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Three-dimensional constitutive model for magneto-mechanical deformation of NiMnGa ferromagnetic shape memory alloy single crystals

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Abstract

Existing experimental results have shown that four types of physical mechanisms, namely, martensite transformation, martensite reorientation, magnetic domain wall motion and magnetization vector rotation, can be activated during the magneto-mechanical deformation of NiMnGa ferromagnetic shape memory alloy (FSMA) single crystals. In this work, based on irreversible thermodynamics, a three-dimensional (3D) single crystal constitutive model is constructed by considering the aforementioned four mechanisms simultaneously. Three types of internal variables, i.e., the volume fraction of each martensite variant, the volume fraction of magnetic domain in each variant and the deviation angle between the magnetization vector, and easy axis are introduced to characterize the magneto-mechanical state of the single crystals. The thermodynamic driving force of each mechanism and the thermodynamic constraints on the constitutive model are obtained from Clausius’s dissipative inequality and constructed Gibbs free energy. Then, thermodynamically consistent kinetic equations for the four mechanisms are proposed, respectively. Finally, the ability of the proposed model to describe the magneto-mechanical deformation of NiMnGa FSMA single crystals is verified by comparing the predictions with corresponding experimental results. It is shown that the proposed model can quantitatively capture the main experimental phenomena. Further, the proposed model is used to predict the deformations of the single crystals under the non-proportional mechanical loading conditions.

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Abbreviations

\( c_{\text{A}} \) :

Specific heat at a constant volume in austenite phase

\( c_{\text{M}} \) :

Specific heat at a constant volume in martensite phase

\( \bar{c} \) :

Effective specific heat

\( {\varvec{e}}_{1} ,\,{\varvec{e}}_{2} ,{\varvec{e}}_{3} \, \) :

Three basis vectors of local coordinate system

\( f^{\zeta } \) :

Resistance function of magnetic domain wall motion

\( f^{\xi } \) :

Transformation and reorientation hardening function

\( G \) :

Gibbs free energy

\( G_{\alpha }^{\text{M}} \) :

Gibbs free energy of the α-th martensite variant

\( G_{\text{mech}}^{{{\text{M}}\alpha }} \) :

Mechanical part of \( G_{\alpha }^{\text{M}} \)

\( G_{\text{th}}^{{{\text{M}}\alpha }} \) :

Thermal part of \( G_{\alpha }^{\text{M}} \)

\( G_{\text{mag}}^{{{\text{M}}\alpha }} \) :

Magnetic part of \( G_{\alpha }^{M} \)

\( G_{{}}^{\text{A}} \) :

Gibbs free energy of austenite phase

\( G_{\text{mech}}^{\text{A}} \) :

Mechanical part of \( G_{{}}^{\text{A}} \)

\( G_{\text{th}}^{\text{A}} \) :

Thermal part of \( G_{{}}^{\text{A}} \)

\( G_{\text{mag}}^{\text{A}} \) :

Magnetic part of \( G_{{}}^{\text{A}} \)

\( {\varvec{H}} \) :

Magnetic field

\( H^{\text{tr}} \) :

Transformation hardening modulus

\( H^{\text{reo}} \) :

Reorientation hardening modulus

\( H^{\text{cri}} \) :

Critical field at which magnetic saturation is achieved through domain wall motion

\( {\varvec{I}} \) :

Fourth-ordered unit tensor

\( K \) :

Initial reorientation resistance

\( K_{I} \) :

Magneto-crystalline anisotropic constant

\( {\varvec{k}} \) :

Second-ordered thermal conductivity tensor

\( k_{{}}^{\alpha ij} \) :

Incidence matrix

\( {\varvec{M}} \) :

Magnetization vector

\( {\varvec{M}}_{M}^{\alpha } \) :

Magnetization vector of the α-th martensite variant

\( M_{\text{sat}}^{\text{M}} \) :

Saturated magnetization of martensite phase

\( {\varvec{M}}_{\text{A}}^{{}} \) :

Magnetization vector of austenite phase

\( M_{\text{sat}}^{\text{A}} \) :

Saturated magnetization of austenite phase

\( {\varvec{m}}_{\alpha 1} , {\varvec{m}}_{\alpha 2} \) :

Magnetization vectors of magnetic domains in the α-th martensite variant

\( {\varvec{S}}_{\text{M}}^{\alpha } \) :

Elastic compliance of the α-th martensite variant

\( {\varvec{S}}_{\text{A}}^{{}} \) :

Elastic compliance of austenite phase

\( {\bar{\varvec{S}}} \) :

Effective elastic compliance

\( T \) :

Temperature

\( T_{0} \) :

Balance temperature

\( u_{\text{A}}^{0} \) :

Internal energy at the referential state in austenite phase

\( u_{\text{M}}^{0} \) :

Internal energy at the referential state in martensite phase

\( Y \) :

Initial transformation resistance

\( \varGamma_{\text{int}}^{{}} \) :

Intrinsic dissipation

\( \varGamma_{\text{flux}}^{{}} \) :

Heat flux dissipation

\( \varGamma_{\text{tr}}^{\alpha } \) :

Energy dissipation caused by transformation

\( \varGamma_{\text{reo}}^{ij} \) :

Energy dissipation caused by reorientation

\( {\varvec{\delta}} \) :

Second-ordered unit tensor

\( {\varvec{\varepsilon}} \) :

Total strain

\( {\varvec{\varepsilon}}^{e} \) :

Elastic strain

\( {\varvec{\varepsilon}}^{\text{in}} \) :

Inelastic strain

\( {\varvec{\varepsilon}}^{\text{tr}} \) :

Transformation strain

\( {\varvec{\varepsilon}}^{\text{reo}} \) :

Reorientation strain

\( {\varvec{\varepsilon}}_{\alpha }^{*} \) :

Eigenstrain of the α-th martensite variant

\( \zeta_{\alpha } , 1 - \zeta_{\alpha } \) :

Volume fractions of magnetic domains in the α-th martensite variant

\( \eta \) :

Entropy

\( \eta_{\text{A}}^{0} \) :

Specific entropy in austenite phase

\( \eta_{\text{M}}^{0} \) :

Specific entropy in martensite phase

\( \Delta \eta_{{}}^{0} \) :

Difference of the specific entropy between martensite and austenite phases

\( \theta_{\alpha i} ,\,\varphi_{\alpha i} \) :

Deviation angles of magnetization vectors

\( \lambda_{ij} \) :

Transition amount from the j-th to i-th martensite variant

\( \mu_{0} \) :

Vacuum permeability

\( \pi_{\alpha }^{\text{tr}} \) :

Thermodynamic driving force of martensite transformation

\( \pi_{ij}^{\text{reo}} \) :

Thermodynamic driving force of martensite reorientation

\( \pi_{\alpha }^{\text{wall}} \) :

Thermodynamic driving force of domain wall motion

\( \pi_{\alpha \beta }^{\theta } , \pi_{\alpha \beta }^{\theta } \) :

Thermodynamic driving force of magnetization vector rotation

\( \xi_{\alpha } \) :

Volume fraction of the α-th martensite variant

\( \xi_{\alpha }^{\text{tr}} \) :

Volume fraction of the α-th martensite variant related to martensite transformation

\( \xi_{\alpha }^{\text{reo}} \) :

Volume fraction of the α-th martensite variant related to martensite reorientation

\( \rho \) :

Density

\( {\varvec{\sigma}} \) :

Stress

References

  1. Karaman, I., Basaran, B., Karaca, H.E., et al.: Energy harvesting using martensite variant reorientation mechanism in a NiMnGa magnetic shape memory alloy. Appl. Phys. Lett. 90, 172505 (2007)

    Article  Google Scholar 

  2. Qu, Y.H., Cong, D.Y., Li, S.H., et al.: Simultaneously achieved large reversible elastocaloric and magnetocaloric effects and their coupling in a magnetic shape memory alloy. Acta Mater. 151, 41–55 (2017)

    Article  Google Scholar 

  3. Karaca, H.E., Karaman, I., Basaran, B., et al.: Magnetic field and stress induced martensite reorientation in NiMnGa ferromagnetic shape memory alloy single crystals. Acta Mater. 54, 233–245 (2006)

    Article  Google Scholar 

  4. Tickle, R., James, R.D.: Magnetic and magnetomechanical properties of Ni2MnGa. J. Magn. Magn. Mater. 195, 627–638 (1999)

    Article  Google Scholar 

  5. Heczko, O.: Magnetic shape memory effect and magnetization reversal. J. Magn. Magn. Mater. 290, 787–794 (2005)

    Article  Google Scholar 

  6. Straka, L., Heczko, O.: Reversible 6% strain of Ni-Mn-Ga martensite using opposing external stress in static and variable magnetic fields. J. Magn. Magn. Mater. 290, 829–831 (2005)

    Article  Google Scholar 

  7. Chen, X., Moumni, Z., He, Y., et al.: A three-dimensional model of magneto-mechanical behaviors of martensite reorientation in ferromagnetic shape memory alloys. J. Mech. Phys. Solids 64, 249–286 (2014)

    Article  MathSciNet  Google Scholar 

  8. Kiefer, B., Karaca, H., Lagoudas, D., et al.: Characterization and modeling of the magnetic field-induced strain and work output in Ni2MnGa magnetic shape memory alloys. J. Magn. Magn. Mater. 312, 164–175 (2007)

    Article  Google Scholar 

  9. Nelson, I., Ciocanel, C., LaMaster, D., et al.: Three dimensional experimental characterization of a NiMnGa alloy. In Proceedings of SPIE 8689, Behavior and Mechanics of Multifunctional Materials and Composites (2013)

  10. LaMaster, D.H., Feigenbaum, H.P., Nelson, I.D., et al.: A full two-dimensional thermodynamic-based model for magnetic shape memory alloys. J. Appl. Mech. 81, 061003 (2014)

    Article  Google Scholar 

  11. Feigenbaum, H.P., Ciocanel, C., Eberle, J.L., et al.: Experimental characterization and modeling of a three-variant magnetic shape memory alloy. Smart Mater. Struct. 25, 104004 (2016)

    Article  Google Scholar 

  12. Zhang, H., Armstrong, A., Müllner, P.: Effects of surface modifications on the fatigue life of unconstrained Ni–Mn–Ga single crystals in a rotating magnetic field. Acta Mater. 155, 175–186 (2018)

    Article  Google Scholar 

  13. Kucza, N.J., Patrick, C.L., Dunand, D.C., et al.: Magnetic-field-induced bending and straining of Ni–Mn–Ga single crystal beams with high aspect ratios. Acta Mater. 95, 284–290 (2015)

    Article  Google Scholar 

  14. Chernenko, V.A., Pons, J., Cesari, E., et al.: Stress-temperature phase diagram of a ferromagnetic Ni–Mn–Ga shape memory alloy. Acta Mater. 53, 5071–5077 (2005)

    Article  Google Scholar 

  15. Seguí, C., Chernenko, V.A., Pons, J., et al.: Low temperature-induced intermartensitic phase transformations in Ni–Mn–Ga single crystal. Acta Mater. 53, 111–120 (2005)

    Article  Google Scholar 

  16. Kim, J.H., Inaba, F., Fukuda, T., et al.: Effect of magnetic field on martensitic transformation temperature in Ni–Mn–Ga ferromagnetic shape memory alloys. Acta Mater. 54, 493–499 (2006)

    Article  Google Scholar 

  17. Heczko, O., L’vov, V.A., Straka, L., et al.: Magnetic indication of the stress-induced martensitic transformation in ferromagnetic Ni–Mn–Ga alloy. J. Magn. Magn. Mater. 302, 387–390 (2006)

    Article  Google Scholar 

  18. Karaman, I., Karaca, H.E., Basaran, B., et al.: Stress-assisted reversible magnetic field-induced phase transformation in Ni2MnGa magnetic shape memory alloys. Scr. Mater. 55, 403–406 (2006)

    Article  Google Scholar 

  19. Karaca, H.E., Karaman, I., Basaran, B., et al.: On the stress-assisted magnetic-field-induced phase transformation in Ni2MnGa ferromagnetic shape memory alloys. Acta Mater. 55, 4253–4269 (2007)

    Article  Google Scholar 

  20. James, R.D., Wuttig, M.: Magnetostriction of martensite. Philos. Mag. A 77, 1273–1299 (1998)

    Article  Google Scholar 

  21. DeSimone, A., James, R.D.: A constrained theory of magnetoelasticity. J. Mech. Phys. Solids 50, 283–320 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. O’Handley, R.C.: Model for strain and magnetization in magnetic shape-memory alloys. J. Appl. Phys. 83, 3263–3270 (1998)

    Article  Google Scholar 

  23. Murray, S.J., O’Handley, R.C., Allen, S.M.: Model for discontinuous actuation of ferromagnetic shape memory alloy under stress. J. Appl. Phys. 89, 1295–1301 (2001)

    Article  Google Scholar 

  24. Straka, L., Heczko, O.: Superelastic response of Ni–Mn–Ga martensite in magnetic fields and a simple model. IEEE Trans. Magn. 39, 3402–3404 (2003)

    Article  Google Scholar 

  25. Kiang, J., Tong, L.: Modelling of magneto-mechanical behaviour of Ni–Mn–Ga single crytals. J. Magn. Magn. Mater. 292, 394–412 (2005)

    Article  Google Scholar 

  26. He, Y.J., Chen, X., Moumni, Z.: Two-dimensional analysis to improve the output stress in ferromagnetic shape memory alloys. J. Appl. Phys. 110, 063905 (2011)

    Article  Google Scholar 

  27. He, Y.J., Chen, X., Moumni, Z.: Reversible-strain criteria of ferromagnetic shape memory alloys under cyclic 3D magneto-mechanical loadings. J. Appl. Phys. 112, 033902 (2012)

    Article  Google Scholar 

  28. Wang, J., Steinmann, P.: A variational approach towards the modeling of magnetic field-induced strains in magnetic shape memory alloys. J. Mech. Phys. Solids 60, 1179–1200 (2012)

    Article  MathSciNet  Google Scholar 

  29. Pei, Y., Fang, D.: A model for giant magnetostrain and magnetization in the martensitic phase of NiMnGa alloys. Smart Mater. Struct. 16, 779 (2007)

    Article  Google Scholar 

  30. Brinson, L.C.: One-dimensional constitutive behavior of shape memory alloys: thermomechanical derivation with non-constant material functions and redefined martensite internal variable. J. Intell. Mater. Syst. Struct. 4, 229–242 (1993)

    Article  Google Scholar 

  31. Zhu, Y., Dui, G.: Micromechanical modeling of the stress-induced superelastic strain in magnetic shape memory alloy. Mech. Mater. 39, 1025–1034 (2007)

    Article  Google Scholar 

  32. Zhu, Y., Dui, G.: Model for field-induced reorientation strain in magnetic shape memory alloy with tensile and compressive loads. J. Alloys Compd. 459, 55–60 (2008)

    Article  Google Scholar 

  33. Zhu, Y., Yu, K.: A model considering mechanical anisotropy of magnetic-field-induced superelastic strain in magnetic shape memory alloys. J. Alloys Compd. 550, 308–313 (2013)

    Article  Google Scholar 

  34. Wang, X., Li, F., Hu, Q.: An anisotropic micromechanical-based model for characterizing the magneto-mechanical behavior of NiMnGa alloys. Smart Mater. Struct. 21, 065021 (2012)

    Article  Google Scholar 

  35. Jin, Y.M.: Domain microstructure evolution in magnetic shape memory alloys: phase-field model and simulation. Acta Mater. 57, 2488–2495 (2009)

    Article  Google Scholar 

  36. Li, L.J., Lei, C.H., Shu, Y.C., et al.: Phase-field simulation of magnetoelastic couplings in ferromagnetic shape memory alloys. Acta Mater. 59, 2648–2655 (2011)

    Article  Google Scholar 

  37. Mennerich, C., Wendler, F., Jainta, M., et al.: A phase-field model for the magnetic shape memory effect. Arch. Mech. 63, 549–571 (2011)

    MathSciNet  Google Scholar 

  38. Peng, Q., He, Y.J., Moumni, Z.: A phase-field model on the hysteretic magneto-mechanical behaviors of ferromagnetic shape memory alloy. Acta Mater. 88, 13–24 (2015)

    Article  Google Scholar 

  39. Peng, Q., Huang, J., Chen, M., et al.: Phase-field simulation of magnetic hysteresis and mechanically induced remanent magnetization rotation in Ni–Mn–Ga ferromagnetic shape memory alloy. Scr. Mater. 127, 49–53 (2017)

    Article  Google Scholar 

  40. Peng, Q., Huang, J., Chen, M.: Effects of demagnetization on magnetic-field-induced strain and microstructural evolution in Ni–Mn–Ga ferromagnetic shape memory alloy by phase-field simulations. Mater. Des. 107, 361–370 (2016)

    Article  Google Scholar 

  41. Hirsinger, L., Lexcellent, C.: Modelling detwinning of martensite platelets under magnetic and (or) stress actions on Ni-Mn-Ga alloys. J. Magn. Magn. Mater. 254, 275–277 (2003)

    Article  Google Scholar 

  42. Kiefer, B., Lagoudas, D.C.: Magnetic field-induced martensitic variant reorientation in magnetic shape memory alloys. Philos. Magn. 85, 4289–4329 (2005)

    Article  Google Scholar 

  43. Kiefer, B., Lagoudas, D.C.: Modeling the coupled strain and magnetization response of magnetic shape memory alloys under magnetomechanical loading. J. Intell. Mater. Syst. Struct. 20, 143–170 (2009)

    Article  Google Scholar 

  44. Gauthier, J.Y., Lexcellent, C., Hubert, A., et al.: Modeling rearrangement process of martensite platelets in a magnetic shape memory alloy Ni2MnGa single crystal under magnetic field and (or) stress action. J. Intell. Mater. Syst. Struct. 18, 289–299 (2007)

    Article  Google Scholar 

  45. Wang, X., Li, F.: A kinetics model for martensite variants rearrangement in ferromagnetic shape memory alloys. J. Appl. Phys. 108, 113921 (2010)

    Article  Google Scholar 

  46. Shirani, M., Kadkhodaei, M.: A modified constitutive model with an enhanced phase diagram for ferromagnetic shape memory alloys. J. Intell. Mater. Syst. Struct. 26, 56–68 (2015)

    Article  Google Scholar 

  47. Shirani, M., Kadkhodaei, M.: A geometrical approach to determine reorientation start and continuation conditions in ferromagnetic shape memory alloys considering the effects of loading history. Smart Mater. Struct. 23, 125008 (2014)

    Article  Google Scholar 

  48. Shirani, M., Kadkhodaei, M.: Constitutive modeling of Ni–Mn–Ga ferromagnetic shape memory alloys under biaxial compression. J. Intell. Mater. Syst. Struct. 27, 1547–1564 (2016)

    Article  Google Scholar 

  49. Auricchio, F., Bessoud, A.L., Reali, A., et al.: A phenomenological model for the magneto-mechanical response of single-crystal magnetic shape memory alloys. Eur J Mech A Solid 52, 1–11 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  50. LaMaster, D.H., Feigenbaum, H.P., Ciocanel, C., et al.: A full 3D thermodynamic-based model for magnetic shape memory alloys. J. Intell. Mater. Syst. Struct. 26, 663–679 (2015)

    Article  Google Scholar 

  51. Mousavi, M.R., Arghavani, J.: A three-dimensional constitutive model for magnetic shape memory alloys under magneto-mechanical loadings. Smart Mater. Struct. 26, 015014 (2017)

    Article  Google Scholar 

  52. Haldar, K., Lagoudas, D.C., Karaman, I.: Magnetic field-induced martensitic phase transformation in magnetic shape memory alloys: modeling and experiments. J. Mech. Phys. Solids 69, 33–66 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  53. Rogovoy, A., Stolbova, O.: Modeling the magnetic field control of phase transition in ferromagnetic shape memory alloys. Int. J. Plast 85, 130–155 (2016)

    Article  Google Scholar 

  54. Martynov, V.V., Kokorin, V.V.: The crystal structure of thermally-and stress-induced martensites in Ni2MnGa single crystals. J. Phys. III, 739–749 (1992)

    Google Scholar 

  55. Heczko, O., Sozinov, A., Ullakko, K.: Giant field-induced reversible strain in magnetic shape memory NiMnGa alloy. IEEE Trans. Magn. 36, 3266–3268 (2000)

    Article  Google Scholar 

  56. Straka, L., Heczko, O., Seiner, H., et al.: Highly mobile twinned interface in 10 M modulated Ni–Mn–Ga martensite: Analysis beyond the tetragonal approximation of lattice. Acta Mater. 59, 7450–7463 (2011)

    Article  Google Scholar 

  57. Tickle, R., James, R.D., Shield, T., et al.: Ferromagnetic shape memory in the NiMnGa system. IEEE Trans. Magn. 35, 4301–4310 (1999)

    Article  Google Scholar 

  58. Heczko, O., Jurek, K., Ullakko, K.: Magnetic properties and domain structure of magnetic shape memory Ni-Mn-Ga alloy. J. Magn. Magn. Mater. 226, 996–998 (2001)

    Article  Google Scholar 

  59. Thamburaja, P.: Constitutive equations for martensitic reorientation and detwinning in shape-memory alloys. J. Mech. Phys. Solids 53, 825–856 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  60. Lagoudas, D., Hartl, D., Chemisky, Y., Machado, L., et al.: Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys. Int. J. Plast. 32, 155–183 (2012)

    Article  Google Scholar 

  61. O’Handley, R.C., Murray, S.J., Marioni, M., et al.: Phenomenology of giant magnetic-field-induced strain in ferromagnetic shape-memory materials. J. Appl. Phys. 87, 4712–4717 (2000)

    Article  Google Scholar 

  62. Anand, L., Gurtin, M.E.: Thermal effects in the superelasticity of crystalline shape-memory materials. J. Mech. Phys. Solids 51, 1015–1058 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  63. Yu, C., Kang, G., Kan, Q.: Crystal plasticity based constitutive model of NiTi shape memory alloy considering different mechanisms of inelastic deformation. Int. J. Plast. 54, 132–162 (2014)

    Article  Google Scholar 

  64. Yu, C., Kang, G., Song, D., et al.: Effect of martensite reorientation and reorientation-induced plasticity on multiaxial transformation ratchetting of super-elastic NiTi shape memory alloy: new consideration in constitutive model. Int. J. Plast. 67, 69–101 (2015)

    Article  Google Scholar 

  65. Anand, L., Kothari, M.: A computational procedure for rate-independent crystal plasticity. J. Mech. Phys. Solids 44, 525–558 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  66. Kiefer, B., Lagoudas, D.C.: Modeling of magnetic SMAs. Shape Memory Alloys. Springer, Berlin (2008)

    MATH  Google Scholar 

  67. Panico, M., Brinson, L.C.: A three-dimensional phenomenological model for martensite reorientation in shape memory alloys. J. Mech. Phys. Solids 55, 2491–2511 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  68. Zhang, S., Chen, X., Moumni, Z., et al.: Thermal effects on high-frequency magnetic-field-induced martensite reorientation in ferromagnetic shape memory alloys: an experimental and theoretical investigation. Int. J. Plast 108, 1–20 (2018)

    Article  Google Scholar 

  69. Zhang, S., Chen, X., Moumni, Z., et al.: Coexistence and compatibility of martensite reorientation and phase transformation in high-frequency magnetic-field-induced deformation of Ni–Mn–Ga single crystal. Int. J. Plast. 110, 110–122 (2018)

    Article  Google Scholar 

  70. Song, D., Kang, G., Kan, Q., et al.: Non-proportional multiaxial transformation ratchetting of super-elastic NiTi shape memory alloy: experimental observations. Mech. Mater. 70, 94–105 (2014)

    Article  Google Scholar 

  71. Arghavani, J., Auricchio, F., Naghdabadi, R., et al.: A 3-D phenomenological constitutive model for shape memory alloys under multiaxial loadings. Int. J. Plast 26, 976–991 (2010)

    Article  MATH  Google Scholar 

  72. Song, D., Kang, G., Kan, Q., et al.: Effects of peak stress and stress amplitude on multiaxial transformation ratchetting and fatigue life of superelastic NiTi SMA micro-tubes: experiments and life-prediction model. Int. J. Fatigue 96, 252–260 (2017)

    Article  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant 11602203), Young Elite Scientist Sponsorship Program by the China Association for Science and Technology (Grant 2016QNRC001), and Fundamental Research Funds for the Central Universities (Grant 2682018CX43).

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Authors and Affiliations

  1. State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu, 610031, China

    Chao Yu & Guozheng Kang

  2. Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, 610031, China

    Chao Yu, Guozheng Kang & Xi Xie

  3. School of Mechatronics Engineering, University of Electronic Science and Technology of China, Chengdu, 610031, China

    Di Song

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  1. Chao Yu
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Yu, C., Kang, G., Song, D. et al. Three-dimensional constitutive model for magneto-mechanical deformation of NiMnGa ferromagnetic shape memory alloy single crystals. Acta Mech. Sin. 35, 563–588 (2019). https://doi.org/10.1007/s10409-018-0816-6

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