# American Institute of Mathematical Sciences

April  2013, 6(2): 293-316. doi: 10.3934/dcdss.2013.6.293

## Free energies and pseudo-elastic transitions for shape memory alloys

 1 Facoltà di Ingegneria, Università e-Campus, 22060 Novedrate (CO) 2 Dipartimento di Matematica, Università di Brescia, 25133 Brescia, Italy, Italy

Received  May 2011 Revised  October 2011 Published  November 2012

A one-dimensional model for a shape memory alloy is proposed. It provides a simplified description of the pseudo-elastic regime, where stress-induced transitions from austenitic to oriented martensitic phases occurs. The stress-strain evolution is ruled by a bilinear rate-independent o.d.e. which also accounts for the fine structure of minor hysteresis loops and applies to the case of single crystals only. The temperature enters the model as a parameter through the yield limit $y$.Above the critical temperature $\theta_A^*$, the austenite-martensite phase transformations are described by a Ginzburg-Landau theory involving an order parameter $φ$, which is related to the anelastic deformation. As usual, the basic ingredient is the Gibbs free energy, $\zeta$, which is a function of the order parameter, the stress and the temperature. Unlike other approaches, the expression of this thermodynamic potential is derived rather then assumed, here. The explicit expressions of the minimum and maximum free energies are obtained by exploiting the Clausius-Duhem inequality, which ensures the compatibility with thermodynamics, and the complete controllability of the system. This allows us to highlight the role of the Ginzburg-Landau equation when phase transitions in materials with hysteresis are involved.
Citation: Alessia Berti, Claudio Giorgi, Elena Vuk. Free energies and pseudo-elastic transitions for shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 293-316. doi: 10.3934/dcdss.2013.6.293
##### References:
 [1] F. Auricchio, Considerations on the constitutive modeling of shape-memory alloys,, in, (2002), 125.   Google Scholar [2] A. Berti, C. Giorgi and E. Vuk, Free energies in one-dimensional models of magnetic transitions with hysteresis,, Il Nuovo Cimento, 125 (2010), 371.   Google Scholar [3] V. Berti, M. Fabrizio and D. Grandi, Phase transitions in shape memory alloys: A non-isothermal Ginzburg-Landau model,, Physica D, 239 (2010), 95.  doi: 10.1016/j.physd.2009.10.005.  Google Scholar [4] V. Berti, M. Fabrizio and D. Grandi, Hysteresis and phase transitions for one-dimensional and three-dimensional models in shape memory alloys,, J. Math. Phys., 51 (2010).  doi: 10.1063/1.3430573.  Google Scholar [5] L. C. Brinson, One-dimensional constitutive behavior of shape memory alloys: Thermomechanical derivation with non-constant material functions and redefined martensite internal variables,, Journal of Intelligent Material Systems and Structures, 4 (1993), 229.  doi: 10.1177/1045389X9300400213.  Google Scholar [6] M. Fabrizio and A. Morro, "Electromagnetism of Continuous Media,", Oxford University Press, (2003).  doi: 10.1093/acprof:oso/9780198527008.003.0010.  Google Scholar [7] M. Fremond, Materiaux a memoire de forme,, C. R. Acad. Sci. Paris Ser. II, 304 (1987), 239.   Google Scholar [8] M. Fremond, "Non-smooth Thermomechanics,", Springer-Verlag, (2002).  doi: 10.1115/1.1497489.  Google Scholar [9] V. I. Levitas and D. L. Preston, Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations. I. Austenite$\leftrightarrow$martensite,, Physical Review B, 66 (2002), 134.   Google Scholar [10] S. Miyazaki, Development and characterization of shape memory alloys,, in, 351 (1996), 69.   Google Scholar [11] I. Müller, Thermodynamics of ideal pseudoelasticity,, Journal de Physique IV, C2-5 (1995), 2.   Google Scholar [12] I. Müller and S. Seelecke, Thermodynamic aspects of shape memory alloys,, Math. Comp. Modelling, 34 (2001), 1307.  doi: 10.1016/S0895-7177(01)00134-0.  Google Scholar [13] F. Nishimura, N. Watanabe, T. Watanabe and K. Tanaka, Transformation conditions in an Fe-based shape memory alloy under tensile-torsional loads: Martensite start surface and austenite start/finish planes,, Material Science and Engineering, A264 (1999), 232.  doi: 10.1016/S0921-5093(98)01093-4.  Google Scholar [14] C. M. Wayman, Shape memory and related phenomena,, Progress in Materials Science, 36 (1992), 203.  doi: 10.1016/0079-6425(92)90009-V.  Google Scholar [15] J. C. Willems, Dissipative dynamical systems - Part I: General theory,, Arch. Rational Mech. Anal., 45 (1972), 321.  doi: 10.1007/BF00276493.  Google Scholar

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##### References:
 [1] F. Auricchio, Considerations on the constitutive modeling of shape-memory alloys,, in, (2002), 125.   Google Scholar [2] A. Berti, C. Giorgi and E. Vuk, Free energies in one-dimensional models of magnetic transitions with hysteresis,, Il Nuovo Cimento, 125 (2010), 371.   Google Scholar [3] V. Berti, M. Fabrizio and D. Grandi, Phase transitions in shape memory alloys: A non-isothermal Ginzburg-Landau model,, Physica D, 239 (2010), 95.  doi: 10.1016/j.physd.2009.10.005.  Google Scholar [4] V. Berti, M. Fabrizio and D. Grandi, Hysteresis and phase transitions for one-dimensional and three-dimensional models in shape memory alloys,, J. Math. Phys., 51 (2010).  doi: 10.1063/1.3430573.  Google Scholar [5] L. C. Brinson, One-dimensional constitutive behavior of shape memory alloys: Thermomechanical derivation with non-constant material functions and redefined martensite internal variables,, Journal of Intelligent Material Systems and Structures, 4 (1993), 229.  doi: 10.1177/1045389X9300400213.  Google Scholar [6] M. Fabrizio and A. Morro, "Electromagnetism of Continuous Media,", Oxford University Press, (2003).  doi: 10.1093/acprof:oso/9780198527008.003.0010.  Google Scholar [7] M. Fremond, Materiaux a memoire de forme,, C. R. Acad. Sci. Paris Ser. II, 304 (1987), 239.   Google Scholar [8] M. Fremond, "Non-smooth Thermomechanics,", Springer-Verlag, (2002).  doi: 10.1115/1.1497489.  Google Scholar [9] V. I. Levitas and D. L. Preston, Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations. I. Austenite$\leftrightarrow$martensite,, Physical Review B, 66 (2002), 134.   Google Scholar [10] S. Miyazaki, Development and characterization of shape memory alloys,, in, 351 (1996), 69.   Google Scholar [11] I. Müller, Thermodynamics of ideal pseudoelasticity,, Journal de Physique IV, C2-5 (1995), 2.   Google Scholar [12] I. Müller and S. Seelecke, Thermodynamic aspects of shape memory alloys,, Math. Comp. Modelling, 34 (2001), 1307.  doi: 10.1016/S0895-7177(01)00134-0.  Google Scholar [13] F. Nishimura, N. Watanabe, T. Watanabe and K. Tanaka, Transformation conditions in an Fe-based shape memory alloy under tensile-torsional loads: Martensite start surface and austenite start/finish planes,, Material Science and Engineering, A264 (1999), 232.  doi: 10.1016/S0921-5093(98)01093-4.  Google Scholar [14] C. M. Wayman, Shape memory and related phenomena,, Progress in Materials Science, 36 (1992), 203.  doi: 10.1016/0079-6425(92)90009-V.  Google Scholar [15] J. C. Willems, Dissipative dynamical systems - Part I: General theory,, Arch. Rational Mech. Anal., 45 (1972), 321.  doi: 10.1007/BF00276493.  Google Scholar
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