2014,23(2):658. 15 Lin Y C, Ding Y, Chen M S, et al. A new phenomenological constitutive model for hot tensile deformation behaviors of a typical Al-Cu-Mg alloy[J].Materials and Design,2013,52(12):118. 16 Medina S F,Hernandez C A. General expression of the Zener-Hollomon parameter as a function of the chemical composition of low alloy and micro-alloyed steels[J].Acta Materialia,1996,44(1):137. 17 Karhausen K F, Roters F. Development and application of constitutive equations for the multiple-stand hot rolling of Al-alloys[J].Journal of Materials Processing Technology,2002,123(1):155. 18 Chen M, Niu Q L, An Q L, et al. Johnson-Cook constitutive equation for titanium alloy TC11[J].Key Engineering Materials,2014,589-590:140. 19 Mirzakhani B. Mathematical modeling of flow behaviour of API-X70 during hot torsion testing[J].Advanced Materials Research,2011,264-265:60. 20 Yu D H. Modeling high-temperature tensile deformation behavior of AZ31B magnesium alloy considering strain effects[J].Materials and Design,2013,51(10):323. 21 Khamei A A, Dehghani K, Mahmudi R. Modeling the hot ductility of AA6061 aluminum alloy after severe plastic deformation[J].Journal of Metals,2015,67(5):966. 22 Castellanos J, Rieiro I, Carsi M, et al. Analysis of adiabatic heating and its influence on the Garofalo equation parameters of a high nitrogen steel[J].Materials Science and Engineering A,2009,517(1-2):191. 23 Zhou M, Clode M P. Hot torsion tests to model the deformation behavior of aluminium alloys at hot working temperatures[J].Journal of Materials Processing Technology,1997,72(1):78. 24 Sellars C M, McTegart W J. On the mechanism of hot deformation[J].Acta Metallurgica,1966,14(9):1136. 25 Li J, Li F G, Cai J, et al. Flow behavior modeling of the 7050 aluminum alloy at elevated temperatures considering the compensation of strain[J].Materials and Design,2012,42(12):369. 26 Ou L, Nie Y F, Zheng Z Q. Strain compensation of the constitutive equation for high temperature flow stress of a Al-Cu-Li alloy[J].Journal of Materials Engineering and Performance,2014,23(1):25. 27 Li H Y, Li Y H, Wei D D, et al. Constitutive equation to predict elevated temperature flow stress of V150 grade oil casing steel[J].Materials Science and Engineering A,2011,530(1):367. 28 Li J B, Liu Y, Wang Y, et al. Constitutive equation and processing map for hot compressed as-cast Ti-43Al-4Nb-1.4W-0.6B alloy[J].Transactions of Nonferrous Metals Society of China,2013,23(11):3383. 29 Saravanan L, Senthilvelan T. Constitutive equation and microstructure evaluation of an extruded aluminum alloy[J].Journal of Mate-rials Research and Technology,2016,5(1):21. 30 Samantaray D, Mandal S, Bhaduti A K. Constitutive analysis to predict high-temperature flow stress in modified 9Cr-1Mo (P91) steel[J].Materials and Design,2010,31:981. 31 Liu Y C, Chen M S, Zhong J. Constitutive modeling for elevated temperature flow behavior of 42CrMo steel[J].Computional Mate-rials Science,2008,42(3):470. 32 Mandal S, Rakesh V, Sivaprasad P V, et al. Constitutive equations to predict high temperature stress in a Ti-modified austenitic stainless steel[J].Materials Science and Engineering A,2009,500(1-2):114. 33 Cai J, Li F G, Liu T Y, et al. Constitutive equations for elevated temperature flow stress of Ti-6Al-4V alloy considering the effect of strain[J].Materials and Design,2011,32(3):1144. 34 Ding Z Y, Hu Q D, Zeng L, et al. Hot deformation characteristic of as-cast high-Cr ultra-super-critical rotor steel with columnar grains[J].International Journal of Minerals, Metallurgy and Materials,2016,23(11):1275. 35 Qian D S, Peng Y Y, Deng J D. Hot deformation behavior and constitutive modeling of Q345E alloy steel under hot compression[J].Journal of Central South University,2017,24(2):284. 36 Johnson G R, Cook W H. A constitutive model and data for metals subjected to large strain, high strain rates and high temperature[C]∥Proceedings of the Seventh International Symposium on Ballistics. Den Haag, Netherlands,1983:541. 37 Shin H, Kim J B. A phenomenological constitutive equation to describe various flow stress behaviors of materials in wide strain rate and temperature regimes[J].Journal of Engineering Materials and Technology,2010,132(2):0210091. 38 Follansbee P S, Kocks U F. A constitutive description of the deforma-tion of copper based on the use of the mechanical threshold stress as an internal state variable[J].Acta Metallurgica,1988,36(1):81. 39 Preston D L, Tonks D L, Wallace D C. Model of plastic deformation for extreme loading conditions[J].Journal of Applied Physics,2003,93(1):211. 40 Follansbee P S. High strain rate deformation in FCC metals and alloys[J].Marcel Dekker Inc (Mechanical Engineering 52),1986,451:155. 41 Hoge K G, Mukherjee A K. The temperature and strain rate depen-dency of the flow stress of tantalum[J].Journal of Materials Science,1977,12(8):1666. 42 Kimj B, Shin H. Comparison of plasticity models for tantalum and a modification of the PTW model for wide ranges of strain, strain rate, and temperature[J].International Journal of Impact Enginee-ring,2009,36(5):746. 43 Shin H, Yoo Y H. Effect of the velocity of a single flying plate on the protection capability against obliquely impacting Long-Rod penetrators[J].Combustion Explosion and Shock Waves,2003,39(5):591. 44 Cai J, Wang K S, Zhai P, et al. A modified Johnson-Cook constitutive equation to predict hot deformation behavior of Ti-6Al-4V alloy[J].Journal of Materials Engineering and Performance,2015,24(1):32. 45 Zerilli P J, Armstrong R W. Dislocation-mechanics-based constitutive relations for material dynamics calculations[J].Journal of Applied Physics,1987,61:1816. 46 Samantaray D, Mandal S, Borah U, et al. A thermo-viscoplastic constitutive model to predict elevated-temperature flow behaviour in a titanium-modified austenitic stainless steel[J].Materials Science and Engineering A,2009,526(1-2):1. 47 Zhao J W, Ding H, Zhao W J, et al. Modelling of the hot deformation behavior of a titanium alloy using constitutive equations and artificial neural network[J].Computational Materials Science,2014,92:47. 48 Hornik K, Stinchcombe M, White H. Multilayer feedforward networks are universal approximators[J].Neural Networks,1989,2(5):359. 49 Genel K, Kurnaz S C, Durman M. Modeling of tribological properties of alumina fiber reinforced zinc-aluminum composites using artificial neural network[J].Materials Science and Engineering A,2003,363(1-2):203. 50 Chen H S, Feng Y, Ma F J, et al. Isothermal compression flow stress prediction of Ti-6Al-3Nb-2Zr-1Mo alloy based on BP-ANN[J].Rare Metal Materials and Engineering,2016,45(6):1549(in Chinese). 陈海生,冯勇,马凡蛟,等.基于BP-ANN网络Ti-6Al-3Nb-2Zr-1Mo合金等温压缩流变应力预测[J].稀有金属材料与工程,2016,45(6):1549. 51 He A, Wang X T, Xie G L, et al. Modified Arrhenius-type constitutive model and artificial neural network-based model for constitutive relationship of 316LN stainless steel during hot deformation[J].Journal of Iron and Steel Research(International),2015,22(8):721. 52 Cabrera J M, Al Omar A, Jonas J J, et al. Modeling the flow beha-vior of a medium carbon microalloyed steel under hot working conditions[J].Metallurgical and Materials Transaction A: Physical Metallurgy and Materials Science,1997,28(11):2233. 53 Mirzadeh H, Cabrera J M, Najafizadeh A. Constitutive relationships for hot deformation of austenite[J].Acta Materialia,2011,59(16):6441. 54 Frost H J, Ashby M F. Deformation-mechanism maps: The plasticity and creep of metals and ceramics[M].New York: Pergamon Press,1982:21. 55 Mirzadeh H. Constitutive description of 7075 aluminum alloy during hot deformation by apparent and physically-based approaches double multivariate nonlinear regression (DMNR) model[J].Journal of Materials Engineering and Performance,2015,24(3):1095. 56 Lenard J G. Modeling hot deformation of steels[M].Berlin:Springer,1989:101. 57 Sellars C M, Tegart W J. Hot workability[J].International Metallurgical Reviews,1972,17:1. 58 Xiao M L, Li F G, Zhao W, et al. Constitutive equation for elevated temperature flow behavior of TiNiNb alloy based on orthogonal analysis[J].Materials and Design,2012,35:184. 59 Wang L, Liu F, Zuo Q, et al. Prediction of flow stress for N08028 alloy under hot working conditions[J].Materials and Design,2013,47(3):737. 60 Wu B, Li M Q, Ma D W. The flow behavior and constitutive equations in isothermal compression of 7050 aluminum alloy[J].Materials Science and Engineering A,2012,542(4):79. 61 Sellars C. The kinetics of softening processes during hot working of austenite[J].Journal of Physics,1985,35(3):239. 62 Zener C, Hollomon H. Effect of strain rate upon plastic flow of steel[J].Journal of Applied Physics,1944,15(1):15. 63 Shen G S, Semiatin S L, Shivpuri R. Modeling microstructural development during the forging of Waspaloy[J].Metallurgical Mate-rials and Transaction A,1995,26(7):1795. 64 Wang Z J, Qiang H F, Wang X R, et al. Constitutive model for a new kind of metastable β titanium alloy during hot deformation[J].Transactions of Nonferrous Metals Society of China,2012,22:634. 65 Gottstein G, Kocks U F. Dynamic recrystallization and dynamic recovery in 〈111〉 single crystals of nickel and copper[J].Acta Materialia,1983,31(1):175. 66 Gottstein G, Shvindlerman L S. Grain boundary migration in metals: Thermodynamics, kinetics, applications, second edition[M].Boca Raton: CRC Press,2009:211. 67 Sellars C M, Whiteman J A. Recrystallization and grain growth in hot rolling[J].Acta Materialia,1979,13(3-4):187. 68 Liang H Q, Guo H Z, Ning Y Q, et al. Analysis on the constitutive relationship of TC18 titanium alloy based on the softening mechanism[J].Acta Metallurgica Sinica,2014,50(7):871. 69 Lin Y C, Li K K, Li H B, et al. New constitutive model for high-temperature deformation behavior of inconel 718 superalloy[J].Materials and Design,2015,74(5):108. 70 Molinari A, Ravichandran G. Constitutive modeling of high-strain-rate deformation in metals based on the evolution of an effective microstructural length[J].Mechanics of Materials,2005,37(7):737. 71 Voce E. The relationship between stress and strain for homogeneous deformation[J].Journal of the Institute Metals,1948,74:537. 72 Xie B S, Cai Q W, Yu W, et al. Prediction for flow stress of 95CrMo hollow steel during hot compression[J].Acta Metallurgica Sinica (English Letters),2017,30(3):250. 73 Wang Y H, Han F B, Lou H C, et al. Internal-atate-variable based constitutive modeling for near β Ti-7Mo-3Al-3Nb-3Cr alloy during hot deformation process[J].Rare Metal Materials and Engineering,2015,44(8):1883. 74 Yu H C, Dong C L, Jiao Z H, et al. High temperature creep and fatigue behavior and life prediction method of a TiAl alloy[J].Acta Metallurgica Sinica 2013,49(11):1311(in Chinese). 于慧臣,董成利,焦泽辉,等.一种合金的高温蠕变和疲劳行为及其寿命预测方法[J].金属学报,2013,49(11):1311. 75 Yin X N, Zhan L H, Zhao J. Establishment of steady creep constitutive equation of 2219 aluminum alloy[J].Transactions of Nonferrous Metals Society of China,2014,24(9):2251(in Chinese). 尹旭妮,湛利华,赵俊.2219铝合金稳态蠕变本构方程的建立[J].中国有色金属学报,2014,24(9):2251. 76 Zong Q. An investigation of the high temperature creep and superplastic deformation of AZ31 magnesium alloy[D].Wuhan: Wuhan University of Science and Technology,2007(in Chinese). 宗钦.AZ31镁合金高温蠕变及超塑性研究[D].武汉:武汉科技大学,007. 77 Xu L, Dai G Z, Huang X M, et al. Foundation and application of Al-Zn-Mg-Cu alloy flow stress constitutive equation in friction screw press die forging[J].Materials and Design,2013,47(5):465. 78 Luo J, Wu B, Li M Q. 3D finite element simulation of microstructure evolution in blade forging of Ti-6Al-4V alloy based on the internal state variable models[J].International Journal of Minerals, Me-tallurgy and Materials,2012,19(3):122. 79 Mandal S, Rakesh V, Sivaprasad P V, et al. Constitutive equations to predict high temperature flow stress in a modified austenitic stainless steel[J].Materials Science and Engineering A,2002,322(1-2):43. 80 Mostafaei M A, Kazeminezhad M. Analyses on the flow stress of an Al-Mg alloy during dynamic recovery[J].Journal of Materials Engineering and Performance,2013,22(3):700. 81 Mostafaei M A, Kazeminezhad M. Hot deformation behavior of hot extruded Al-6Mg alloy[J].Materials Science and Engineering A,2012,535(2):216. 82 Meng G, Li B L, Li H M, et al. Hot deformation behavior of an Al-5.7wt% Mg alloy with erbium[J].Materials Science and Engineering A,2009,516(1-2):131. 83 Gholamzade A, Karimitaheri A. The prediction of hot flow behavior of Al-6%Mg alloy[J].Mechanics Research Communications,2009,36(2):252. 84 Ebrahimi R, Najafizadeh A. A new method for evaluation of friction in bulk metal forming[J].Journal of Materials Processing and Technology,2004,152(2):136. 85 Goetz R L, Semiatin S L. The adiabatic correction factor for defor-mation heating during the uniaxial compression test[J].Journal of Materials Engineering and Performance,2001,10(6):710. 86 Zhao D. Temperature correction in compression tests[J].Journal of Materials Processing and Technology,1993,36(4):467. 87 Fan J K, Kou H C, Lai M J, et al. Characterization of hot deformation behavior of a new near beta titanium alloy: Ti-7333[J].Mate-rials and Design,2013,49(8):945.