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Thermodynamics of shape-memory alloys under electric current

  • Tomáš RoubíčekEmail author
  • Giuseppe Tomassetti
Article

Abstract

Phase transformation in shape-memory alloys is known to cause electric resistivity variation that, under electric current, may conversely influence Joule heat production and thus eventually the martensitic transformation itself. A thermodynamically consistent general continuum-mechanical model at large strains is presented. In special cases, a proof of the existence of a weak solution is outlined, using a semidiscretization in time.

Mathematics Subject Classification (2000)

35K55 74A15 74N10 80A17 

Keywords

Generalized standard materials Heat equation Joule heat Electrically conductive media Martensitic transformation Weak solution 

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References

  1. 1.
    Aiki T.: Weak solutions for Falk’s model of shape memory alloys. Math. Methods Appl. Sci. 23, 299–319 (2000)zbMATHCrossRefMathSciNetGoogle 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)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Arndt M., Griebel M., Novák V., Roubíček T., Šittner P.: Martensitic/austenitic transformation in NiMnGa: simulation and experimental approaches. Int. J. Plast. 22, 1943–1961 (2006)zbMATHCrossRefGoogle Scholar
  4. 4.
    Arndt M., Griebel M., Roubíček T.: Modelling and numerical simulation of martensitic transformation in shape memory alloys. Cont. Mech. Thermodyn. 15, 463–485 (2003)zbMATHCrossRefGoogle Scholar
  5. 5.
    Ball J.M., James R.D.: Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. Royal Soc. Lond. A 338, 389–450 (1992)zbMATHCrossRefGoogle Scholar
  6. 6.
    Bhattacharya K.: Microstructure of Martensite. Why it Forms and how it Gives Rise to the Shape-Memory Effect. Oxford University Press, New York (2003)Google Scholar
  7. 7.
    Boccardo L., Dall’aglio A., Gallouët T., Orsina L.: Nonlinear parabolic equations with measure data. J. Funct. Anal. 147, 237–258 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Boccardo L., Gallouët T.: Non-linear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87, 149–169 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Brokate M., Sprekels J.: Hystreresis and Phase Transitions. Springer, New York (1996)Google Scholar
  10. 10.
    Chen Z., Hoffmann K.-H.: On a one-dimensional nonlinear thermovisoelastic model for structural phase transitions in shape memory alloys. J. Differ. Equ. 12, 325–350 (1994)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Coleman B.D., Noll W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mech. Anal. 13, 167–178 (1963)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    DalMaso G., Francfort G.A., Toader R.: Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal. 176, 165–225 (2005)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Falk F.: Model free energy, mechanics and thermodynamics of shape memory alloys. Acta Metall. 28, 1773–1780 (1980)CrossRefGoogle Scholar
  14. 14.
    Faulkner M.G., Amalraj J.J., Bhattacharyya A.: Experimental determination of thermal and electrical properties of NiTi shape memory wires. Smart Mater. Struct. 9, 632–639 (2000)CrossRefGoogle Scholar
  15. 15.
    Francfort G., Mielke A.: An existence result for a rate-independent material model in the case of nonconvex energies. J. Reine u. Angew. Math. 595, 55–91 (2006)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Frémond M.: Non-Smooth Thermomechanics. Springer, Berlin (2002)zbMATHGoogle Scholar
  17. 17.
    Fried E., Gurtin M.E.: Dynamic solid–solid transitions with phase characterized by an order parameter. Phys. D. Nonlinear Phenom. 72, 287–308 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Garcke H.: Travelling wave solutions as dynamic phase transitions in shape memory alloys. J. Differ. Equ. 121, 203–231 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Gori F., Carnevale D., Doro Altan A., Nicosia S., Pennestrì E.: A new hysteretic behavior in the electrical resistivity of flexinol shape memory alloys versus temperature. Int. J. Thermophys. 27, 866–879 (2006)CrossRefGoogle Scholar
  20. 20.
    Halphen B., Nguyen Q.S.: Sur les matériaux standards généralisés. J. Mécanique 14, 39–63 (1975)zbMATHGoogle Scholar
  21. 21.
    He Z., Gall K.R., Brinson L.C.: Use of electric reistance testing to redefine the transformation kinetics and phase diagram for shape-mmory alloys. Metall. Mater. Trans. A 37A, 579–581 (2006)CrossRefGoogle Scholar
  22. 22.
    Hoffmann K.-H., Zochowski A.: Existence of solutions to some non-linear thermoelastic systems with viscosity. Math. Methods Appl. Sci. 15, 187–204 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Hormann K., Zimmer J.: On Landau theory and symmetric landscape for phase transitions. J. Mech. Phys. Solids 55, 1385–1409 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    James R.D., Hane K.F.: Martensitic transformations and shape-memory materials. Acta Mater. 48, 197–222 (2000)CrossRefGoogle Scholar
  25. 25.
    Kružík M.: Numerical approach to double-well problem. SIAM J. Numer. Anal. 35, 1833–1849 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Kružík M., Mielke A., Roubíček T.: Modelling of microstructure and its evolution in shape-memory-alloy single- crystals, in particular in CuAlNi. Meccanica 40, 389–418 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Mielke A.: Evolution of rate-independent systems. In: Dafermos, C., Feireisl, E. (eds) Handbook of Differential equations: Evolutionary Differntial Equations, pp. 461–559. Elsevier, Amsterdam (2005)Google Scholar
  28. 28.
    Mielke A. et al.: A mathematical framework for generalized standard materials in rate-independent case. In: Helmig, R. (eds) Multifield Problems in Fluid and Solid Mechanics, pp. 491–529. Springer, Berlin (2006)Google Scholar
  29. 29.
    Mielke A., Theil F.: On rate-independent hysteresis models. Nonlinear Differ. Equ. Appl. 11, 151–189 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Mielke A., Timofte A.: Two-scale homogenization for evolutionar variational inequalities via energetic formulation. SIAM J. Math. Anal. 39, 642–668 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Niezgódka M., Sprekels J.: Existence of solutions for a mathematical model of structural phase transitions in shape memory alloys. Math. Methods Appl. Sci. 10, 197–223 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Novák V., Šittner P., Dayananda G.N., Braz-Fernandes F.M., Mahesh K.K.: Electric resistance variation of NiTi shape memory alloy wires in thermomechanical tests: experiments and simulation. Mater. Sci. Eng. A 481–482, 127–133 (2008)Google Scholar
  34. 34.
    Onsager L.: Reciprocal relations in irreversible processes. Phys. Rev. II 37, 405–426 (1931) 38, 2265–2279 (1931Google Scholar
  35. 35.
    Öttinger H.C.: Beyond Equilibrium Thermodynamics. Wiley, Hoboken, NJ (2002)Google Scholar
  36. 36.
    Pawłow I.: Three-dimensional model of thermomechanical evolution of shape memory material. Control Cybern. 29, 341–365 (2000)zbMATHGoogle Scholar
  37. 37.
    Pawłow I., Zaja̧czkowski W.M.: Global existence to a three-dimensional non-linear thermoelasticity system arising in shape memory materials. Math. Methods Appl. Sci. 28, 407–442 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Pawłow I., Zochowski A.: Existence and uniqueness of solutions for a three-dimensional thermoelastic system. Diss. Math. (Rozprawy Mat.) 406, 46 (2002)Google Scholar
  39. 39.
    Plecháč P., Roubíček T.: Visco-elasto-plastic model for martensitic phase transformation in shape-memory alloys . Math. Methods Appl. Sci. 25, 1281–1298 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Pitteri M., Zanzotto G.: Continuum Models for Phase Transitions and Twinning in Crystals. Chapman&Hall, Boca Raton (2003)zbMATHGoogle Scholar
  41. 41.
    Podio-Guidugli P.: Contact interactions, stress, and material symmetry, for nonsimple elastic materials Theor. Appl. Mech. 28–29, 261–276 (2002)MathSciNetGoogle Scholar
  42. 42.
    Podio-Guidugli P.: Models of phase segregation and diffusion of atomic species on a lattice. Ric. Mater. 55, 105–118 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Rajagopal K.R., Roubíček T.: On the effect of dissipation in shape-memory alloys. Nonlinear Anal. Real World Appl. 4, 581–597 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
  45. 45.
    Roubíček T.: Dissipative evolution of microstructure in shape memory alloys. In: Bungartz, H.-J., Hoppe, R.H.W., Zenger, C. (eds) Lectures on Applied Mathematics, pp. 45–63. Springer, Berlin (2000)Google Scholar
  46. 46.
    Roubíček T.: Models of microstructure evolution in shape memory materials. In: Ponte Castañeda, P., Telega, J.J., Gambin, B. (eds) Nonlinear Homogenization and Applications to Composites, Polycrystals and Smart Materials, NATO Science Series II/170, pp. 269–304. Kluwer, Dordrecht (2004)Google Scholar
  47. 47.
    Roubíček T.: Nonlinear Partial Differential Equations with Applications. Birkhäuser, Basel (2005)zbMATHGoogle Scholar
  48. 48.
    Roubíček, T.: Modelling of thermodynamics of martensitic transformation in shape-memory alloys. Disc. Cont. Dynam. Syst. 892–902 (2007)Google Scholar
  49. 49.
    Roubíček T.: Rate independent processes in viscous solids at small strains. Math. Methods Appl. Sci. 32, 825–862 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Roubíček, T.: Thermodynamics of rate independent processes in viscous solids at small strains. SIAM J. Math. Anal. (to appear)Google Scholar
  51. 51.
    Sprekels J.: Global existence for thermomechanical processes with nonconvex free energies of Ginzburg-Landau form. J. Math. Anal. Appl. 141, 333–348 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Sprekels J., Zheng S.: Global solutions to the equations of a Ginzburg-Landau theory for structural phase transitions in shape memory alloys. Phys. D 39, 39–54 (1989)CrossRefMathSciNetGoogle Scholar
  53. 53.
    Thamburaja P., Nikabdullah N.: A macroscopic constitutive model for shape-memory alloys: theory and finite-element simulations. Comput. Methods Appl. Mech. Eng. 198, 1074–1086 (2009)CrossRefGoogle Scholar
  54. 54.
    Trimarco C.: The structure of material forces in electromagnetic materials. Rend. Sem. Mat. Univ. Politec. Torino 58, 237–244 (2000)zbMATHMathSciNetGoogle Scholar
  55. 55.
    Uchil J., Mahesh K.K., Ganesh Kumara K.: Electrical resistivity and strain recovery studies on the effect of thermal cycling under constant strass on R-phase in NiTi shape memory alloy. Phys. B 324, 419–428 (2002)CrossRefGoogle Scholar
  56. 56.
    Wu X.D., Fan Y.Z., Wu J.S.: A study on the variations of the eletrical resistance for NiTi shape memory alloy wires during the thermomechanical loading. Mater. Des. 21, 511–515 (2000)Google Scholar
  57. 57.
    Zimmer J.: Global existence of a nonlinear system in thermoviscoelasticity with nonconvex energy. J. Math. Anal. Appl. 292, 589–604 (2004)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Mathematical InstituteCharles UniversityPrague 8Czech Republic
  2. 2.Institute of Thermomechanics of the ASCRPrague 8Czech Republic
  3. 3.Dipartimento di Ingegneria CivileUniversità di Roma “Tor Vergata”RomeItaly

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