Review on cyclic plasticity of magnesium alloys: Experiments and constitutive models

Guo-zheng Kang; Hang Li

doi:10.1007/s12613-020-2216-8

Volume 28 Issue 4

Apr. 2021

Turn off MathJax

Article Contents

Article Navigation > International Journal of Minerals, Metallurgy and Materials > 2021 > 28(4): 567-589

Guo-zheng Kangand Hang Li, Review on cyclic plasticity of magnesium alloys: Experiments and constitutive models, Int. J. Miner. Metall. Mater., 28(2021), No. 4, pp. 567-589. https://doi.org/10.1007/s12613-020-2216-8

Cite this article as:

Guo-zheng Kangand Hang Li, Review on cyclic plasticity of magnesium alloys: Experiments and constitutive models, Int. J. Miner. Metall. Mater., 28(2021), No. 4, pp. 567-589. https://doi.org/10.1007/s12613-020-2216-8

Citation:

PDF( 2480 KB)

Invited Review

Review on cyclic plasticity of magnesium alloys: Experiments and constitutive models

Guo-zheng Kang^1,2, and
Hang Li¹

+ Author Affiliations

Corresponding author:
Guo-zheng Kang E-mail: guozhengkang@home.swjtu.edu.cn
Received: 9 August 2020; Revised: 26 October 2020; Accepted: 27 October 2020; Available online: 4 November 2020

Abstract

Abstract

Fatigue analysis has always been a concern in the design and assessment of Mg alloy structure components subjected to cyclic loading, and research on the cyclic plasticity is fundamental to investigate the corresponding fatigue failure. Thus, this work reviews the progress in the cyclic plasticity of Mg alloys. First, the existing macroscopic and microscopic experimental results of Mg alloys are summarized. Then, corresponding macroscopic phenomenological constitutive models and crystal plasticity-based models are reviewed. Finally, some conclusions and recommended topics on the cyclic plasticity of Mg alloys are provided to boost the further development and application of Mg alloys.
- magnesium alloy;
- cyclic plasticity;
- macroscopic experiments;
- microscopic observations;
- constitutive model

FullText(HTML)

Supplements(0)

References(127)

References

[1]	E. Aghion, B. Bronfin, and D. Eliezer, The role of the magnesium industry in protecting the environment, J. Mater. Process. Technol., 117(2001), No. 3, p. 381. doi: 10.1016/S0924-0136(01)00779-8
[2]	D. Eliezer, E. Aghion, and F.H. (Sam) Froes, Magnesium science, technology and applications, Adv. Perform. Mater., 5(1998), No. 3, p. 201. doi: 10.1023/A:1008682415141
[3]	B.L. Mordike and T. Ebert, Magnesium: Properties—applications—potential, Mater. Sci. Eng. A, 302(2001), No. 1, p. 37. doi: 10.1016/S0921-5093(00)01351-4
[4]	T.M. Pollock, Weight loss with magnesium alloys, Science, 328(2010), No. 5981, p. 986. doi: 10.1126/science.1182848
[5]	E.A. Ball and P.B. Prangnell, Tensile-compressive yield assymetries in high strength wrought magnesium alloys, Scr. Metall. Mater., 31(1994), No. 2, p. 111. doi: 10.1016/0956-716X(94)90159-7
[6]	M.R. Barnett, Z. Keshavarz, A.G. Beer, and D. Atwell, Influence of grain size on the compressive deformation of wrought Mg–3Al–1Zn, Acta Mater., 52(2004), No. 17, p. 5093. doi: 10.1016/j.actamat.2004.07.015
[7]	X.Y. Lou, M. Li, R.K. Boger, S.R. Agnew, and R.H. Wagoner, Hardening evolution of AZ31B Mg sheet, Int. J. Plast., 23(2007), No. 1, p. 44. doi: 10.1016/j.ijplas.2006.03.005
[8]	J.P. Nobre, U. Noster, M. Kornmeier, A.M. Dias, and B. Scholtes, Deformation asymmetry of AZ31 wrought magnesium alloy, Key Eng. Mater., 230-232(2002), p. 267. doi: 10.4028/www.scientific.net/KEM.230-232.267
[9]	Y. Xiong, Q. Yu, and Y.Y. Jiang, An experimental study of cyclic plastic deformation of extruded ZK60 magnesium alloy under uniaxial loading at room temperature, Int. J. Plast., 53(2014), p. 107. doi: 10.1016/j.ijplas.2013.07.008
[10]	H. Li, G.Z. Kang, C. Yu, and Y.J. Liu, Experimental investigation on temperature-dependent uniaxial ratchetting of AZ31B magnesium alloy, Int. J. Fatigue, 120(2019), p. 33. doi: 10.1016/j.ijfatigue.2018.10.020
[11]	A. Gryguc, S.K. Shaha, S.B. Behravesh, H. Jahed, M. Wells, B. Williams, and X. Su, Monotonic and cyclic behaviour of cast and cast-forged AZ80 Mg, Int. J. Fatigue, 104(2017), p. 136. doi: 10.1016/j.ijfatigue.2017.06.038
[12]	J. Albinmousa, H. Jahed, and S. Lambert, Cyclic behaviour of wrought magnesium alloy under multiaxial load, Int. J. Fatigue, 33(2011), No. 8, p. 1127. doi: 10.1016/j.ijfatigue.2011.01.009
[13]	J. Albinmousa, H. Jahed, and S. Lambert, Cyclic axial and cyclic torsional behaviour of extruded AZ31B magnesium alloy, Int. J. Fatigue, 33(2011), No. 11, p. 1403. doi: 10.1016/j.ijfatigue.2011.04.012
[14]	S. Begum, D.L. Chen, S. Xu, and A.A. Luo, Strain-controlled low-cycle fatigue properties of a newly developed extruded magnesium alloy, Metall. Mater. Trans. A, 39(2008), No. 12, p. 3014. doi: 10.1007/s11661-008-9677-0
[15]	C. Chen, T.M. Liu, C.L. Lv, L.W. Lu, and D.Z. Luo, Study on cyclic deformation behavior of extruded Mg–3Al–1Zn alloy, Mater. Sci. Eng. A, 539(2012), p. 223. doi: 10.1016/j.msea.2012.01.084
[16]	L.J. Chen, C.Y. Wang, W. Wu, Z. Liu, G.M. Stoica, L. Wu, and P.K. Liaw, Low-cycle fatigue behavior of an as-extruded AM50 magnesium alloy, Metall. Mater. Trans. A, 38(2007), No. 13, p. 2235. doi: 10.1007/s11661-007-9181-y
[17]	T. Hama, Y. Kariyazaki, N. Hosokawa, H. Fujimoto, and H. Takuda, Work-hardening behaviors of magnesium alloy sheet during in-plane cyclic loading, Mater. Sci. Eng. A, 551(2012), p. 209. doi: 10.1016/j.msea.2012.05.009
[18]	S. Dong, Q. Yu, Y.Y. Jiang, J. Dong, F.H. Wang, L. Jin, and W.J. Ding, Characteristic cyclic plastic deformation in ZK60 magnesium alloy, Int. J. Plast., 91(2017), p. 25. doi: 10.1016/j.ijplas.2017.01.005
[19]	Y. Xiong, Q. Yu, and Y.Y. Jiang, Cyclic deformation and fatigue of extruded AZ31B magnesium alloy under different strain ratios, Mater. Sci. Eng. A, 649(2016), p. 93. doi: 10.1016/j.msea.2015.09.084
[20]	A.H. Pahlevanpour, S.M.H. Karparvarfard, S.K. Shaha, S.B. Behravesh, S. Adibnazari, and H. Jahed, Anisotropy in the quasi-static and cyclic behavior of ZK60 extrusion: Characterization and fatigue modeling, Mater. Des., 160(2018), p. 936. doi: 10.1016/j.matdes.2018.10.026
[21]	C. Wang, T.J. Luo, J.X. Zhou, and Y.S. Yang, Anisotropic cyclic deformation behavior of extruded ZA81M magnesium alloy, Int. J. Fatigue, 96(2017), p. 178. doi: 10.1016/j.ijfatigue.2016.11.020
[22]	S.M.H. Karparvarfard, S.K. Shaha, S.B. Behravesh, H. Jahed, and B.W. Williams, Fatigue characteristics and modeling of cast and cast-forged ZK60 magnesium alloy, Int. J. Fatigue, 118(2019), p. 282. doi: 10.1016/j.ijfatigue.2018.03.019
[23]	S.M.A.K. Mohammed, D.J. Li, X.Q. Zeng, and D.L. Chen, Cyclic deformation behavior of a high zinc-containing cast magnesium alloy, Int. J. Fatigue, 125(2019), p. 1. doi: 10.1016/j.ijfatigue.2019.03.015
[24]	K. Máthis, P. Beran, J. Čapek, and P. Lukáš, In-situ neutron diffraction and acoustic emission investigation of twinning activity in magnesium, J. Phys. Conf. Ser., 340(2012), art. No. 12096. doi: 10.1088/1742-6596/340/1/012096
[25]	S.H. Park, J.H. Lee, B.G. Moon, and B.S. You, Tension–compression yield asymmetry in as-cast magnesium alloy, J. Alloys Compd., 617(2014), p. 277. doi: 10.1016/j.jallcom.2014.07.164
[26]	S. Begum, D.L. Chen, S. Xu, and A.A. Luo, Effect of strain ratio and strain rate on low cycle fatigue behavior of AZ31 wrought magnesium alloy, Mater. Sci. Eng. A, 517(2009), No. 1-2, p. 334. doi: 10.1016/j.msea.2009.04.051
[27]	C. Wang, T.J. Luo, and Y.S. Yang, Low cycle fatigue behavior of the extruded AZ80 magnesium alloy under different strain amplitudes and strain rates, J. Magnesium Alloys, 4(2016), No. 3, p. 181. doi: 10.1016/j.jma.2016.07.002
[28]	G. Chen, J.W. Gao, Y. Cui, H. Gao, X. Guo, and S.Z. Wu, Effects of strain rate on the low cycle fatigue behavior of AZ31B magnesium alloy processed by SMAT, J. Alloys Compd., 735(2018), p. 536. doi: 10.1016/j.jallcom.2017.11.141
[29]	J.H. Kim, D. Kim, Y.S. Lee, M.G. Lee, K. Chung, H.Y. Kim, and R.H. Wagoner, A temperature-dependent elasto-plastic constitutive model for magnesium alloy AZ31 sheets, Int. J. Plast., 50(2013), p. 66. doi: 10.1016/j.ijplas.2013.04.001
[30]	K. Piao, J.K. Lee, J.H. Kim, H.Y. Kim, K. Chung, F. Barlat, and R.H. Wagoner, A sheet tension/compression test for elevated temperature, Int. J. Plast., 38(2012), p. 27. doi: 10.1016/j.ijplas.2012.03.009
[31]	L. Jiang, J.J. Jonas, R.K. Mishra, A.A. Luo, A.K. Sachdev, and S. Godet, Twinning and texture development in two Mg alloys subjected to loading along three different strain paths, Acta Mater., 55(2007), No. 11, p. 3899. doi: 10.1016/j.actamat.2007.03.006
[32]	F.H. Wang, M.L. Feng, Y.Y. Jiang, J. Dong, and Z.Y. Zhang, Cyclic shear deformation and fatigue of extruded Mg–Gd–Y magnesium alloy, J. Mater. Sci. Technol., 39(2020), p. 74. doi: 10.1016/j.jmst.2019.08.025
[33]	J.X. Zhang, Q. Yu, Y.Y. Jiang, and Q.Z. Li, An experimental study of cyclic deformation of extruded AZ61A magnesium alloy, Int. J. Plast., 27(2011), No. 5, p. 768. doi: 10.1016/j.ijplas.2010.09.004
[34]	Q. Yu, J.X. Zhang, Y.Y. Jiang, and Q.Z. Li, Multiaxial fatigue of extruded AZ61A magnesium alloy, Int. J. Fatigue, 33(2011), No. 3, p. 437. doi: 10.1016/j.ijfatigue.2010.09.020
[35]	H. Li, G.Z. Kang, Y.J Liu, and H. Jiang, Non-proportionally multiaxial cyclic deformation of AZ31 magnesium alloy: Experimental observations, Mater. Sci. Eng. A, 671(2016), p. 70. doi: 10.1016/j.msea.2016.06.043
[36]	A.A. Roostaei and H. Jahed, Multiaxial cyclic behaviour and fatigue modelling of AM30 Mg alloy extrusion, Int. J. Fatigue, 97(2017), p. 150. doi: 10.1016/j.ijfatigue.2016.12.037
[37]	A. Gryguć, S.B. Behravesh, S.K. Shaha, H. Jahed, M. Wells, B. Williams, and X. Su, Multiaxial cyclic behaviour of extruded and forged AZ80 Mg alloy, Int. J. Fatigue, 127(2019), p. 324. doi: 10.1016/j.ijfatigue.2019.06.015
[38]	J.L. Chaboche, A review of some plasticity and viscoplasticity constitutive theories, Int. J. Plast., 24(2008), No. 10, p. 1642. doi: 10.1016/j.ijplas.2008.03.009
[39]	G.Z. Kang, Ratchetting: Recent progresses in phenomenon observation, constitutive modeling and application, Int. J. Fatigue, 30(2008), No. 8, p. 1448. doi: 10.1016/j.ijfatigue.2007.10.002
[40]	O. Nobutada, Recent topics in constitutive modeling of cyclic plasticity and viscoplasticity, Appl. Mech. Rev., 43(1990), No. 11, p. 283. doi: 10.1115/1.3119155
[41]	G.Z. Kang, Y. Chao, Y.J. Liu, and G.F. Quan, Uniaxial ratchetting of extruded AZ31 magnesium alloy: Effect of mean stress, Mater. Sci. Eng. A, 607(2014), p. 318. doi: 10.1016/j.msea.2014.04.023
[42]	Z.F. Yan, D.H. Wang, X.L. He, W.X. Wang, H.X. Zhang, P. Dong, C.H. Li, Y.L. Li, J. Zhou, Z. Liu, and L.Y. Sun, Deformation behaviors and cyclic strength assessment of AZ31B magnesium alloy based on steady ratcheting effect, Mater. Sci. Eng. A, 723(2018), p. 212. doi: 10.1016/j.msea.2018.03.023
[43]	Y.C. Lin, Z.H. Liu, X.M. Chen, and J. Chen, Uniaxial ratcheting and fatigue failure behaviors of hot-rolled AZ31B magnesium alloy under asymmetrical cyclic stress-controlled loadings, Mater. Sci. Eng. A, 573(2013), p. 234. doi: 10.1016/j.msea.2013.03.004
[44]	Y.C. Lin, X.M. Chen, and G. Chen, Uniaxial ratcheting and low-cycle fatigue failure behaviors of AZ91D magnesium alloy under cyclic tension deformation, J. Alloys Compd., 509(2011), No. 24, p. 6838. doi: 10.1016/j.jallcom.2011.03.129
[45]	L. Meng, S. Hallais, A. Tanguy, W.F. Chen, and M.L. Feng, The effect of stress rate on ratchetting behavior of rolled AZ31B magnesium alloy at 393 K and room temperature, Mater. Res. Express, 6(2019), No. 8, art. No. 086510. doi: 10.1088/2053-1591/ab19e9
[46]	U. Noster and B. Scholtes, Cyclic deformation behavior of magnesium alloys AZ31 and AZ91 in the temperature range 20–300°C, Mater. Sci. Forum., 419-422(2003), p. 103. doi: 10.4028/www.scientific.net/MSF.419-422.103
[47]	F. Castro and Y.Y. Jiang, Fatigue of extruded AZ31B magnesium alloy under stress- and strain-controlled conditions including step loading, Mech. Mater., 108(2017), p. 77. doi: 10.1016/j.mechmat.2017.03.002
[48]	M.H. Yoo, Slip, twinning, and fracture in hexagonal close-packed metals, Metall. Trans. A, 12(1981), No. 3, p. 409. doi: 10.1007/BF02648537
[49]	A. Couret and D. Caillard, An in situ study of prismatic glide in magnesium—I. The rate controlling mechanism, Acta Metall., 33(1985), No. 8, p. 1447. doi: 10.1016/0001-6160(85)90045-8
[50]	R.E. Reed-Hill and W.D. Robertson, Deformation of magnesium single crystals by nonbasal slip, JOM, 9(1957), No. 4, p. 496. doi: 10.1007/BF03397907
[51]	P.J.F. Stohr and J.P. Poirier, Etude en microscopie electronique du glissement pyramidal $ \left\{ {11\bar 22} \right\}\left\langle {11\bar 23} \right\rangle $ dans le magnesium, Philos. Mag., 25(1972), No. 6, p. 1313. doi: 10.1080/14786437208223856
[52]	T. Obara, H. Yoshinga, and S. Morozumi, $ \left\{ {11\bar 22} \right\}\left\langle {11\bar 23} \right\rangle $ Slip system in magnesium, Acta Metall., 21(1973), No. 7, p. 845. doi: 10.1016/0001-6160(73)90141-7
[53]	S. Ando, M. Tanaka, and H. Tonda, Pyramidal slip in magnesium alloy single crystals, Mater. Sci. Forum., 419-422(2003), p. 87. doi: 10.4028/www.scientific.net/MSF.419-422.87
[54]	E. Lilleodden, Microcompression study of Mg (0001) single crystal, Scripta Mater., 62(2010), No. 8, p. 532. doi: 10.1016/j.scriptamat.2009.12.048
[55]	C.M. Byer, B. Li, B.Y. Cao, and K.T. Ramesh, Microcompression of single-crystal magnesium, Scripta Mater., 62(2010), No. 8, p. 536. doi: 10.1016/j.scriptamat.2009.12.017
[56]	K.Y. Xie, Z. Alam, A. Caffee, and K.J. Hemker, Pyramidal I slip in c-axis compressed Mg single crystals, Scripta Mater., 112(2016), p. 75. doi: 10.1016/j.scriptamat.2015.09.016
[57]	C. Guillemer, M. Clavel, and G. Cailletaud, Cyclic behavior of extruded magnesium: Experimental, microstructural and numerical approach, Int. J. Plast., 27(2011), No. 12, p. 2068. doi: 10.1016/j.ijplas.2011.06.002
[58]	W. Wu, Y.F. Gao, N. Li, C.M. Parish, W.J. Liu, P.K. Liaw, and K. An, Intragranular twinning, detwinning, and twinning-like lattice reorientation in magnesium alloys, Acta Mater., 121(2016), p. 15. doi: 10.1016/j.actamat.2016.08.058
[59]	L. Lu, B.X. Bie, Q.H. Li, T. Sun, K. Fezzaa, X.L. Gong, and S.N. Luo, Multiscale measurements on temperature-dependent deformation of a textured magnesium alloy with synchrotron x-ray imaging and diffraction, Acta Mater., 132(2017), p. 389. doi: 10.1016/j.actamat.2017.04.065
[60]	L. Lu, J.W. Huang, D. Fan, B.X. Bie, T. Sun, K. Fezzaa, X.L. Gong, and S.N. Luo, Anisotropic deformation of extruded magnesium alloy AZ31 under uniaxial compression: A study with simultaneous in situ synchrotron x-ray imaging and diffraction, Acta Mater., 120(2016), p. 86. doi: 10.1016/j.actamat.2016.08.029
[61]	S. Dong, Q. Yu, Y.Y. Jiang, J. Dong, F.H. Wang, and W.J. Ding, Electron backscatter diffraction observations of twinning–detwinning evolution in a magnesium alloy subjected to large strain amplitude cyclic loading, Mater. Des., 65(2015), p. 762. doi: 10.1016/j.matdes.2014.09.079
[62]	S.M. Yin, F. Yang, X.M. Yang, S.D. Wu, S.X. Li, and G.Y. Li, The role of twinning–detwinning on fatigue fracture morphology of Mg–3%Al–1%Zn alloy, Mater. Sci. Eng. A, 494(2008), No. 1-2, p. 397. doi: 10.1016/j.msea.2008.04.056
[63]	Q. Yu, J. Wang, Y.Y. Jiang, R.J. McCabe, N. Li, and C.N. Tomé, Twin–twin interactions in magnesium, Acta Mater., 77(2014), p. 28. doi: 10.1016/j.actamat.2014.05.030
[64]	Q. Sun, T. Xia, L. Tan, J. Tu, M. Zhang, M.H. Zhu, and X.Y. Zhang, Influence of $ \left\{ {10\bar 12} \right\} $ twin characteristics on detwinning in Mg–3Al–1Zn alloy, Mater. Sci. Eng. A, 735(2018), p. 243. doi: 10.1016/j.msea.2018.08.051
[65]	L. Wu, S.R. Agnew, D.W. Brown, G.M. Stoica, B. Clausen, A. Jain, D.E. Fielden, and P.K. Liaw, Internal stress relaxation and load redistribution during the twinning–detwinning-dominated cyclic deformation of a wrought magnesium alloy, ZK60A, Acta Mater., 56(2008), No. 14, p. 3699. doi: 10.1016/j.actamat.2008.04.006
[66]	D. Sarker, J. Friedman, and D.L. Chen, Influence of pre-strain on de-twinning activity in an extruded AM30 magnesium alloy, Mater. Sci. Eng. A, 605(2014), p. 73. doi: 10.1016/j.msea.2014.03.046
[67]	B.M. Morrow, R.J. McCabe, E.K. Cerreta, and C.N. Tomé, In-situ TEM observation of twinning and detwinning during cyclic loading in Mg, Metall. Mater. Trans. A, 45(2013), No. 1, p. 36.
[68]	L.C. Lv, Y.C. Xin, H.H. Yu, R. Hong, and Q. Liu, The role of dislocations in strain hardening of an extension twinning predominant deformation, Mater. Sci. Eng. A, 636(2015), p. 389. doi: 10.1016/j.msea.2015.04.007
[69]	Q. Ma, H. El Kadiri, A.L. Oppedal, J.C. Baird, B. Li, M.F. Horstemeyer, and S.C. Vogel, Twinning effects in a rod-textured AM30 Magnesium alloy, Int. J. Plast., 29(2012), p. 60. doi: 10.1016/j.ijplas.2011.08.001
[70]	S. Dong, Y.Y. Jiang, J. Dong, F.H. Wang, and W.J. Ding, Cyclic deformation and fatigue of extruded ZK60 magnesium alloy with aging effects, Mater. Sci. Eng. A, 615(2014), p. 262. doi: 10.1016/j.msea.2014.07.074
[71]	P. Chen, B. Li, D. Culbertson, and Y.Y. Jiang, Negligible effect of twin-slip interaction on hardening in deformation of a Mg–3Al–1Zn alloy, Mater. Sci. Eng. A, 729(2018), p. 285. doi: 10.1016/j.msea.2018.05.067
[72]	H.H. Yu, Y.C. Xin, Y. Cheng, B. Guan, M.Y. Wang, and Q. Liu, The different hardening effects of tension twins on basal slip and prismatic slip in Mg alloys, Mater. Sci. Eng. A, 700(2017), p. 695. doi: 10.1016/j.msea.2017.06.034
[73]	J. Jeong, M. Alfreider, R. Konetschnik, D. Kiener, and S.H. Oh, In-situ TEM observation of $ \left\{ {10\bar 12} \right\} $ twin-dominated deformation of Mg pillars: Twinning mechanism, size effects and rate dependency, Acta Mater., 158(2018), p. 407. doi: 10.1016/j.actamat.2018.07.027
[74]	M.H. Yoo, Interaction of slip dislocations with twins in hcp metals, Trans. Metall. Soc. AIME, 245(1969), p. 2051.
[75]	F.L. Wang and S.R. Agnew, Dislocation transmutation by tension twinning in magnesium alloy AZ31, Int. J. Plast., 81(2016), p. 63. doi: 10.1016/j.ijplas.2016.01.012
[76]	F.L. Wang, C.D. Barrett, R.J. McCabe, H. El Kadiri, L. Capolungo, and S.R. Agnew, Dislocation induced twin growth and formation of basal stacking faults in $ \left\{ {10\bar 12} \right\} $ twins in pure Mg, Acta Mater., 165(2019), p. 471. doi: 10.1016/j.actamat.2018.12.003
[77]	F. Wang, K. Hazeli, K.D. Molodov, C.D. Barrett, T. Al-Samman, D.A. Molodov, A. Kontsos, K.T. Ramesh, H. El Kadiri, and S.R. Agnew, Characteristic dislocation substructure in $ \left\{ {10\bar 12} \right\} $ twins in hexagonal metals, Scripta Mater., 143(2018), p. 81. doi: 10.1016/j.scriptamat.2017.09.015
[78]	Y. Chino, K. Kimura, and M. Mabuchi, Twinning behavior and deformation mechanisms of extruded AZ31 Mg alloy, Mater. Sci. Eng. A, 486(2008), No. 1-2, p. 481. doi: 10.1016/j.msea.2007.09.058
[79]	C.F. Gu, L.S. Toth, and M. Hoffman, Twinning effects in a polycrystalline magnesium alloy under cyclic deformation, Acta Mater., 62(2014), p. 212. doi: 10.1016/j.actamat.2013.09.048
[80]	F. Kabirian, A.S. Khan, and T. Gnäupel-Herlod, Visco-plastic modeling of mechanical responses and texture evolution in extruded AZ31 magnesium alloy for various loading conditions, Int. J. Plast., 68(2015), p. 1. doi: 10.1016/j.ijplas.2014.10.012
[81]	K.D. Molodov, T. Al-Samman, and D.A. Molodov, Profuse slip transmission across twin boundaries in magnesium, Acta Mater., 124(2017), p. 397. doi: 10.1016/j.actamat.2016.11.022
[82]	D.F. Shi, T.M. Liu, T.Y. Wang, D.W. Hou, S.Q. Zhao, and S. Hussain, { $$} Twins across twin boundaries traced by in situ EBSD, J. Alloys Compd., 690(2017), p. 699. doi: 10.1016/j.jallcom.2016.08.076
[83]	M. Zecevic, I.J. Beyerlein, and M. Knezevic, Activity of pyramidal I and II <c+a> slip in Mg alloys as revealed by texture development, J. Mech. Phys. Solids, 111(2018), p. 290. doi: 10.1016/j.jmps.2017.11.004
[84]	N.T. Nguyen, M.G. Lee, J.H. Kim, and H.Y. Kim, A practical constitutive model for AZ31B Mg alloy sheets with unusual stress–strain response, Finite. Elem. Anal. Des., 76(2013), p. 39. doi: 10.1016/j.finel.2013.08.008
[85]	C.A. Lee, M.G. Lee, O.S. Seo, N.T. Nguyen, J.H. Kim, and H.Y. Kim, Cyclic behavior of AZ31B Mg: Experiments and non-isothermal forming simulations, Int. J. Plast., 75(2015), p. 39. doi: 10.1016/j.ijplas.2015.06.005
[86]	A.A. Roostaei and H. Jahed, A cyclic small-strain plasticity model for wrought Mg alloys under multiaxial loading: Numerical implementation and validation, Int. J. Mech. Sci., 145(2018), p. 318. doi: 10.1016/j.ijmecsci.2018.07.024
[87]	M. Li, X.Y. Lou, J.H. Kim, and R.H. Wagoner, An efficient constitutive model for room-temperature, low-rate plasticity of annealed Mg AZ31B sheet, Int. J. Plast., 26(2010), No. 6, p. 820. doi: 10.1016/j.ijplas.2009.11.001
[88]	O. Cazacu and F. Barlat, A criterion for description of anisotropy and yield differential effects in pressure-insensitive metals, Int. J. Plast., 20(2004), No. 11, p. 2027. doi: 10.1016/j.ijplas.2003.11.021
[89]	O. Cazacu, B. Plunkett, and F. Barlat, Orthotropic yield criterion for hexagonal closed packed metals, Int. J. Plast., 22(2006), No. 7, p. 1171. doi: 10.1016/j.ijplas.2005.06.001
[90]	B. Plunkett, R.A. Lebensohn, O. Cazacu, and F. Barlat, Anisotropic yield function of hexagonal materials taking into account texture development and anisotropic hardening, Acta Mater., 54(2006), No. 16, p. 4159. doi: 10.1016/j.actamat.2006.05.009
[91]	B. Plunkett, O. Cazacu, and F. Barlat, Orthotropic yield criteria for description of the anisotropy in tension and compression of sheet metals, Int. J. Plast., 24(2008), No. 5, p. 847. doi: 10.1016/j.ijplas.2007.07.013
[92]	J.W. Yoon, Y.S. Lou, J. Yoon, and M.V. Glazoff, Asymmetric yield function based on the stress invariants for pressure sensitive metals, Int. J. Plast., 56(2014), p. 184. doi: 10.1016/j.ijplas.2013.11.008
[93]	D.G. Tari, M.J. Worswick, U. Ali, and M.A. Gharghouri, Mechanical response of AZ31B magnesium alloy: Experimental characterization and material modeling considering proportional loading at room temperature, Int. J. Plast., 55(2014), p. 247. doi: 10.1016/j.ijplas.2013.10.006
[94]	W. Muhammad, M. Mohammadi, J.D. Kang, R.K. Mishra, and K. Inal, An elasto-plastic constitutive model for evolving asymmetric/anisotropic hardening behavior of AZ31B and ZEK100 magnesium alloy sheets considering monotonic and reverse loading paths, Int. J. Plast., 70(2015), p. 30. doi: 10.1016/j.ijplas.2015.03.004
[95]	M.G. Lee, R.H. Wagoner, J.K. Lee, K. Chung, and H.Y. Kim, Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheets, Int. J. Plast., 24(2008), No. 4, p. 545. doi: 10.1016/j.ijplas.2007.05.004
[96]	M.G. Lee, S.J. Kim, R.H. Wagoner, K. Chung, and H.Y. Kim, Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheets: Application to sheet springback, Int. J. Plast., 25(2009), No. 1, p. 70. doi: 10.1016/j.ijplas.2007.12.003
[97]	F. Barlat, J.J. Gracio, M.G. Lee, E.F. Rauch, and G. Vincze, An alternative to kinematic hardening in classical plasticity, Int. J. Plast., 27(2011), No. 9, p. 1309. doi: 10.1016/j.ijplas.2011.03.003
[98]	J. Lee, S.J. Kim, Y.S. Lee, J.Y. Lee, D. Kim, and M.G. Lee, Distortional hardening concept for modeling anisotropic/asymmetric plastic behavior of AZ31B magnesium alloy sheets, Int. J. Plast., 94(2017), p. 74. doi: 10.1016/j.ijplas.2017.02.002
[99]	W.J. He, T. Lin, and Q. Liu, Experiments and constitutive modeling of deformation behavior of a magnesium sheet during two-step loading, Int. J. Solids Struct., 147(2018), p. 52. doi: 10.1016/j.ijsolstr.2018.04.009
[100]	G.I. Taylor, Plastic Strain in Metals, J. Jpn. Inst. Met., 62(1938), p. 307.
[101]	E. Kröner, Zur plastischen verformung des vielkristalls, Acta Metall., 9(1961), No. 2, p. 155. doi: 10.1016/0001-6160(61)90060-8
[102]	B. Budiansky and T.T. Wu, Theoretical prediction of plastic strains of polycrystals, [in] Proceedings of the 4th US National Congress of Applied Mechanics, California, 1962, p. 1175.
[103]	P. Van Houtte, Simulation of the rolling and shear texture of brass by the Taylor theory adapted for mechanical twinning, Acta Metall., 26(1978), No. 4, p. 591. doi: 10.1016/0001-6160(78)90111-6
[104]	C.N. Tomé, R.A. Lebensohn, and U.F. Kocks, A model for texture development dominated by deformation twinning: Application to zirconium alloys, Acta Metall. Mater., 39(1991), No. 11, p. 2667. doi: 10.1016/0956-7151(91)90083-D
[105]	R.A. Lebensohn and C.N. Tome, A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: Application to zirconium alloys, Acta Metall. Mater., 41(1993), No. 9, p. 2611. doi: 10.1016/0956-7151(93)90130-K
[106]	S.R. Agnew, M.H. Yoo, and C.N. Tomé, Application of texture simulation to understanding mechanical behavior of Mg and solid solution alloys containing Li or Y, Acta Mater., 49(2001), No. 20, p. 4277. doi: 10.1016/S1359-6454(01)00297-X
[107]	S.H. Choi, E.J. Shin, and B.S. Seong, Simulation of deformation twins and deformation texture in an AZ31 Mg alloy under uniaxial compression, Acta Mater., 55(2007), No. 12, p. 4181. doi: 10.1016/j.actamat.2007.03.015
[108]	S.R. Kalidindi, Incorporation of deformation twinning in crystal plasticity models, J. Mech. Phys. Solids, 46(1998), No. 2, p. 267. doi: 10.1016/S0022-5096(97)00051-3
[109]	A. Staroselsky and L. Anand, A constitutive model for hcp materials deforming by slip and twinning: application to magnesium alloy AZ31B, Int. J. Plast., 19(2003), No. 10, p. 1843. doi: 10.1016/S0749-6419(03)00039-1
[110]	H. Abdolvand and M.R. Daymond, Internal strain and texture development during twinning: Comparing neutron diffraction measurements with crystal plasticity finite-element approaches, Acta Mater., 60(2012), No. 5, p. 2240. doi: 10.1016/j.actamat.2012.01.016
[111]	H. Abdolvand and M.R. Daymond, Multi-scale modeling and experimental study of twin inception and propagation in hexagonal close-packed materials using a crystal plasticity finite element approach—Part I: Average behavior, J. Mech. Phys. Solids, 61(2013), No. 3, p. 783. doi: 10.1016/j.jmps.2012.10.013
[112]	H. Abdolvand and M.R. Daymond, Multi-scale modeling and experimental study of twin inception and propagation in hexagonal close-packed materials using a crystal plasticity finite element approach; part II: Local behavior, J. Mech. Phys. Solids, 61(2013), No. 3, p. 803. doi: 10.1016/j.jmps.2012.10.017
[113]	T. Hama, T. Suzuki, S. Hatakeyama, H. Fujimoto, and H. Takuda, Role of twinning on the stress and strain behaviors during reverse loading in rolled magnesium alloy sheets, Mater. Sci. Eng. A, 725(2018), p. 8. doi: 10.1016/j.msea.2018.03.124
[114]	Q. Liu, A. Roy, and V.V. Silberschmidt, Temperature-dependent crystal-plasticity model for magnesium: A bottom-up approach, Mech. Mater., 113(2017), p. 44. doi: 10.1016/j.mechmat.2017.07.008
[115]	H.J. Zhang, A. Jérusalem, E. Salvati, C. Papadaki, K.S. Fong, X. Song, and A.M. Korsunsky, Multi-scale mechanisms of twinning-detwinning in magnesium alloy AZ31B simulated by crystal plasticity modeling and validated via in situ synchrotron XRD and in situ SEM–EBSD, Int. J. Plast., 119(2019), p. 43. doi: 10.1016/j.ijplas.2019.02.018
[116]	J. Zhang and S.P. Joshi, Phenomenological crystal plasticity modeling and detailed micromechanical investigations of pure magnesium, J. Mech. Phys. Solids, 60(2012), No. 5, p. 945. doi: 10.1016/j.jmps.2012.01.005
[117]	J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. R. Soc. London,Ser. A,Math. Phys. Sci., 241(1957), No. 1226, p. 376.
[118]	R. Hill, Continuum micro-mechanics of elastoplastic polycrystals, J. Mech. Phys. Solids, 13(1965), No. 2, p. 89. doi: 10.1016/0022-5096(65)90023-2
[119]	J.W. Hutchinson, Bounds and self-consistent estimates for creep of polycrystalline materials, Proc. R. Soc. Lond. A, 348(1976), No. 1652, p. 101. doi: 10.1098/rspa.1976.0027
[120]	H. Wang, P.D. Wu, J. Wang, and C.N. Tomé, A crystal plasticity model for hexagonal close packed (HCP) crystals including twinning and de-twinning mechanisms, Int. J. Plast., 49(2013), p. 36. doi: 10.1016/j.ijplas.2013.02.016
[121]	H. Wang, P.D. Wu, and J. Wang, Modeling inelastic behavior of magnesium alloys during cyclic loading–unloading, Int. J. Plast., 47(2013), p. 49. doi: 10.1016/j.ijplas.2013.01.007
[122]	H. Qiao, S.R. Agnew, and P.D. Wu, Modeling twinning and detwinning behavior of Mg alloy ZK60A during monotonic and cyclic loading, Int. J. Plast., 65(2015), p. 61. doi: 10.1016/j.ijplas.2014.08.010
[123]	H. Qiao, X.Q. Guo, A.L. Oppedal, H. El Kadiri, P.D. Wu, and S.R. Agnew, Twin-induced hardening in extruded Mg alloy AM30, Mater. Sci. Eng. A, 687(2017), p. 17. doi: 10.1016/j.msea.2016.12.123
[124]	G. Cailletaud and P. Pilvin, Utilisation de modèles polycristallins pour le calcul par éléments finis, Rev. Européenne des Éléments Finis, 3(1994), No. 4, p. 515.
[125]	C. Yu, G.Z. Kang, and Q.H. Kan, Crystal plasticity based constitutive model for uniaxial ratchetting of polycrystalline magnesium alloy, Comput. Mater. Sci., 84(2014), p. 63. doi: 10.1016/j.commatsci.2013.11.054
[126]	C.O. Frederick and P.J. Armstrong, A mathematical representation of the multiaxial Bauschinger effect, Mater. High Temp., 24(2007), No. 1, p. 1. doi: 10.3184/096034007X207589
[127]	H. Li, G.Z. Kang, and C. Yu, Modeling uniaxial ratchetting of magnesium alloys by a new crystal plasticity considering dislocation slipping, twinning and detwinning mechanisms, Int. J. Mech. Sci., 179(2020), art. No. 105660. doi: 10.1016/j.ijmecsci.2020.105660