Skip to main content
SpringerLink
Log in

On the modeling of equilibrium twin interfaces in a single-crystalline magnetic shape memory alloy sample. I: theoretical formulation

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

In this paper, a variational approach is proposed to study the response of a single-crystalline magnetic shape memory alloy (MSMA) sample subject to external forces and magnetic fields. Especially, some criteria are derived to model the (quasi-static) movements of twin interfaces in the sample. By considering the compatibility condition, twin interfaces between two martensite variants are found to be flat planes with given normal vectors. To adopt the variational method, a total energy functional for the whole magneto-mechanical system is proposed. By calculating the variations of the total energy functional with respect to the independent variables, the equilibrium equations and the evolution laws for the internal variables can be derived. By further considering the variation of the total energy functional with respect to the variant distribution, some criteria for twin interface movements can be derived. The governing system of the current model is then formulated by composing the equilibrium equations, the evolution laws for the internal variables and the twin interface movement criteria. To show the validity of the governing system, some analytical results are constructed under certain simplified conditions, which can be used to simulate the magneto-mechanical response of the MSMA sample.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Webster P.J., Ziebeck K.R.A., Town S.L., Peak M.S.: Magnetic order and phase transformation in Ni 2 MnGa. Philos. Mag. B 49, 295–310 (1984)

    Article  ADS  Google Scholar 

  2. Ullakko K., Huang J.K., Kantner C., O’Handley R.C., Kokorin V.V.: Large magnetic-field-induced strains in Ni 2 MnGa single crystals. Appl. Phys. Lett. 69, 1966–1968 (1996)

    Article  ADS  Google Scholar 

  3. Tickle R., James R.D.: Magnetic and magnetomechanical properties of Ni 2 MnGa. J. Magn. Magn. Mater. 195, 627–638 (1999)

    Article  ADS  Google Scholar 

  4. Heczko O., Straka L.: Temperature dependence and temperature limits of magnetic shape memory effect. J. Appl. Phys. 94, 7139–7143 (2003)

    Article  ADS  Google Scholar 

  5. Müllner P., Chernenko V.A., Kostorz G.: Stress-induced twin rearrangement resulting in change of magnetization in a Ni–Mn–Ga ferromagnetic martensite. Scripta Materialia 49, 129–133 (2003)

    Article  Google Scholar 

  6. Sozinov, A., Likhachev, A.A., Lanska, N., Söderberg, O., Ullakko, K., Lindroos, V.K.: Effect of crystal structure on magnetic-field-induced strain in Ni–Mn–Ga. In: Proceedings of SPIE, Symposium on Smart Structures and Materials, vol. 5053, pp. 586–594 (2003)

  7. Chernenko V.A., Lvov V.A., Müllner P., Kostorz G., Takagi T.: Magnetic-field-induced superelasticity of ferromagnetic thermoelastic martensites: experiment and modeling. Phys. Rev. B 69, 134410 (2004)

    Article  ADS  Google Scholar 

  8. Couch, R.N.: Development of Magnetic Shape Memory Alloy Actuators for a Swashplateless Helicopter Rotor. Doctoral Dissertation, University of Maryland (2006)

  9. Karaca H.E., Karaman I., Basaran B., Chumlyakov Y.I., Maier H.J.: Magnetic field and stress induced martensite reorientation in NiMnGa ferromagnetic shape memory alloy single crystals. Acta Materialia 54, 233–245 (2006)

    Article  Google Scholar 

  10. Straka, L.: Magnetic and Magneto-Mechanical Properties of Ni–Mn–Ga Magnetic Shape Memory Alloys. Doctoral Dissertation, Helsinki University of Technology (2007)

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Murrey S.J., Marioni M., Allen S.M., O’Handley R.C., Lograsso T.A.: 6 % Magnetic-field-induced strain by twin-boundary motion in ferromagnetic Ni–Mn–Ga. Appl. Phys. Lett. 77, 886–888 (2000)

    Article  ADS  Google Scholar 

  13. Straka L., Novák V., Landa M., Heczko O.: Acoustic emission of Ni–Mn–Ga magnetic shape memory alloy in different straining modes. Mater. Sci. Eng. A 374, 263–269 (2004)

    Article  Google Scholar 

  14. Lai Y.W., Scheerbaum N., Hinz D., Gutfleisch O., Schäfer R., Schultz L., McCord J.: Absence of magnetic domain wall motion during magnetic field induced twin boundary motion in bulk magnetic shape memory alloys. Appl. Phys. Lett. 90, 192504 (2007)

    Article  ADS  Google Scholar 

  15. Ge Y., Zárubová N., Dlabáček Z., Aaltio I., Söderberg O., Hannula S.P.: In-situ TEM straining of tetragonal martensite of Ni–Mn–Ga alloy. ESOMAT 2009, 04007 (2009)

    Google Scholar 

  16. Lai, Y.W.: Magnetic Microstructure and Actuation Dynamics of NiMnGa Magnetic Shape Memory Materials. Ph.D. thesis, Technical University of Dresden (2009)

  17. Straka L., Hänninen H., Lanska N., Sozinov A.: Twin interaction and large magnetoelasticity in Ni–Mn–Ga single crystals. J. Appl. Phys. 109, 063504 (2011)

    Article  ADS  Google Scholar 

  18. Ge Y., Heczko O., Söderberg O., Lindroos V.K.: Various magnetic domain structures in a Ni–Mn–Ga martensite exhibiting magnetic shape memory effect. J. Appl. Phys. 96, 2159–2163 (2004)

    Article  ADS  Google Scholar 

  19. Ge Y., Heczko O., Söderberg O., Hannula S.P.: Direct optical observation of magnetic domains in Ni–Mn–Ga martensite. Appl. Phys. Lett. 89, 082502 (2006)

    Article  ADS  Google Scholar 

  20. Scheerbaum N., Lai Y.W., Leisegang T., Thomas M., Liu J., Khlopkov K., McCord J., Fähler S., Träger R., Meyer D.C., Schultz L., Gutfleisch O.: Constraint-dependent twin variant distribution in Ni 2 MnGa single crystal, polycrystals and thin film: an EBSD study. Acta Mater. 58, 4629–4638 (2010)

    Article  Google Scholar 

  21. Hirsinger, L., Lexcellent, C.: Modelling detwinning of martensite platelets under magnetic and (or) stress actions in Ni–Mn–Ga alloys. J. Magn. Magn. Mater. 254–255, 275–277 (2003)

    Google Scholar 

  22. Kiefer, B., Lagoudas, D.C.: Magnetic field-induced martensitic variant reorientation in magnetic shape memory alloys. Philosophical Magazine Special Issue: Recent Advances in Theoretical Mechanics, in Honor of SES 2003 A.C. Eringen Medalist G.A. Maugin, 85(33–35), 4289–4329 (2005)

  23. Kiefer B., Karaca H.E., Lagoudas D.C., Karaman I.: Characterization and modeling of the magnetic field-induced strain and work output in Ni 2 MnGa shape memory alloys. J. Magn. Magn. Mater. 312, 164–175 (2007)

    Article  ADS  Google Scholar 

  24. Sarawate N.N., Dapino M.J.: Magnetomechanical characterization and unified energy model for the quasistatic behavior of ferromagnetic shape memory Ni–Mn–Ga. Smart Mater. Struct. 19, 035001 (2010)

    Article  ADS  Google Scholar 

  25. Haldar K., Kiefer B., Lagoudas D.C.: Finite element analysis of the demagnetization effect and stress inhomogeneities in magnetic shape memory alloy samples. Philos. Mag. 91, 4126–4157 (2011)

    Article  ADS  Google Scholar 

  26. Müllner P., Ullakko K.: The force of a magnetic/electric field on a twinning dislocation. Physica Status Solidi 208, R1–R2 (1998)

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  28. Likhachev A.A., Ullakko K.: Magnetic-field-controlled twin boundaries motion and giant magneto-mechanical effects in Ni–Mn–Ga shape memory alloy. Phys. Lett. A 275, 142–151 (2000)

    Article  ADS  Google Scholar 

  29. James R.D.: Configurational forces in magnetism with application to the dynamics of a small-scale ferromagnetic shape memory cantilever. Continuum Mech. Thermodyn. 14, 55–86 (2002)

    Article  ADS  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  31. 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  ADS  MathSciNet  Google Scholar 

  32. Wang J., Steinmann P., Dai H.H.: Analytical study on the stress-induced phase or variant transformation in slender shape memory alloy samples. Meccanica 48, 943–970 (2012)

    Article  MathSciNet  Google Scholar 

  33. Wang, J., Steinmann, P.: Finite element simulation of the magneto-mechanical response of a magnetic shape memory alloy sample. Philos. Mag., doi:10.1080/14786435.2013.782443 (2013)

  34. Shield T.W.: Magnetomechanical testing machine for ferromagnetic shape-memory alloys. Rev. Sci. Instrum. 74, 4077–4088 (2003)

    Article  ADS  Google Scholar 

  35. DeSimone A.: Coarse-grained models of materials with non-convex free-energy: two case studies. Comput. Methods Appl. Mech. Eng. 193, 5129–5141 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  36. Wu P.P., Ma X.Q., Zhang J.X., Chen L.Q.: Phase-field simulations of stress-strain behaviour in ferromagnetic shape memory alloy NiMnGa. J. Appl. Phys. 104, 073906 (2008)

    Article  ADS  Google Scholar 

  37. Stefanelli U.: Magnetic control of magnetic shape-memory single crystals. Physica B 407, 1316–1321 (2012)

    Article  ADS  Google Scholar 

  38. Abeyaratne R.: An admissibility condition for equilibrium shocks in finite elasticity. J. Elast. 13, 175–184 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  39. Rajagopal K.R., Srinivasa A.R.: On the thermomechanics of shape memory wires. Z. Angew. Math. Phys. 50, 459–496 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  40. Kröner E.: Allgemeine Kontinuumstheorie der Versetzyngen und Eigenspannungen. Arch. Ration. Mech. Anal. 4, 273–334 (1960)

    Article  MATH  Google Scholar 

  41. Lee E.H.: Elastic-plastic deformations in finite strains. J. Appl. Mech. ASME 36, 1–60 (1969)

    Article  ADS  MATH  Google Scholar 

  42. Steinmann P.: Views on multiplicative elastoplasticity and the continuum theory of dislocations. Int. J. Eng. Sci. 34, 1717–1735 (1996)

    Article  MATH  Google Scholar 

  43. Mielke A., Theil F., Levitas V.I.: A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Ration. Mech. Anal. 162, 137–177 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  44. Eshelby J.D.: The elastic energy-momentum tensor. J. Elast. 5, 321–335 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  45. Fomethe A., Maugin G.A.: Material forces in thermoelastic ferromagnets. Continuum Mech. Thermodyn. 8, 275–292 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  46. Straka L., Hänninen H., Soroka A., Sozinov A.: Ni–Mn–Ga single crystals with very low twinning stress. J. Phys. Conf. Ser. 303, 012079 (2011)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Chair of Applied Mechanics, University of Erlangen-Nuremberg, Egerlandstr. 5, 91058, Erlangen, Germany

    Jiong Wang & Paul Steinmann

Authors
  1. Jiong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paul Steinmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiong Wang.

Additional information

Communicated by Andreas Öchsner.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, J., Steinmann, P. On the modeling of equilibrium twin interfaces in a single-crystalline magnetic shape memory alloy sample. I: theoretical formulation. Continuum Mech. Thermodyn. 26, 563–592 (2014). https://doi.org/10.1007/s00161-013-0319-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-013-0319-4

Keywords

Search

Navigation