Acta Mechanica Solida Sinica

, Volume 27, Issue 2, pp 181–194 | Cite as

Constitutive Modeling of Rolled Shape Memory Alloy Sheets Taking into Account Pre-Texture and Anisotropic Hardening

  • Yuping Zhu
  • Yuanbing Wang
  • Guansuo Dui


The drawing or rolling process endows polycrystal shape memory alloy with a crystallographic texture, which can result in macroscopic anisotropy. The main purpose of this work is to develop a constitutive model to predict the thermomechanical behavior of shape memory alloy sheets, which accounts for the crystallographic texture. The total macroscopic strain is decomposed into elastic strain and macro-transformation strain under isothermal condition. Considering the transformation strain in local grains and the orientation distribution function of crystallographic texture, the macro-transformation strain and the effective elastic modulus of textured polycrystal shape memory alloy are developed by using tensor expressions. The kinetic equation is established to calculate the volume fraction of the martensite transformation under given stress. Furthermore, the Hill’s quadratic model is developed for anisotropic transformation hardening of textured SMA sheets. All the calculation results are in good agreement with experimental data, which show that the present model can accurately describe the macro-anisotropic behaviors of textured shape memory alloy sheets.

Key Words

shape memory alloy constitutive model transformation yield function texture anisotropic hardening 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Gall, K. and Sehitoglu, H., The role of texture in tension-compression asymmetry in poly- crystalline NiTi. International Journal of Plasticity, 1999, 15(1): 69–92.CrossRefGoogle Scholar
  2. 2.
    Gall, K., Lim, T.J., McDowell, D.L., Sehitoglu, H. and Chumlyakov, Y.I., The role of intergranular constraint on the stress-induced martensitic transformation in textured polycrystalline NiTi. International Journal of Plasticity, 2000, 16(4): 1189–1214.CrossRefGoogle Scholar
  3. 3.
    Miyazaki, S., No, V.H., Kitamura, K., Khantachawana, A. and Hosoda, H., Texture of Ti-Ni rolled thin plates and sputter-deposited thin films. International Journal of Plasticity, 2000, 16(4): 1135–1154.CrossRefGoogle Scholar
  4. 4.
    Gao, S. and Yi, S., Experimental study on the anisotropic behavior of textured NiTi pseudoelastic shape memory alloys. Materials Science and Engineering A, 2003, 362(1–2): 107–111.CrossRefGoogle Scholar
  5. 5.
    Banumathy, S., Mandal, R.K. and Singh, A.K., Texture and anisotropy of a hot rolled Ti-16Nb alloy. Journal of Alloys and Compounds, 2010, 500 (2): L26–L30.CrossRefGoogle Scholar
  6. 6.
    Geng, Y., Jin, M., Ren, W., Zhang, W. and Jin, X., Effects of aging treatment on martensitic transformation of Fe-Ni-Co-Al-Ta-B alloys. Journal of Alloys and Compounds, doi: 10.1016/j.jallcom.2012.03.033.CrossRefGoogle Scholar
  7. 7.
    Yuan, W.Q. and Yi, S., Pseudo-elastic strain estimation of textured TiNi shape memory alloys. Materials Science and Engineering A, 1999, 271(1–2): 439–448.CrossRefGoogle Scholar
  8. 8.
    Eucken, S., Hirsch, J. and Hornbogen, E., Texture and microstructure of meltspun shape memory alloys. Textures and Microstructures, 1988, 8–9: 415–426.CrossRefGoogle Scholar
  9. 9.
    Mulder, J.H., Thoma, P.E. and Beyer, J.Z., Anisotropy of the shape memory effect in tension of cold-rolled Ti50.8 Ni49.2 (at.%) sheet. Zeitschrift für Metallkunde, 1993, 84(7): 501–508.Google Scholar
  10. 10.
    Inoue, H., Miwa, N. and Inakazu, N., Texture and shape memory strain in TiNi alloy sheets. Acta Materialia, 1996, 44(12): 4825–4834.CrossRefGoogle Scholar
  11. 11.
    Zhao, L., Willemse, P.E., Mulder, J.H., Beyer, J. and Wei, W., Texture development and transformation strain of a cold-rolled Ti50-Ni45-Cu5 alloy. Scripta Materialia, 1998, 39(9): 1317–1323.CrossRefGoogle Scholar
  12. 12.
    Sittner, P., Liu, Y. and Novak, V., On the origin of Lüders-like deformation of NiTi shape memory alloys. Journal of the Mechanics and Physics of Solids, 2005, 53(8): 1719–1746.CrossRefGoogle Scholar
  13. 13.
    Chang, S.H. and Wu, S.K., Textures in cold-rolled and annealed Ti50Ni50 shape memory alloy. Scripta Materialia, 2004, 50(7): 937–941.CrossRefGoogle Scholar
  14. 14.
    Sittner, P., Neov, D., Lukas, P. and Toebbens, D.M., Neutron diffraction Sstudies of the stress effect on texture transformations in NiTi shape memory alloys. Journal of Neutron Research, 2004, 12(1–3): 15–20.CrossRefGoogle Scholar
  15. 15.
    Sutou, Y., Koeda, N., Omori, T., Kainuma, R. and Ishida, K., Effects of aging on stress-induced martensitic transformation in ductile Cu-Al-Mn–based shape memory alloys. Acta Materialia, 2009, 57(19): 5759–5770.CrossRefGoogle Scholar
  16. 16.
    Liu, Y., Xie, Z.L., Humbeeck, Van.J. and Delaey, L., Effect of texture orientation on the martensite deformation of NiTi shape memory alloy sheet. Acta Materialia, 1999, 47(2): 645–660.CrossRefGoogle Scholar
  17. 17.
    Paula, A.S., Canejo, J.H.P.G., Mahesh, K.K., Silva, R.J.C., Braz Fernandes, F.M., Martins, R.M.S., Cardos, A.M. A. and Schell, N., Study of the textural evolution in Ti-rich NiTi using synchrotron radiation. Nuclear Instruments and Methods in Physics Research B, 2006, 246(1): 206–210.CrossRefGoogle Scholar
  18. 18.
    Eggeler, G., Wagner, M., Khalil-Allafi, J. and Baruj, A., Hard X-ray studies of stress-induced phase transformations of superelastic NiTi shape memory alloys under uniaxial load. Materials Science and Engineering A, 2008, 481–482(1): 414–419Google Scholar
  19. 19.
    Murasawa, G., Kitamura, K., Yoneyama, S., Miyazaki, S., Miyata, K., Nishioka, A. and Koda, T., Macroscopic stress–strain curve, local strain band behavior and the texture of NiTi thin sheets. Smart Materials and Structures, 2009, 18(5): 055003.CrossRefGoogle Scholar
  20. 20.
    Böhlke, T. and Bertram, A., The evolution of Hoohe’s law due to texture development in FCC polycrystas. International Journal of Solids and Structures, 2001, 38(52): 9437–9459.CrossRefGoogle Scholar
  21. 21.
    Solas, D.E. and Tomé, C.N., Texture and strain localization prediction using a N-site polycrystal model. International Journal of Plasticity, 2001, 17(5): 737–753.CrossRefGoogle Scholar
  22. 22.
    Lia, S., Hoferlin, E., Bael, A.V., Houtte, P.V. and Teodosiu, C., Finite element modeling of plastic anisotropy induced by texture and strain-path change. International Journal of Plasticity, 2003, 19(5): 647–674.CrossRefGoogle Scholar
  23. 23.
    Houtte, P.V., Kanjarla, A.K., Bael, A.V., Seefeldt, M. and Delannay, L., Multiscale modelling of the plastic anisotropy and deformation texture of polycrystalline materials. European Journal of Mechanics A/Solids, 2006, 25(4): 634–648.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Nikolova, S., Lebensohnb, R.A. and Raabea, D., Self-consistent modeling of large plastic deformation, texture and morphology evolution in semi-crystalline polymers. Journal of the Mechanics and Physics of Solids, 2006, 54(7): 1350–1375.CrossRefGoogle Scholar
  25. 25.
    Plunket, B., Lebensohn, R.A., Cazacu, O. and Barlat, F., Anisotropic yield function of hexagonal materials taking into account texture development and anisotropic hardening. Acta Materialia, 2006, 54(16): 4159–4169.CrossRefGoogle Scholar
  26. 26.
    Kim, J.H., Lee, M.G., Barlat, F., Wagoner, R.H. and Chung, K., An elasto-plastic constitutive model with plastic strain rate potentials for anisotropic cubic metals. International Journal of Plasticity, 2008, 24(12): 2298–2334.CrossRefGoogle Scholar
  27. 27.
    Lademo, O-G., Pedersen, K.O., Berstad, T., Furu, T. and Hopperstad, O.S., An experimental and numerical study on the formability of textured AlZnMg alloys. European Journal of Mechanics A/Solids, 2008, 27(2): 116–140.CrossRefGoogle Scholar
  28. 28.
    Huang, M. and Zheng, T., Orientation-dependent function for properties of polycrystals and its applications. Acta Mechanica, 2009, 207(3–4): 135–143.CrossRefGoogle Scholar
  29. 29.
    Chen, Y., Lee, W.B. and Nakamachi, E., Crystallographic homogenization finite element method and its application on simulationof evolutionof plastic deformation induced texture. Acta Mechanica Solida Sinica, 2010, 23(1): 36–48.CrossRefGoogle Scholar
  30. 30.
    Shu, Y. and Bhattacharya, K., The influence of texture on the shape memory effect in polycrystals. Acta Materialia, 1998, 46(15): 5457–5473.CrossRefGoogle Scholar
  31. 31.
    Thamburaja, P. and Anand, L., Polycrystalline shape-memory materials: effect of crystallographic texture. Journal of the Mechanics and Physics of Solids, 2001, 49(4): 709–737.CrossRefGoogle Scholar
  32. 32.
    Sadjadpour, A. and Bhattacharya, K., A micromechanics-inspired constitutive model for shape- memory alloys. Smart Materials and Structures, 2007, 16(5): 1751–1765.CrossRefGoogle Scholar
  33. 33.
    Hill, R., A theory of the yielding and plastic flow of anisotropic metals. Proceedings of the Royal Society of London A, 1948, 193(2): 281–297.MathSciNetCrossRefGoogle Scholar
  34. 34.
    Bunge, H.J., Texture Analysis in Materials Science. London: Butterworths, 1982.Google Scholar
  35. 35.
    Lu, Z.K. and Weng, G.J., A self-consistent model for the stress-strain behaveior of shape memory alloy polycrystals. Acta Materialia, 1998, 46(15): 5423–5433.CrossRefGoogle Scholar
  36. 36.
    Yuan, W.Q. and Wang, J.N., Anisotropy of the phase-transformation plasticity in textured CuZnAl shape-memory sheets. Journal of Materials Processing Technology, 2002, 123(1): 31–35.CrossRefGoogle Scholar
  37. 37.
    Li, H., Yin, F., Sawaguchi, T., Ogawa, K., Zhao, X. and Tsuzaki, K., Texture evolution analysis of warm-rolled Fe-28Mn-6Si-5Cr shape memory alloy. Materials Science and Engineering A, 2008, 494(1–2): 217–226.CrossRefGoogle Scholar
  38. 38.
    Lubarda, V. A. and Krajcinovic, D., Damage tensors and the crack density distribution. International Journal of Solids and Structures, 1993, 30(20): 2859–2877.CrossRefGoogle Scholar
  39. 39.
    Qidwai, M.A. and Lagoudas, D.C., On thermomechanics and transformation surfaces of poly–crystalline NiTi shape memory alloy material. International Journal of Plasticity, 2000, 16(10–11): 1309–1343.CrossRefGoogle Scholar
  40. 40.
    Wang, Z., Zhu, Y.P., Dui, G.S. and Cui, H.N., Micromechanical analysis for mechanical property of shape memory alloy. Journal Beijing Jiaotong University, 2008, 32(1): 119–122, 126. (In Chinese)Google Scholar
  41. 41.
    Bunge, H.J., Kiewel, R., Reinert, Th. and Fritsche, L., Elastic properties of polycrystals—influence of texture and stereology. Journal of the Mechanics and Physics of Solids, 2000, 48(1): 29–66.MathSciNetCrossRefGoogle Scholar
  42. 42.
    Lu, Z.K. and Weng, G.J., Martensitic transformation and stress-strain relations of shape- memory alloys. Journal of the Mechanics and Physics of Solids, 1997, 45(11–12): 1905–1928.CrossRefGoogle Scholar
  43. 43.
    Khachaturyan, A.G., Theory of Structural Transformations in Solids. New York: John Wiley & Sons, 1983.Google Scholar
  44. 44.
    Mercier, O., Melton, K.N., Gremaud, G. and Hgi, J., Single-crystal elastic constants of the equiatomic NiTi alloy near the martensitic transformation. Journal of Applied Physics, 1980, 51(3): 1833–1834.CrossRefGoogle Scholar
  45. 45.
    Comstock, R. J. and Wert, J. A., Evaluation of a model of stress-induced martensite formation in NiTi sheet. Zeitschrift für Metallkunde, 1997, 8(6): 887–895.Google Scholar
  46. 46.
    Darrieulat, M. and Montheillet, F., A texture based continuum approach for predicting the plastic behaviour of rolled sheet. International Journal of Plasticity, 2003, 19(4): 517–546.CrossRefGoogle Scholar
  47. 47.
    Cazacu, O. and Barlat, F., A criterion for description of anisotropy and yield differential effects in pressure-insensitive metals. International Journal of Plasticity, 2004, 20(11): 2027–2045.CrossRefGoogle Scholar
  48. 48.
    Yoon, J.W., Barlat, F., Gracio, J.J. and Rauch, E., Anisotropic strain hardening behavior in simple shear for cube textured aluminum alloy sheets. International Journal of Plasticity, 2005, 21(12): 2426–2447.CrossRefGoogle Scholar
  49. 49.
    Hu, W.L., Constitutive modeling of orthotropic sheet metals by presenting hardeninginduced anisotropy. International Journal of Plasticity, 2007, 23(4): 620–639.MathSciNetCrossRefGoogle Scholar
  50. 50.
    Soare, S., Yoon, J.W. and Cazacu, O., On the use of homogeneous polynomials to develop anisotropic yield functions with applications to sheet forming. International Journal of Plasticity, 2008, 24(6): 915–944.CrossRefGoogle Scholar
  51. 51.
    Brenner, R., Lebensohn, R.A. and Castelnau, O., Elastic anisotropy and yield surface estimates of polycrystals. International Journal of Solids and Structures, 2009, 46(17): 3018–3026.CrossRefGoogle Scholar
  52. 52.
    Nixon, M.E., Cazacu, O. and Lebensohn, R.A., Anisotropic response of high-purity a-titanium: Experimental characterization and constitutive modeling. International Journal of Plasticity, 2010, 26(4): 516–532.CrossRefGoogle Scholar
  53. 53.
    Mura, T., Micromechanics of Defects in Solids. Dordrecht: Martinus Nijhoff Publishers, 1987.CrossRefGoogle Scholar
  54. 54.
    Gavazzi, A.C. and Lagoudas, D.C., On the numerical evaluation of Eshelby’s tensor and its application to elastoplastic fibrous composites. Computational. Mechanics, 1990, 7(1): 13–19.CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2014

Authors and Affiliations

  1. 1.Institute of Mechanics & EngineeringJiangsu UniversityZhenjiangChina
  2. 2.Institute of MechanicsBeijing Jiaotong UniversityBeijingChina

Personalised recommendations