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The structure of shock and interphase layers for a heat conducting Maxwellian rate-type approach to solid–solid phase transitions

Part I: thermodynamics and admissibility

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Abstract

We consider a thermoelastic model for phase transforming materials which can adequately describe the evolution with respect to the temperature of the hysteresis loop both in compression and tension tests. The specificity of this model is that the Grüneisen coefficient changes its sign. The model is augmented by considering a dissipative mechanism governed by a Maxwellian rate-type constitutive equation that can describe stress relaxation phenomena toward equilibrium between phases. Existence and uniqueness of traveling wave solutions are investigated. One derives that the admissibility condition induced by the Maxwellian rate-type approach, coupled or not with Fourier heat conduction law is related to the chord criterion with respect to the Hugoniot locus. We investigate the structure of profile layers, and we focus on their thermodynamic properties. The influence of the exothermic or endothermic character of phase transitions on the inner structure of interphase layers is captured. A phenomenon of temperature overshoot/undershoot with respect to the front state temperature and Hugoniot back state temperature inside an interphase layer is revealed.

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References

  1. Drumheller D.S.: Introduction to Wave Propagation in Nonlinear Fluids and Solids. Cambridge University Press, Cambridge (1998)

    Book  Google Scholar 

  2. Menikoff R., Plohr J.B.: The Riemann problem for fluid flow of real materials. Rev. Mod. Phys. 61, 75–130 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Liu T.-P.: The entropy condition and the admissibility of shocks. J. Math. Anal. Appl. 53, 78–88 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  4. Slemrod M.: Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch. Ration. Mech. Anal. 81, 301–315 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  5. Slemrod M.: Dynamic phase transitions in a van der Waals fluid. J. Diff. Eqs. 52, 1–23 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  6. Pego R.L.: Nonexistence of a shock layer in gas dynamics with a nonconvex equation of state. Arch. Ration. Mech. Anal. 94, 165–178 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ngan S-C., Truskinovsky L.: Thermal trapping and kinetics of martensitic phase boundaries. J. Mech. Phys. Solids 47, 141–172 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Weyl H.: Shock waves in arbitrary fluids. Comm. Pure Appl. Math. 2, 103–122 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gilbarg D.: The existence and limit behavior of the one-dimensional shock layer. Am. J. Math. 73(2), 256–274 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hamad H.: On the structure of an inviscid shock wave. Acta Mech. 138, 61–73 (1999)

    Article  MATH  Google Scholar 

  11. Barker L.M.: Fine structure of compression and release wave shapes in aluminum measured by velocity interferometer technique. In Berger, J. (ed.) Behavior of Dense Media Under High Dynamic Pressures, pp. 483–504. Gordon and Breach, New York (1968)

  12. Molinari A., Ravichandran G.: Fundamental structure of steady plastic shock waves in metals. J. Appl. Phys. 95, 1718–1732 (2004)

    Article  Google Scholar 

  13. Dunn J.E., Fosdick R.L.: Steady, structured shock waves. Part 1:Thermoelastic materials. Arch. Ration. Mech. Anal. 104, 295–365 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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 

  15. Shaw J.A., Kyriakides S.: On the nucleation and propagation of phase transformation fronts in a NiTi alloy. Acta Mater. 45, 683–700 (1997)

    Article  Google Scholar 

  16. Abeyaratne R., Knowles J.K.: Evolution of Phase Transitions: A Continuum Theory. Cambridge University Press, Cambridge (2006)

    Book  Google Scholar 

  17. Truskinovsky L.: Dynamics of non-equilibrium phase boundaries in a heat-conducting non-linearly elastic medium. J. Appl. Math. Mech. (PMM USSR) 51, 777–784 (1987)

    Article  MathSciNet  Google Scholar 

  18. Abeyaratne R., Knowles J.K.: Kinetic relations and the propagation of phase boundaries in solids. Arch. Ration. Mech. Anal. 114, 119–154 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  19. Vainchtein A.: Dynamics of non-isothermal martensitic phase transitions and hysteresis. Int. J. Solids Struct. 39, 3387–3408 (2002)

    Article  MATH  Google Scholar 

  20. Făciu C., Mihăilescu-Suliciu M.: On modelling phase propagation in SMAs by a Maxwellian thermo-viscoelastic approach. Int. J. Solids Struct. 39, 3811–3830 (2002)

    Article  MATH  Google Scholar 

  21. Ngan S.-C, Truskinovsky L.: Thermo-elastic aspects of dynamic nucleation. J. Mech. Phys. Solids 50, 1193–1229 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Turteltaub S.: Viscosity and strain gradient effects on the kinetics of propagating phase boundaries in solids. J. Elast. 46, 53–90 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. Vainchtein A.: Non-isothermal kinetics of a moving phase boundary. Continum Mech. Thermodyn. 15, 1–19 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Umantsev A.: Thermodynamic stability of phases and transition kinetics under adiabatic conditions. J. Chem. Phys. 96, 605–617 (1992)

    Article  Google Scholar 

  25. Bruno O.P., Leo P.H., Reitlich F.: Free boundary conditions at austenite-martensite boundary. Phys. Rev. Lett. 74, 746–749 (1995)

    Article  Google Scholar 

  26. Abeyaratne R., Knowles J.K.: Dynamics of propagating phase boundaries: adiabatic theory for thermoelastic solids. Physica D 79, 269–288 (1994)

    MathSciNet  MATH  Google Scholar 

  27. Abeyaratne R., Knowles J.K.: Impact-induced phase transitions in thermoelastic solids. Phil. Trans. R. Soc. Lond. A 355, 843–867 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. Knowles J.K.: On the structure of the Hugoniot relation for a shock-induced martensitic transformation. Shock Waves 17, 421–432 (2008)

    Article  MATH  Google Scholar 

  29. Shaw J.A.: Simulations of localized thermo-mechanical behavior in a NiTi shape memory alloy. Int. J. Plast. 16, 541–562 (2000)

    Article  MATH  Google Scholar 

  30. Abeyaratne R., Kim S-J., Knowles J.K.: A one-dimensional continuum model for shape memory alloys. Int. J. Solids Struct. 31, 2229–2249 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  31. Liu T.-P.: Shock waves in the nonisentopic gas flow. J. Diff. Eqs. 22, 442–452 (1976)

    Article  MATH  Google Scholar 

  32. Oleinik O.: Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation. Usp. Mat. Nauk 14, 165–170 (1959)

    MathSciNet  Google Scholar 

  33. Lax P.D.: Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math. 10, 537–566 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  34. Gall K., Sehitoglu H., Chumlyakov Y.I., Kireeva I.V.: Tension-compression asymmetry of the stress-strain response in aged single crystal and polycrystalline NiTi. Acta Mater. 47, 1203–1217 (1999)

    Article  Google Scholar 

  35. Făciu C., Molinari A.: The structure of shock and interphase layers for a heat conducting Maxwellian rate-type approach to solid-solid phase transitions. Part II: Numerical study for a SMA model. Acta Mech. doi:10.1007/s00707-013-0847-9

  36. Făciu C., Molinari A.: On the longitudinal impact of two phase transforming bars. Elastic versus a rate-type approach. Part I: the elastic case. Part II: the rate-type case. Int. J. Solids Struct. 43, 497–522 and 523–550 (2006)

    Google Scholar 

  37. Uchil J., Mohanchandra K., Ganesh Kumara K., Mahesh K.K., Murali T.P.: Thermal expansion in various phases of Nitinol using TMA. Physica B 270, 289–297 (1999)

    Article  Google Scholar 

  38. Coleman B.D., Noll W.: On the thermostatics of continuous media. Arch. Rat. Mech. Anal. 4, 97–128 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  39. Liu T.-P.: Hyperbolic conservation laws with relaxation. Comm. Math. Phys. 108, 153–175 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  40. Pego R.L.: Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability. Arch. Ration. Mech. Anal. 87, 353–394 (1987)

    Article  MathSciNet  Google Scholar 

  41. Landau, L.D., Lifschitz, E.M.: Mécanique des Fluides, Collection Physique Théorique, vol.6, Editions Mir 2eme édn. Moscou (1989)

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Authors and Affiliations

  1. “Simion Stoilow” Institute of Mathematics of the Romanian Academy Research Unit No. 6, P.O. Box 1-764, 014700, Bucharest, Romania

    Cristian Făciu

  2. Laboratoire d’Etude des Microstructures et de Mécanique des Matériaux, UMR-CNRS 7239, Labex Damas Université de Lorraine, Ile du Saulcy, 57045, Metz, France

    Alain Molinari

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  1. Cristian Făciu
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  2. Alain Molinari
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Correspondence to Cristian Făciu.

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Făciu, C., Molinari, A. The structure of shock and interphase layers for a heat conducting Maxwellian rate-type approach to solid–solid phase transitions. Acta Mech 224, 2577–2610 (2013). https://doi.org/10.1007/s00707-013-0846-x

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  • DOI: https://doi.org/10.1007/s00707-013-0846-x

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