Thermomechanical couplings in shape memory alloy materials


In this work, we address several theoretical and computational issues which are related to the thermomechanical modeling of shape memory alloy materials. More specifically, in this paper we revisit a non-isothermal version of the theory of large deformation generalized plasticity which is suitable for describing the multiple and complex mechanisms occurring in these materials during phase transformations. We also discuss the computational implementation of a generalized plasticity-based constitutive model, and we demonstrate the ability of the theory in simulating the basic patterns of the experimentally observed behavior by a set of representative numerical examples.

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  1. 1.

    Agelet de Saracibar, C., Cervera, M., Chiumenti, M.: On the constitutive modeling of coupled thermomechanical phase-change problems. Int. J. Plast. 17, 1565–1622 (2001)

    Article  Google Scholar 

  2. 2.

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

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    Andani, T.A., Alipour, A., Elahinia, M.: Coupled rate-dependent superelastic behavior of shape memory alloy bars induced by combined axial-torsional loading: a semi-analytic modeling. J. Intell. Mater. Syst. Struct. 24, 1995–2007 (2013)

    Article  Google Scholar 

  4. 4.

    Armero, F., Simo, J.C.: A priori stability estimates and unconditionally stable product formula algorithms for non-linear coupled thermoplasticity. Int. J. Plast. 9, 149–182 (1993)

    Article  Google Scholar 

  5. 5.

    Auricchio, F., Fugazza, D., DesRoches, R.: Rate-dependent thermo-mechanical modeling of superelastic shape-memory alloys for seismic applications. J. Intell. Mater. Syst. Struct. 19, 47–61 (2008)

    Article  Google Scholar 

  6. 6.

    Ball, J.M., James, R.D.: Fine phase mixtures and minimizers of energy. Arch. Rat. Mech. Anal. 100, 13–52 (1987)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Boyd, J.G., Lagoudas, D.C.: A thermodynamic constitutive model for shape memory alloy materials. Part I. The monolithic shape memory alloy. Int. J. Plast. 12, 805–842 (1994)

    Article  Google Scholar 

  8. 8.

    Boyd, J.C., Lagoudas, D.C.: Thermodynamical response of shape memory composites. J. Intell. Mater. Syst. Struct. 5, 333–346 (1994)

    Article  Google Scholar 

  9. 9.

    Christ, D., Reese, S.: A finite element model for shape-memory alloys considering thermomechanical couplings at large strains. Int. J. Solids Struct. 46, 3694–3709 (2009)

    Article  Google Scholar 

  10. 10.

    Earman, J.: Laws, symmetry and symmetry breaking: Invariance, conservation principles and objectivity. Philos. Sci. 71, 1227–1241 (2004)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Fosdick, R.L., Serrin, J.: Global properties of continuum thermodynamic processes. Arch. Rat. Mech. Anal. 59, 97–109 (1975)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Ganghoffer, J.F.: Symmetries in mechanics: from field theories to master responses in the constitutive modeling of materials. In: Ganghoffer, J.F., Mladenov, I. (eds.) Similarity and Symmetry Methods, Applications in Elasticity and Mechanics of Materials, pp. 271–351. Springer, New York (2014)

    Google Scholar 

  13. 13.

    Grabe, C., Bruhns, O.T.: On the viscous and strain rate dependent behavior of polycrystalline NiTi. Int. J. Solids Struct. 45, 1876–1895 (2008)

    Article  Google Scholar 

  14. 14.

    Holzapfel, G.A.: Nonlinear Solid Mechanics. A Continuum Approach for Engineering. Wiley, West Sussex (2000)

    MATH  Google Scholar 

  15. 15.

    Huo, Y., Müller, I.: Nonequilibrium thermodynamics of pseudoelasticity. Cont. Mech. Thermodyn. 5, 163–204 (1993)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Kamlah, M., Haupt, P.: On the macroscopic description of stored energy and self heating during plastic deformation. Int. J. Plast. 13, 893–911 (1998)

    Article  Google Scholar 

  17. 17.

    Lagoudas, D.C., Bo, Z., Qidwai, M.A.: A unified constitutive model for SMA and finite element analysis of active metal matrix composites. Mech. Compos. Mater. Struct. 3, 153–179 (1996)

    Article  Google Scholar 

  18. 18.

    Leclercq, S., Lexcellent, C.: A general macroscopic description of the thermomechanical behavior of shape memory alloys. J. Mech. Phys. Solids 44, 953–980 (1996)

    ADS  Article  Google Scholar 

  19. 19.

    Leo, P.H., Shield, T.W., Bruno, O.P.: Transient heat transfer effects on the pseudoelastic behavior of shape-memory wires. Acta Metall. Mater. 41, 2477–2485 (1993)

    Article  Google Scholar 

  20. 20.

    Lu, Z.K., Weng, G.J.: Martensitic transformation and stress–strain relations of shape-memory alloys. J. Mech. Phys. Solids 45, 1905–1921 (1997)

    ADS  Article  Google Scholar 

  21. 21.

    Lubliner, J.: Non-isothermal generalized plasticity. In: Bui, H.D., Nyugen, Q.S. (eds.) Thermomechanical Couplings in solids, pp. 121–133. North-Holland, Amsterdam (1987)

    Google Scholar 

  22. 22.

    Lubliner, J., Auricchio, F.: Generalized plasticity and shape memory alloys. Int. J. Solids Struct. 33, 991–1004 (1996)

    Article  Google Scholar 

  23. 23.

    Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Dover Publications, New York (1994)

    MATH  Google Scholar 

  24. 24.

    McKelvey, A.L., Ritchie, R.O.: On the temperature dependence of the superelastic strength and the prediction of the theoretical uniaxial transformation strain in Nititol. Philos. Mag. 80, 1759–1768 (2000)

    ADS  Article  Google Scholar 

  25. 25.

    Meyers, A., Xiao, H., Bruhns, O.: Elastic stress ratcheting and corotational stress rates. Tech. Mech. 23, 92–102 (2003)

    Google Scholar 

  26. 26.

    Mirzaeifar, R., DesRoches, R., Yavari, A.: Analysis of the rate-dependent coupled thermomechanical response of shape memory alloy bars and wires in tension. Cont. Mech. Thermodyn. 23, 363–385 (2011)

    Article  Google Scholar 

  27. 27.

    Morin, C., Moumni, Z., Zaki, W.: Thermomechanical coupling in shape memory alloys under cyclic loadings: experimental analysis and constitutive modeling. Int. J. Plast. 27, 1959–1980 (2011)

    Article  Google Scholar 

  28. 28.

    Müller, I.: On the size of the hysteresis in pseudoelasticity. Cont. Mech. Thermodyn. 1, 125–142 (1989)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Müller, C., Bruhns, O.T.: A thermodynamic finite-strain model for pseudoelastic shape memory alloys. Int. J. Plast. 22, 1658–1682 (2006)

    Article  Google Scholar 

  30. 30.

    Naghdi, P.M.: A critical review of the state of finite plasticity. Z. Angew. Math. Phys. 41, 315–387 (1990)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Olson, G.B., Cohen, M.: Kinetics of strain-induced martensitic nucleation. Metall. Trans. A 6A, 791–795 (1975)

    ADS  Article  Google Scholar 

  32. 32.

    Panoskaltsis, V.P.: Mechanics of shape memory alloys—constitutive modeling and numerical implications. In: Dr. Fernandes, F.M.B. (ed.) Shape Memory Alloys—Processing, Characterization and Applications. ISBN: 978-953-51-1084-2, In Tech doi:10.5772/52228 (2013)

  33. 33.

    Panoskaltsis, V.P., Bahuguna, S., Soldatos, D.: On the thermomechanical modeling of shape memory alloys. Int. J. Non-Linear Mech. 39, 709–722 (2004)

    Article  Google Scholar 

  34. 34.

    Panoskaltsis, V.P., Soldatos, D., Triantafyllou, S.P.: Generalized plasticity theory for phase transformations. In: Guagliano, M. (ed.) 11th International Conference on the Mechanical Behavior of Materials, Milano, Italy, 5–9 June 2011, pp. 3104–3108. Procedia Engineering (2011)

  35. 35.

    Panoskaltsis, V.P., Soldatos, D., Triantafyllou, S.P.: A new model for shape memory alloy materials under general states of deformation and temperature conditions. In: Boudouvis, A.G., Stavroulakis, G.E. (eds.) 7th GRACM International Congress on Computational Mechanics, Athens, Greece (2011)

  36. 36.

    Panoskaltsis, V.P., Polymenakos, L.C., Soldatos, D.: The concept of physical metric in the thermomechanical modeling of phase transformations with emphasis on shape memory alloy materials. ASME J. Eng. Mater. Technol. 135(2), 021016 (2013). doi:10.1115/1.4023780

    Article  Google Scholar 

  37. 37.

    Panoskaltsis, V.P., Soldatos, D.: A phenomenological constitutive model of non-conventional elastic response. Int. J. Appl. Mech. 5, 1350035 (2013)

    Article  Google Scholar 

  38. 38.

    Panoskaltsis, V.P., Soldatos, D., Triantafyllou, S.P.: On phase transformations in shape memory alloy materials and large deformation generalized plasticity. Cont. Mech. Thermodyn. 26, 811–831 (2014)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Panoskaltsis, V.P., Polymenakos, L.C., Soldatos, D.: Large deformation constitutive theory for a two-phase shape memory alloy. Eng. Trans. 62, 355–380 (2014)

    Google Scholar 

  40. 40.

    Peyroux, R., Chrysochoos, A., Light, C., Löbel, M.: Thermomechanical couplings and pseudoelasticity of shape memory alloys. Int. J. Eng. Sci. 36, 489–509 (1998)

    Article  Google Scholar 

  41. 41.

    Rahuadj, R., Ganghoffer, J.F., Cunat, C.: A thermodynamic approach with internal variables using Lagrange formalism. Part I: general framework. Mech. Res. Commun. 30, 109–117 (2003)

    Article  Google Scholar 

  42. 42.

    Rahuadj, R., Ganghoffer, J.F., Cunat, C.: A thermodynamic approach with internal variables using Lagrange formalism. Part 2: continuous symmetries in the case of the time–temperature equivalence. Mech. Res. Commun. 30, 119–123 (2003)

    Article  Google Scholar 

  43. 43.

    Raniecki, B., Lexcellent, C., Tanaka, K.: Thermodynamic models of pseudoelastic behaviour of shape memory alloys. Arch. Mech. 44(3), 261–284 (1992)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Raniecki, B., Lexcellent, C.: RL models of pseudoelasticity and their specification for some shape memory solids. Eur. J. Mech. A Solids 12(1), 21–50 (1994)

    MATH  Google Scholar 

  45. 45.

    Raniecki, B., Lexcellent, C.: Thermodynamics of isotropic pseudoelasticity in shape memory alloys. Eur. J. Mech. A Solids 17, 185–205 (1998)

    Article  Google Scholar 

  46. 46.

    Romero, I.: A characterization of conserved quantities in non-equilibrium thermodynamics. Entropy 15, 5580–5596 (2013)

    ADS  Article  Google Scholar 

  47. 47.

    Rosakis, P., Rosakis, A.J., Ravichandran, G., Hodowany, J.: A thermodynamic internal variable model for partition of plastic work into heat and stored energy in metals. J. Mech. Phys. Solids 48, 581–607 (2000)

    ADS  MathSciNet  Article  Google Scholar 

  48. 48.

    Simo, J.C., Miehe, C.: Associative couple thermoplasticity at finite strains: formulation, numerical analysis and implementation. Comput. Methods Appl. Mech. Eng. 98, 41–104 (1992)

    ADS  Article  Google Scholar 

  49. 49.

    Speicher, M., Hodgson, D.E., DesRoches, R., Leon, R.T.: Shape memory alloy tension/compression device for seismic retrofit of buildings. J. Mater. Eng. Perform. 18, 746–753 (2009)

    Article  Google Scholar 

  50. 50.

    Smallman, R.E., Bishop, R.J.: Modern Physical Metallurgy and Materials Engineering, 6th edn. Butterworth-Heinemann, Stoneham (2000)

    Google Scholar 

  51. 51.

    Stumpf, H., Hoppe, U.: The application of tensor analysis on manifolds to nonlinear continuum mechanics—invited survey article. Z. Agnew. Math. Mech. 77, 327–339 (1997)

    Article  Google Scholar 

  52. 52.

    Thamburaja, P.: A finite-deformation-based theory for shape-memory alloys. Int. J. Plast. 26, 1195–1219 (2010)

    Article  Google Scholar 

  53. 53.

    Yavari, A., Marsden, J.E., Ortiz, M.: On spatial and material covariant balance laws in elasticity. J. Math. Phys. 47, 1–53 (2006)

    MathSciNet  Article  Google Scholar 

  54. 54.

    Yin, Y.M., Weng, G.J.: Micromechanical study of thermomechanical characteristics of polycrystals shape-memory alloy films. Thin Solid Films 376, 198–207 (2000)

    ADS  Article  Google Scholar 

  55. 55.

    Yu, C., Kang, G., Kan, Q., Zhu, Y.: Rate dependent cyclic deformation of super-elastic NiTi shape memory alloy: thermo-mechanical coupled and physical mechanism-based constitutive model. Int. J. Plast. 72, 60–90 (2015)

    Article  Google Scholar 

  56. 56.

    Zaki, W., Morin, C., Moumni, Z.: A simple 1D model with thermomechanical coupling for superelastic SMAs. In: IOP Conference Series: Materials Science and Engineering. 10, 021149 (2010)

  57. 57.

    Ziołkowski, A.: Three-dimensional phenomenological thermodynamical model of pseudoelasticity of shape memory alloys at finite strains. Cont. Mech. Thermodyn. 19, 379–398 (2007)

    MathSciNet  Article  Google Scholar 

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Correspondence to V. P. Panoskaltsis.

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Communicated by Andreas Öchsner.

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Soldatos, D., Triantafyllou, S.P. & Panoskaltsis, V.P. Thermomechanical couplings in shape memory alloy materials. Continuum Mech. Thermodyn. 29, 805–834 (2017).

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  • Shape memory alloys
  • Shape memory effect
  • Pseudoelasticity
  • Generalized plasticity
  • Invariance
  • Balance energy equation
  • Thermomechanical state equations
  • Isothermal split
  • Thermomechanical couplings