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Thermomechanical modeling of nonlinear internal hysteresis due to incomplete phase transformation in pseudoelastic shape memory alloys

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Abstract

This paper presents a thermomechanical model for pseudoelastic shape memory alloys (SMAs) accounting for internal hysteresis effect due to incomplete phase transformation. The model is developed within the finite-strain framework, wherein the deformation gradient is multiplicatively decomposed into thermal dilation, rigid body rotation, elastic and transformation parts. Helmholtz free energy density comprises three components: the reversible thermodynamic process , the irreversible thermodynamic process and the physical constraints of both. In order to capture the multiple internal hysteresis loops in SMA, two internal variables representing the transition points of the forward and reverse phase transformation, \(\phi _s^f\) and \(\phi _s^r\), are introduced to describe the incomplete phase transformation process. Evolution equations of the internal variables are derived and linked to the phase transformation. Numerical implementation of the model features an Euler discretization and a cutting-plane algorithm. After validation of the model against the experimental data, numerical examples are presented, involving a SMA-based vibration system and a crack SMA specimen subjected to partial loading–unloading case. Simulation results well demonstrate the internal hysteresis and free vibration behavior of SMA.

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References

  1. Wang, J., Moumni, Z., Zhang, W., Xu, Y., Zaki, W.: A 3D finite-strain-based constitutive model for shape memory alloys accounting for thermomechanical coupling and martensite reorientation. Smart Mater. Struct. (2017). https://doi.org/10.1088/1361-665X/aa6c17

    Article  Google Scholar 

  2. Yu, C., Kang, G., Kan, Q.: A micromechanical constitutive model for grain size dependent thermo-mechanically coupled inelastic deformation of super-elastic NiTi shape memory alloy. Int. J. Plast. 105, 99–127 (2018). https://doi.org/10.1016/j.ijplas.2018.02.005

    Article  Google Scholar 

  3. Moumni, Z., Zhang, Y., Wang, J., Gu, X.: A global approach for the fatigue of shape memory alloys. Shape Memory Superelast. 4, 385–401 (2018). https://doi.org/10.1007/s40830-018-00194-2

    Article  Google Scholar 

  4. Dhala, S., Mishra, S., Tewari, A., Alankar, A.: Modeling of finite deformation of pseudoelastic NiTi shape memory alloy considering various inelasticity mechanisms. Int. J. Plast. 115, 216–237 (2019). https://doi.org/10.1016/j.ijplas.2018.11.018

    Article  Google Scholar 

  5. Birman, V.: Review of mechanics of shape memory alloy structures. Appl. Mech. Rev. 50, 629–646 (1997)

    Article  Google Scholar 

  6. Savi, M.A., Paiva, A.: Describing internal subloops due to incomplete phase transformations in shape memory alloys. Arch. Appl. Mech. 74, 637–647 (2005). https://doi.org/10.1007/s00419-005-0385-6

    Article  MATH  Google Scholar 

  7. Tang, C., Wang, T., Huang, W., Sun, L., Gao, X.: Temperature sensors based on the temperature memory effect in shape memory alloys to check minor over-heating. Sens. Actuators A 238, 337–343 (2016). https://doi.org/10.1016/j.sna.2015.11.033

    Article  Google Scholar 

  8. Dhanalakshmi, K.: Shape memory alloy wire for self-sensing servo actuation. Mech. Syst. Signal Process. 83, 36–52 (2017). https://doi.org/10.1016/j.ymssp.2016.05.042

    Article  Google Scholar 

  9. Formentini, M., Lenci, S.: An innovative building envelope (kinetic façade) with Shape Memory Alloys used as actuators and sensors. Autom. Constr. 85, 220–231 (2018). https://doi.org/10.1016/j.autcon.2017.10.006

    Article  Google Scholar 

  10. Rizzello, G., Mandolino, M.A., Schmidt, M., Naso, D., Seelecke, S.: An accurate dynamic model for polycrystalline shape memory alloy wire actuators and sensors. Smart Mater. Struct. 28, 25020 (2019)

    Article  Google Scholar 

  11. Tang, T., Felicelli, S.D.: Micromechanical investigations of polymer matrix composites with shape memory alloy reinforcement. Int. J. Eng. Sci. 94, 181–194 (2015). https://doi.org/10.1016/j.ijengsci.2015.05.008

    Article  MathSciNet  MATH  Google Scholar 

  12. Song, S.-H., Lee, J.-Y., Rodrigue, H., Choi, I.-S., Kang, Y.J., Ahn, S.-H.: 35 Hz shape memory alloy actuator with bending-twisting mode. Sci. Rep. 6, 21118 (2016). https://doi.org/10.1038/srep21118

    Article  Google Scholar 

  13. Mohd Jani, J., Leary, M., Subic, A.: Designing shape memory alloy linear actuators: a review. J. Intell. Mater. Syst. Struct. 28, 1699–1718 (2017). https://doi.org/10.1177/1045389X16679296

    Article  Google Scholar 

  14. Huang, X., Kumar, K., Jawed, M.K., Mohammadi Nasab, A., Ye, Z., Shan, W., Majidi, C.: Highly dynamic shape memory alloy actuator for fast moving soft robots. Adv. Mater. Technol. 4, 1800540 (2019). https://doi.org/10.1002/admt.201800540

    Article  Google Scholar 

  15. Wang, J., Zhang, W., Zhu, J., Xu, Y., Gu, X., Moumni, Z.: Finite element simulation of thermomechanical training on functional stability of shape memory alloy wave spring actuator. J. Intell. Mater. Syst. Struct. 30, 1239–1251 (2019). https://doi.org/10.1177/1045389X19831356

    Article  Google Scholar 

  16. Xue, L., Dui, G., Liu, B., Xin, L.: A phenomenological constitutive model for functionally graded porous shape memory alloy. Int. J. Eng. Sci. 78, 103–113 (2014). https://doi.org/10.1016/j.ijengsci.2014.02.013

    Article  Google Scholar 

  17. Mohd Jani, J., Leary, M., Subic, A., Gibson, M.A.: A review of shape memory alloy research, applications and opportunities. Mater. Des. 56, 1078–1113 (2014). https://doi.org/10.1016/j.matdes.2013.11.084

    Article  Google Scholar 

  18. Qian, H., Li, H., Song, G.: Experimental investigations of building structure with a superelastic shape memory alloy friction damper subject to seismic loads. Smart Mater. Struct. 25, 125026 (2016). https://doi.org/10.1088/0964-1726/25/12/125026

    Article  Google Scholar 

  19. Dutta, S.C., Majumder, R.: Shape memory alloy (SMA) as a potential damper in structural vibration control. In: Advances in Manufacturing Engineering and Materials. Springer, pp. 485–492 (2019). https://doi.org/10.1007/978-3-319-99353-9_51

  20. Bogue, R.: Shape-memory materials: a review of technology and applications. Assem. Autom. 29, 214–219 (2009). https://doi.org/10.1108/01445150910972895

    Article  Google Scholar 

  21. Zhang, Y., Moumni, Z., Zhu, J., Zhang, W.: Effect of the amplitude of the training stress on the fatigue lifetime of NiTi shape memory alloys. Scripta Mater. 149, 66–69 (2018). https://doi.org/10.1016/j.scriptamat.2018.02.012

    Article  Google Scholar 

  22. Abdullah, E., Gaikwad, P., Azid, N., Abdul Majid, D., Mohd Rafie, A.: Temperature and strain feedback control for shape memory alloy actuated composite plate. Sens. Actuators A 283, 134–140 (2018). https://doi.org/10.1016/j.sna.2018.09.059

    Article  Google Scholar 

  23. Spindler, C., Juhre, D.: Development of a shape memory alloy actuator using generative manufacturing. Int. J. Adv. Manuf. Technol. 97, 4157–4166 (2018). https://doi.org/10.1007/s00170-018-2153-0

    Article  Google Scholar 

  24. Tanaka, K., Nishimura, F., Tobushi, H.: Phenomenological Analysis on Subloops in Shape Memory Alloys Due to Incomplete Transformations. J. Intell. Mater. Syst. Struct. 5, 487–493 (1994). https://doi.org/10.1177/1045389X9400500404

    Article  Google Scholar 

  25. Lexcellent, C., Tobushi, H.: Internal loops in pseudoelastic behaviour of Ti–Ni shape memory alloys: experiment and modelling. Meccanica 30, 459–466 (1995). https://doi.org/10.1007/BF01557078

    Article  Google Scholar 

  26. Tobushi, H., Endo, M., Ikawa, T., Shimada, D.: Pseudoviscoelastic behavior of TiNi shape memory alloys under stress-controlled subloop loadings. Arch. Mech. 55, 519–530 (2003)

    Google Scholar 

  27. Pieczyska, E.A., Tobushi, H., Nowacki, W.K., Gadaj, S.P., Sakuragi, T.: Subloop deformation behavior of TiNi shape memory alloy subjected to stress-controlled loadings. Mater. Trans. 48, 2679–2686 (2007). https://doi.org/10.2320/matertrans.MRA2007097

    Article  Google Scholar 

  28. Takeda, K., Tobushi, H., Miyamoto, K., Pieczyska, E.A.: Superelastic deformation of TiNi shape memory alloy subjected to various subloop loadings. Mater. Trans. 53, 217–223 (2012). https://doi.org/10.2320/matertrans.M2011288

    Article  Google Scholar 

  29. Rao, A., Ruimi, A., Srinivasa, A.R.: Internal loops in superelastic shape memory alloy wires under torsion: experiments and simulations/predictions. Int. J. Solids Struct. 51, 4554–4571 (2014). https://doi.org/10.1016/j.ijsolstr.2014.09.002

    Article  Google Scholar 

  30. Boyd, J., Lagoudas, D.: A thermodynamical constitutive model for shape memory materials. Part I. The monolithic shape memory alloy. Int. J. Plast 12, 805–842 (1996). https://doi.org/10.1016/S0749-6419(96)00030-7

    Article  MATH  Google Scholar 

  31. Bo, Z., Lagoudas, D.C.: Thermomechanical modeling of polycrystalline SMAs under cyclic loading, part IV: modeling of minor hysteresis loops. Int. J. Eng. Sci. 37, 1205–1249 (1999). https://doi.org/10.1016/S0020-7225(98)00116-5

    Article  MATH  Google Scholar 

  32. Ortin, J., Delaey, L.: Hysteresis in shape-memory alloys. Int. J. Non Linear Mech. 37, 1275–1281 (2002). https://doi.org/10.1016/S0020-7462(02)00027-6

    Article  MATH  Google Scholar 

  33. Matsuzaki, Y., Funami, K., Naito, H.: Inner loops of pseudoelastic hysteresis of shape memory alloys: Preisach approach, vol. 4699, pp. 355–364 (2002). https://doi.org/10.1117/12.474993

  34. Ikeda, T., Nae, F.A., Naito, H., Matsuzaki, Y.: Constitutive model of shape memory alloys for unidirectional loading considering inner hysteresis loops. Smart Mater. Struct. 13, 916–925 (2004). https://doi.org/10.1088/0964-1726/13/4/030

    Article  Google Scholar 

  35. Bernardini, D., Pence, T.J.: Uniaxial modeling of multivariant shape-memory materials with internal sublooping using dissipation functions. Meccanica 40, 339–364 (2005). https://doi.org/10.1007/s11012-005-2103-4

    Article  MathSciNet  MATH  Google Scholar 

  36. Kishore Kumar, M., Sakthivel, K., Sivakumar, S.M., Lakshmana Rao, C., Srinivasa, A.: Thermomechanical modeling of hysteresis in SMAs using the dissipationless reference response. Smart Mater. Struct. (2007). https://doi.org/10.1088/0964-1726/16/1/S04

    Article  Google Scholar 

  37. Heintze, O., Seelecke, S.: A coupled thermomechanical model for shape memory alloys-From single crystal to polycrystal. Mater. Sci. Eng. A 481–482, 389–394 (2008). https://doi.org/10.1016/j.msea.2007.08.028

    Article  Google Scholar 

  38. Müller, I.: Pseudo-elastic hysteresis in shape memory alloys. Phys. B 407, 1314–1315 (2012). https://doi.org/10.1016/j.physb.2011.06.088

    Article  Google Scholar 

  39. Viet, N.V., Zaki, W., Umer, R., Xu, Y.: Mathematical model for superelastic shape memory alloy springs with large spring index. Int. J. Solids Struct. 185, 159–169 (2020)

    Article  Google Scholar 

  40. Scalet, G., Niccoli, F., Garion, C., Chiggiato, P., Maletta, C., Auricchio, F.: A three-dimensional phenomenological model for shape memory alloys including two-way shape memory effect and plasticity. Mech. Mater. 136, 103085 (2019)

    Article  Google Scholar 

  41. Müller, C., Bruhns, O.T.: A thermodynamic finite-strain model for pseudoelastic shape memory alloys. Int. J. Plast. 22, 1658–1682 (2006). https://doi.org/10.1016/j.ijplas.2006.02.010

    Article  MATH  Google Scholar 

  42. Teeriaho, J.P.: An extension of a shape memory alloy model for large deformations based on an exactly integrable Eulerian rate formulation with changing elastic properties. Int. J. Plast. 43, 153–176 (2013). https://doi.org/10.1016/j.ijplas.2012.11.009

    Article  Google Scholar 

  43. Xiao, H.: An explicit, straightforward approach to modeling SMA pseudoelastic hysteresis. Int. J. Plast. 53, 228–240 (2014). https://doi.org/10.1016/j.ijplas.2013.08.010

    Article  Google Scholar 

  44. Reese, S., Christ, D.: Finite deformation pseudo-elasticity of shape memory alloys: constitutive modelling and finite element implementation. Int. J. Plast. 24, 455–482 (2008). https://doi.org/10.1016/j.ijplas.2007.05.005

    Article  MATH  Google Scholar 

  45. Arghavani, J., Auricchio, F., Naghdabadi, R.: A finite strain kinematic hardening constitutive model based on Hencky strain: general framework, solution algorithm and application to shape memory alloys. Int. J. Plast. 27, 940–961 (2011). https://doi.org/10.1016/j.ijplas.2010.10.006

    Article  MATH  Google Scholar 

  46. Paranjape, H.M., Manchiraju, S., Anderson, P.M.: A phase field: finite element approach to model the interaction between phase transformations and plasticity in shape memory alloys. Int. J. Plast. 80, 1–18 (2016). https://doi.org/10.1016/j.ijplas.2015.12.007

    Article  Google Scholar 

  47. Wang, J., Moumni, Z., Zhang, W., Zaki, W.: A thermomechanically coupled finite deformation constitutive model for shape memory alloys based on Hencky strain. Int. J. Eng. Sci. 117, 51–77 (2017). https://doi.org/10.1016/j.ijengsci.2017.05.003

    Article  MathSciNet  MATH  Google Scholar 

  48. Dunne, F., Petrinic, N.: Introduction to Computational Plasticity. Oxford University Press, New York (2005)

    MATH  Google Scholar 

  49. Anand, L., Ames, N.M., Srivastava, V., Chester, S.A.: A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part I: formulation. Int. J. Plast. 25, 1474–1494 (2009). https://doi.org/10.1016/j.ijplas.2008.11.004

    Article  MATH  Google Scholar 

  50. Wang, J., Moumni, Z., Zhang, W.: A thermomechanically coupled finite-strain constitutive model for cyclic pseudoelasticity of polycrystalline shape memory alloys. Int. J. Plast. 97, 194–221 (2017). https://doi.org/10.1016/j.ijplas.2017.06.003

    Article  Google Scholar 

  51. Qidwai, M.A., Lagoudas, D.C.: On thermomechanics and transformation surfaces of polycrystalline NiTi shape memory alloy material. Int. J. Plast. 16, 1309–1343 (2000). https://doi.org/10.1016/S0749-6419(00)00012-7

    Article  MATH  Google Scholar 

  52. Lagoudas, D., Hartl, D., Chemisky, Y., Machado, L., Popov, P.: Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys. Int. J. Plast. 32–33, 155–183 (2012). https://doi.org/10.1016/j.ijplas.2011.10.009

    Article  Google Scholar 

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Funding

This study was funded by National Natural Science Foundation of China (11802241).

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Authors and Affiliations

  1. Unmanned System Research Institute, Northwestern Polytechnical University, Xi’an, 710072, China

    Jun Wang, Xiaojun Gu & Jihong Zhu

  2. State IJR Center of Aerospace Design and Additive Manufacturing, Northwestern Polytechnical University, Xi’an, 710072, China

    Jun Wang, Xiaojun Gu, Yingjie Xu, Jihong Zhu & Weihong Zhang

  3. Shaanxi Engineering Laboratory of Aerospace Structure Design and Application, Northwestern Polytechnical University, Xi’an, 710072, China

    Yingjie Xu & Jihong Zhu

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  1. Jun Wang
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Wang, J., Gu, X., Xu, Y. et al. Thermomechanical modeling of nonlinear internal hysteresis due to incomplete phase transformation in pseudoelastic shape memory alloys. Nonlinear Dyn 103, 1393–1414 (2021). https://doi.org/10.1007/s11071-020-06121-4

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