# An integrated approach to model strain localization bands in magnesium alloys

- 520 Downloads
- 1 Citations

## Abstract

Strain localization bands (SLBs) that appear at early stages of deformation of magnesium alloys have been recently associated with heterogeneous activation of deformation twinning. Experimental evidence has demonstrated that such “Lüders-type” band formations dominate the overall mechanical behavior of these alloys resulting in sigmoidal type stress–strain curves with a distinct plateau followed by pronounced anisotropic hardening. To evaluate the role of SLB formation on the local and global mechanical behavior of magnesium alloys, an integrated experimental/computational approach is presented. The computational part is developed based on custom subroutines implemented in a finite element method that combine a plasticity model with a stiffness degradation approach. Specific inputs from the characterization and testing measurements to the computational approach are discussed while the numerical results are validated against such available experimental information, confirming the existence of load drops and the intensification of strain accumulation at the time of SLB initiation.

## Keywords

Strain localization Magnesium alloys Twinning Digital image correlation Finite element method## Notes

### Acknowledgements

The corresponding author acknowledges the financial support provided by the National Science Foundation through the CMMI #1434506 award to Drexel University. He also acknowledges the technical support received under the National Aeronautics and Space Administration Space Act Agreement, No. SAA1-19439 with Langley Research Center. The results reported were obtained by using computational resources supported by Drexel’s University Research Computing Facility. This investigation was also supported by the funds received in terms of fellowship to M. Cabal from the Greater Philadelphia Region Louis Strokes Alliance for Minority Participation.

## References

- 1.Pietruszczak S, Mróz Z (1981) Finite element analysis of deformation of strain-softening materials. Int J Numer Meth Eng 17(3):327–334. doi: 10.1002/nme.1620170303 CrossRefzbMATHGoogle Scholar
- 2.Tvergaard V, Needleman A, Lo KK (1981) Flow localization in the plane strain tensile test. J Mech Phys Solids 29(2):115–142. doi: 10.1016/0022-5096(81)90019-3 CrossRefzbMATHGoogle Scholar
- 3.Ortiz M, Leroy Y, Needleman A (1987) A finite element method for localized failure analysis. Comput Methods Appl Mech Eng 61(2):189–214. doi: 10.1016/0045-7825(87)90004-1 CrossRefzbMATHGoogle Scholar
- 4.Belytschko T, Fish J, Engelmann BE (1988) A finite element with embedded localization zones. Comput Methods Appl Mech Eng 70(1):59–89. doi: 10.1016/0045-7825(88)90180-6 CrossRefzbMATHGoogle Scholar
- 5.Batra RC, Kim CH (1992) Analysis of shear banding in twelve materials. Int J Plast 8(4):425–452. doi: 10.1016/0749-6419(92)90058-K CrossRefGoogle Scholar
- 6.Ramakrishnan N, Atluri SN (1994) Simulation of shear band formation in plane strain tension and compression using FEM. Mech Mater 17(2–3):307–317. doi: 10.1016/0167-6636(94)90068-X CrossRefGoogle Scholar
- 7.Shaw JA, Kyriakides S (1997) Initiation and propagation of localized deformation in elasto-plastic strips under uniaxial tension. Int J Plast 13(10):837–871. doi: 10.1016/S0749-6419(97)00062-4 CrossRefGoogle Scholar
- 8.Tsukahara H, Iung T (1998) Finite element simulation of the Piobert-Lüders behavior in an uniaxial tensile test. Mater Sci Eng A 248(1–2):304–308. doi: 10.1016/S0921-5093(97)00857-5 CrossRefGoogle Scholar
- 9.Koplik J, Needleman A (1988) Void growth and coalescence in porous plastic solids. Int J Solids Struct 24(8):835–853. doi: 10.1016/0020-7683(88)90051-0 CrossRefGoogle Scholar
- 10.Kuroda M, Tvergaard V (2007) Effects of texture on shear band formation in plane strain tension/compression and bending. Int J Plast 23(2):244–272. doi: 10.1016/j.ijplas.2006.03.014 CrossRefzbMATHGoogle Scholar
- 11.Jia N, Roters F, Eisenlohr P, Kords C, Raabe D (2012) Non-crystallographic shear banding in crystal plasticity FEM simulations: example of texture evolution in \(\alpha \)-brass. Acta Mater 60(3):1099–1115. doi: 10.1016/j.actamat.2011.10.047 CrossRefGoogle Scholar
- 12.Jia N, Eisenlohr P, Roters F, Raabe D, Zhao X (2012) Orientation dependence of shear banding in face-centered-cubic single crystals. Acta Mater 60(8):3415–3434. doi: 10.1016/j.actamat.2012.03.005 CrossRefGoogle Scholar
- 13.Paul H, Morawiec A, Driver JH, Bouzy E (2009) On twinning and shear banding in a Cu-8 at.% Al alloy plane strain compressed at 77 K. Int J Plast 25(8):1588–1608. doi: 10.1016/j.ijplas.2008.10.003 CrossRefzbMATHGoogle Scholar
- 14.Hazeli K, Cuadra J, Vanniamparambil PA, Kontsos A (2013) In situ identification of twin-related bands near yielding in a magnesium alloy. Scripta Mater 68(1):83–86. doi: 10.1016/j.scriptamat.2012.09.009 CrossRefGoogle Scholar
- 15.Aydıner CC, Telemez MA (2014) Multiscale deformation heterogeneity in twinning magnesium investigated with in situ image correlation. Int J Plast 56:203–218. doi: 10.1016/j.ijplas.2013.12.001 CrossRefGoogle Scholar
- 16.Hazeli K, Cuadra J, Streller F, Barr CM, Taheri ML, Carpick RW, Kontsos A (2015) Three-dimensional effects of twinning in magnesium alloys. Scripta Mater 100:9–12. doi: 10.1016/j.scriptamat.2014.12.001 CrossRefGoogle Scholar
- 17.Barnett MR, Nave MD, Ghaderi A (2012) Yield point elongation due to twinning in a magnesium alloy. Acta Mater 60(4):1433–1443. doi: 10.1016/j.actamat.2011.11.022 CrossRefGoogle Scholar
- 18.Changizian P, Zarei-Hanzaki A, Ghambari M, Imandoust A (2013) Flow localization during severe plastic deformation of AZ81 magnesium alloy: micro-shear banding phenomenon. Mater Sci Eng A 582:8–14. doi: 10.1016/j.msea.2013.05.069 CrossRefGoogle Scholar
- 19.Changizian P, Zarei-Hanzaki A, Abedi HR (2012) On the recrystallization behavior of homogenized AZ81 magnesium alloy: the effect of mechanical twins and \(\gamma \) precipitates. Mater Sci Eng, A 558:44–51. doi: 10.1016/j.msea.2012.07.054 CrossRefGoogle Scholar
- 20.Chun YB, Davies CHJ (2011) Texture effect on microyielding of wrought magnesium alloy AZ31. Mater Sci Eng A 528(9):3489–3495. doi: 10.1016/j.msea.2011.01.046 CrossRefGoogle Scholar
- 21.Fatemi-Varzaneh SM, Zarei-Hanzaki A, Cabrera JM (2011) Shear banding phenomenon during severe plastic deformation of an AZ31 magnesium alloy. J Alloy Compd 509(9):3806–3810. doi: 10.1016/j.jallcom.2011.01.019 CrossRefGoogle Scholar
- 22.Li X, Yang P, Wang LN, Meng L, Cui F (2009) Orientational analysis of static recrystallization at compression twins in a magnesium alloy AZ31. Mater Sci Eng A 517(1–2):160–169. doi: 10.1016/j.msea.2009.03.045 CrossRefGoogle Scholar
- 23.ABAQUS (2013) User’s manual version 6.13. Dassault Systems, PawtucketGoogle Scholar
- 24.Cazacu O, Plunkett B, Barlat F (2006) Orthotropic yield criterion for hexagonal closed packed metals. Int J Plast 22(7):1171–1194. doi: 10.1016/j.ijplas.2005.06.001
- 25.Lee M-G, Wagoner RH, Lee JK, Chung K, Kim HY (2008) Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheets. Int J Plast 24(4):545–582. doi: 10.1016/j.ijplas.2007.05.004 CrossRefzbMATHGoogle Scholar
- 26.Ramaswamy S, Aravas N (1998) Finite element implementation of gradient plasticity models Part II: gradient-dependent evolution equations. Comput Methods Appl Mech Eng 163(1–4):33–53. doi: 10.1016/S0045-7825(98)00027-9 CrossRefzbMATHGoogle Scholar
- 27.de Borst R, Sluys LJ LJ (1993) Fundamental issues in finite element analyses of localization of deformation. Eng Comput 10(2):99–121. doi: 10.1108/eb023897 CrossRefGoogle Scholar
- 28.Needleman A (1988) Material rate dependence and mesh sensitivity in localization problems. Comput Methods Appl Mech Eng 67(1):69–85. doi: 10.1016/0045-7825(88)90069-2 CrossRefzbMATHGoogle Scholar
- 29.Niezgoda SR, Kanjarla AK, Beyerlein IJ, Tomé CN (2014) Stochastic modeling of twin nucleation in polycrystals: an application in hexagonal close-packed metals. Int J Plast 56:119–138. doi: 10.1016/j.ijplas.2013.11.005 CrossRefGoogle Scholar
- 30.Yoo MH (1981) Slip, twinning, and fracture in hexagonal close-packed metals. Metall Trans A 12(3):409–418. doi: 10.1007/bf02648537 MathSciNetCrossRefGoogle Scholar
- 31.Beyerlein IJ, Capolungo L, Marshall PE, McCabe RJ, Tomé CN (2010) Statistical analyses of deformation twinning in magnesium. Phil Mag 90(16):2161–2190. doi: 10.1080/14786431003630835 CrossRefGoogle Scholar
- 32.Yu Q, Jiang Y, Wang J (2015) Cyclic deformation and fatigue damage in single-crystal magnesium under fully reversed strain-controlled tension-compression in the [1 0 0] direction. Scripta Mater 96:41–44. doi: 10.1016/j.scriptamat.2014.10.020 CrossRefGoogle Scholar
- 33.Mo C, Wisner B, Cabal M, Hazeli K, Ramesh K, El Kadiri H, Al-Samman T, Molodov K, Molodov D, Kontsos A (2016) Acoustic emission of deformation twinning in magnesium. Materials 9(8):662CrossRefGoogle Scholar
- 34.Liu X, Jonas JJ, Li LX, Zhu BW (2013) Flow softening, twinning and dynamic recrystallization in AZ31 magnesium. Mater Sci Eng, A 583:242–253. doi: 10.1016/j.msea.2013.06.074 CrossRefGoogle Scholar
- 35.Sun HB, Yoshida F, Ma X, Kamei T, Ohmori M (2003) Finite element simulation on the propagation of Lüders band and effect of stress concentration. Mater Lett 57(21):3206–3210. doi: 10.1016/S0167-577X(03)00036-3 CrossRefGoogle Scholar
- 36.Oliver J (1996) Modelling strong discontinuities in solid mechanics vis strain softening constitutive equations. Part 1: fundamentals. Int J Numer Methods Eng 39(21):3575–3600CrossRefzbMATHGoogle Scholar
- 37.Hielscher R, Schaeben H (2008) A novel pole figure inversion method: specification of the MTEX algorithm. J Appl Crystallogr 41(6):1024–1037. doi: 10.1107/S0021889808030112 CrossRefGoogle Scholar