Free energies and pseudo-elastic transitions for shape memory alloys

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A new "flexible" 3D macroscopic model for shape memory alloys

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

Free energies and pseudo-elastic transitions for shape memory alloys

Alessia Berti ^1, , Claudio Giorgi ^2, and Elena Vuk ^2,

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

Abstract
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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.

Keywords: hysteresis loops, Shape memory alloys, stress-rate materials, austenite-martensite transition, Ginzburg-Landau theory., pseudo-elasticity, free energy potentials.

Mathematics Subject Classification: Primary: 74N30; Secondary: 74C05, 74F05, 80A2.

Citation: Alessia Berti, Claudio Giorgi, Elena Vuk. Free energies and pseudo-elastic transitions for shape memory alloys. Discrete and 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 "Shape Memory Alloys: Advances in Modelling and Applications" (eds. F. Auricchio, L. Faravelli, G. Magonette and V. Torra), CMINE: Barcelona, (2002), 125-187.
[2]	A. Berti, C. Giorgi and E. Vuk, Free energies in one-dimensional models of magnetic transitions with hysteresis, Il Nuovo Cimento, B, 125 (2010), 371-394.
[3]	V. Berti, M. Fabrizio and D. Grandi, Phase transitions in shape memory alloys: A non-isothermal Ginzburg-Landau model, Physica D, 239 (2010), 95-102. doi: 10.1016/j.physd.2009.10.005.
[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), 062901. doi: 10.1063/1.3430573.
[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), 229-242. doi: 10.1177/1045389X9300400213.
[6]	M. Fabrizio and A. Morro, "Electromagnetism of Continuous Media," Oxford University Press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198527008.003.0010.
[7]	M. Fremond, Materiaux a memoire de forme, C. R. Acad. Sci. Paris Ser. II, 304 (1987), 239-244.
[8]	M. Fremond, "Non-smooth Thermomechanics," Springer-Verlag, Berlin, 2002. doi: 10.1115/1.1497489.
[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), 134-206.
[10]	S. Miyazaki, Development and characterization of shape memory alloys, in "CISM Courses and Lectures: Shape Memory Alloys," 351. Springer: Wien, NewYork, (1996), 69-143.
[11]	I. Müller, Thermodynamics of ideal pseudoelasticity, Journal de Physique IV, C2-5 (1995), 423-431.
[12]	I. Müller and S. Seelecke, Thermodynamic aspects of shape memory alloys, Math. Comp. Modelling, 34 (2001), 1307-1355. doi: 10.1016/S0895-7177(01)00134-0.
[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), 232-244. doi: 10.1016/S0921-5093(98)01093-4.
[14]	C. M. Wayman, Shape memory and related phenomena, Progress in Materials Science, 36 (1992), 203-224. doi: 10.1016/0079-6425(92)90009-V.
[15]	J. C. Willems, Dissipative dynamical systems - Part I: General theory, Arch. Rational Mech. Anal., 45 (1972), 321-351. doi: 10.1007/BF00276493.

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References:

[1]	F. Auricchio, Considerations on the constitutive modeling of shape-memory alloys, in "Shape Memory Alloys: Advances in Modelling and Applications" (eds. F. Auricchio, L. Faravelli, G. Magonette and V. Torra), CMINE: Barcelona, (2002), 125-187.
[2]	A. Berti, C. Giorgi and E. Vuk, Free energies in one-dimensional models of magnetic transitions with hysteresis, Il Nuovo Cimento, B, 125 (2010), 371-394.
[3]	V. Berti, M. Fabrizio and D. Grandi, Phase transitions in shape memory alloys: A non-isothermal Ginzburg-Landau model, Physica D, 239 (2010), 95-102. doi: 10.1016/j.physd.2009.10.005.
[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), 062901. doi: 10.1063/1.3430573.
[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), 229-242. doi: 10.1177/1045389X9300400213.
[6]	M. Fabrizio and A. Morro, "Electromagnetism of Continuous Media," Oxford University Press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198527008.003.0010.
[7]	M. Fremond, Materiaux a memoire de forme, C. R. Acad. Sci. Paris Ser. II, 304 (1987), 239-244.
[8]	M. Fremond, "Non-smooth Thermomechanics," Springer-Verlag, Berlin, 2002. doi: 10.1115/1.1497489.
[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), 134-206.
[10]	S. Miyazaki, Development and characterization of shape memory alloys, in "CISM Courses and Lectures: Shape Memory Alloys," 351. Springer: Wien, NewYork, (1996), 69-143.
[11]	I. Müller, Thermodynamics of ideal pseudoelasticity, Journal de Physique IV, C2-5 (1995), 423-431.
[12]	I. Müller and S. Seelecke, Thermodynamic aspects of shape memory alloys, Math. Comp. Modelling, 34 (2001), 1307-1355. doi: 10.1016/S0895-7177(01)00134-0.
[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), 232-244. doi: 10.1016/S0921-5093(98)01093-4.
[14]	C. M. Wayman, Shape memory and related phenomena, Progress in Materials Science, 36 (1992), 203-224. doi: 10.1016/0079-6425(92)90009-V.
[15]	J. C. Willems, Dissipative dynamical systems - Part I: General theory, Arch. Rational Mech. Anal., 45 (1972), 321-351. doi: 10.1007/BF00276493.

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