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

show all references

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

[1]

Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. A rate-independent model for permanent inelastic effects in shape memory materials. Networks & Heterogeneous Media, 2011, 6 (1) : 145-165. doi: 10.3934/nhm.2011.6.145

[2]

Leonid Berlyand, Volodymyr Rybalko, Nung Kwan Yip. Renormalized Ginzburg-Landau energy and location of near boundary vortices. Networks & Heterogeneous Media, 2012, 7 (1) : 179-196. doi: 10.3934/nhm.2012.7.179

[3]

Michel Frémond, Elisabetta Rocca. A model for shape memory alloys with the possibility of voids. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1633-1659. doi: 10.3934/dcds.2010.27.1633

[4]

Michela Eleuteri, Luca Lussardi. Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials. Evolution Equations & Control Theory, 2014, 3 (3) : 411-427. doi: 10.3934/eect.2014.3.411

[5]

Hassen Aydi, Ayman Kachmar. Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint. II. Communications on Pure & Applied Analysis, 2009, 8 (3) : 977-998. doi: 10.3934/cpaa.2009.8.977

[6]

John Murrough Golden. Constructing free energies for materials with memory. Evolution Equations & Control Theory, 2014, 3 (3) : 447-483. doi: 10.3934/eect.2014.3.447

[7]

Hans G. Kaper, Bixiang Wang, Shouhong Wang. Determining nodes for the Ginzburg-Landau equations of superconductivity. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 205-224. doi: 10.3934/dcds.1998.4.205

[8]

Mickaël Dos Santos, Oleksandr Misiats. Ginzburg-Landau model with small pinning domains. Networks & Heterogeneous Media, 2011, 6 (4) : 715-753. doi: 10.3934/nhm.2011.6.715

[9]

Fanghua Lin, Ping Zhang. On the hydrodynamic limit of Ginzburg-Landau vortices. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 121-142. doi: 10.3934/dcds.2000.6.121

[10]

Claudio Giorgi. Phase-field models for transition phenomena in materials with hysteresis. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 693-722. doi: 10.3934/dcdss.2015.8.693

[11]

Diego Grandi, Ulisse Stefanelli. The Souza-Auricchio model for shape-memory alloys. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 723-747. doi: 10.3934/dcdss.2015.8.723

[12]

Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. Thermal control of the Souza-Auricchio model for shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 369-386. doi: 10.3934/dcdss.2013.6.369

[13]

Linxiang Wang, Roderick Melnik. Dynamics of shape memory alloys patches with mechanically induced transformations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1237-1252. doi: 10.3934/dcds.2006.15.1237

[14]

Shuji Yoshikawa, Irena Pawłow, Wojciech M. Zajączkowski. A quasilinear thermoviscoelastic system for shape memory alloys with temperature dependent specific heat. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1093-1115. doi: 10.3934/cpaa.2009.8.1093

[15]

Tomáš Roubíček. Modelling of thermodynamics of martensitic transformation in shape-memory alloys. Conference Publications, 2007, 2007 (Special) : 892-902. doi: 10.3934/proc.2007.2007.892

[16]

Ferdinando Auricchio, Elena Bonetti. A new "flexible" 3D macroscopic model for shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 277-291. doi: 10.3934/dcdss.2013.6.277

[17]

Sandra Carillo. Materials with memory: Free energies & solution exponential decay. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1235-1248. doi: 10.3934/cpaa.2010.9.1235

[18]

Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345

[19]

Leonid Berlyand, Petru Mironescu. Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain. Networks & Heterogeneous Media, 2008, 3 (3) : 461-487. doi: 10.3934/nhm.2008.3.461

[20]

Leonid Berlyand, Volodymyr Rybalko. Homogenized description of multiple Ginzburg-Landau vortices pinned by small holes. Networks & Heterogeneous Media, 2013, 8 (1) : 115-130. doi: 10.3934/nhm.2013.8.115

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]