August  2015, 8(4): 693-722. doi: 10.3934/dcdss.2015.8.693

Phase-field models for transition phenomena in materials with hysteresis

1. 

DICATAM, Università di Brescia, Via Valotti 9, 25133 Brescia

Received  March 2014 Revised  July 2014 Published  October 2014

Non-isothermal phase-field models of transition phenomena in materials with hysteresis are considered within the framework of the Ginzburg-Landau theory. Our attempt is to capture the relation between phase-transition and hysteresis (either mechanical or magnetic). All models are required to be compatible with thermodynamics and to fit well the shape of the major hysteresis loop. Focusing on uniform cyclic processes, numerical simulations at different temperatures are performed.
Citation: 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
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, (2002), 125.

[2]

A. Berti, C. Giorgi and E. Vuk, Free energies in one-dimensional models of magnetic transitions with hysteresis,, Nuovo Cimento Soc. Ital. Fis. B, 125 (2010), 371.

[3]

A. Berti, C. Giorgi and E. Vuk, Free energies and pseudo-elastic transitions for shape memory alloys,, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 293.

[4]

A. Berti, C. Giorgi and E. Vuk, Hysteresis and thermally-induced transitions in ferro-magnetic materials,, Appl. Math. Model., 132 (2014), 73. doi: 10.1007/s10440-014-9906-z.

[5]

V. Berti, M. Fabrizio and C. Giorgi, A three-dimensional phase transition model in ferromagnetism: Existence and uniqueness,, J. Math. Anal. Appl., 355 (2009), 661. doi: 10.1016/j.jmaa.2009.01.065.

[6]

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.

[7]

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.

[8]

G. Bertotti and I. D. Mayergoyz, The Science of Hysteresis: Mathematical Modeling and Applications,, Volume 1, (2006).

[9]

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.

[10]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions,, Springer, (1996). doi: 10.1007/978-1-4612-4048-8.

[11]

W. F. Brown, Jr., Micromagnetics,, Krieger Pub Co, (1978).

[12]

J. M. D. Coey, Magnetism and Magnetic Materials,, Cambridge University Press, (2009).

[13]

B. D. Cullity and C. D. Graham, Introduction to Magnetic Materials,, $2^{nd}$ edition, (2008). doi: 10.1002/9780470386323.

[14]

M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to non-isothermal phase-field evolution in continuum physics,, Physica D, 214 (2006), 144. doi: 10.1016/j.physd.2006.01.002.

[15]

M. Fabrizio, C. Giorgi and A. Morro, Phase transition in ferromagnetism,, Internat. J. Engrg. Sci., 47 (2009), 821. doi: 10.1016/j.ijengsci.2009.05.010.

[16]

M. Fabrizio, G. Matarazzo and M. Pecoraro, Hysteresis loops for para-ferromagnetic phase transitions,, Z. Angew. Math. Phys., 62 (2011), 1013. doi: 10.1007/s00033-011-0160-1.

[17]

M. Fabrizio and A. Morro, Electromagnetism of Continuous Media,, Oxford University Press, (2003). doi: 10.1093/acprof:oso/9780198527008.001.0001.

[18]

M. Frémond, Matériaux à mémoire de forme,, C.R. Acad. Sci. Paris Ser. II, 304 (1987), 239.

[19]

M. Frémond, Non-smooth Thermomechanics,, Springer-Verlag, (2002). doi: 10.1007/978-3-662-04800-9.

[20]

C. Giorgi, Continuum thermodynamics and phase-field models,, Milan J. Math., 77 (2009), 67. doi: 10.1007/s00032-009-0101-z.

[21]

N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group,, Addison-Wesley, (1992).

[22]

D. Grandi, M. Maraldi and L. Molari, A macroscale phase-field model for shape memory alloys with non-isothermal effects: Influence of strain rate and environmental conditions on the mechanical response,, Internat. J. Engrg. Sci., 50 (2012), 31.

[23]

L. D. Landau and V. L. Ginzburg, On the theory of superconductivity,, in Collected papers of L. D. Landau (ed. D. ter Haar), (1965).

[24]

L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics,, Pergamon, (1980).

[25]

L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media,, $2^{nd}$ edition, (1984).

[26]

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.

[27]

S. Miyazaki, Development and characterization of shape memory alloys,, in Shape Memory Alloys, (1996), 69. doi: 10.1007/978-3-7091-4348-3_2.

[28]

A. H. Morrish, The Physical Principles of Magnetism,, Wiley-IEEE Press, (2001). doi: 10.1109/9780470546581.

[29]

A. Visintin, Differential Models of Hysteresis,, Series: Applied Mathematical Sciences, (1994). doi: 10.1007/978-3-662-11557-2.

[30]

C. M. Wayman, Shape memory and related phenomena,, Progress in Materials Science, 36 (1992), 203. doi: 10.1016/0079-6425(92)90009-V.

[31]

J. C. Willems, Dissipative dynamical systems - Part I: General theory,, Arch. Rational Mech. Anal., 45 (1972), 321. doi: 10.1007/BF00276493.

show all references

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, (2002), 125.

[2]

A. Berti, C. Giorgi and E. Vuk, Free energies in one-dimensional models of magnetic transitions with hysteresis,, Nuovo Cimento Soc. Ital. Fis. B, 125 (2010), 371.

[3]

A. Berti, C. Giorgi and E. Vuk, Free energies and pseudo-elastic transitions for shape memory alloys,, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 293.

[4]

A. Berti, C. Giorgi and E. Vuk, Hysteresis and thermally-induced transitions in ferro-magnetic materials,, Appl. Math. Model., 132 (2014), 73. doi: 10.1007/s10440-014-9906-z.

[5]

V. Berti, M. Fabrizio and C. Giorgi, A three-dimensional phase transition model in ferromagnetism: Existence and uniqueness,, J. Math. Anal. Appl., 355 (2009), 661. doi: 10.1016/j.jmaa.2009.01.065.

[6]

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.

[7]

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.

[8]

G. Bertotti and I. D. Mayergoyz, The Science of Hysteresis: Mathematical Modeling and Applications,, Volume 1, (2006).

[9]

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.

[10]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions,, Springer, (1996). doi: 10.1007/978-1-4612-4048-8.

[11]

W. F. Brown, Jr., Micromagnetics,, Krieger Pub Co, (1978).

[12]

J. M. D. Coey, Magnetism and Magnetic Materials,, Cambridge University Press, (2009).

[13]

B. D. Cullity and C. D. Graham, Introduction to Magnetic Materials,, $2^{nd}$ edition, (2008). doi: 10.1002/9780470386323.

[14]

M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to non-isothermal phase-field evolution in continuum physics,, Physica D, 214 (2006), 144. doi: 10.1016/j.physd.2006.01.002.

[15]

M. Fabrizio, C. Giorgi and A. Morro, Phase transition in ferromagnetism,, Internat. J. Engrg. Sci., 47 (2009), 821. doi: 10.1016/j.ijengsci.2009.05.010.

[16]

M. Fabrizio, G. Matarazzo and M. Pecoraro, Hysteresis loops for para-ferromagnetic phase transitions,, Z. Angew. Math. Phys., 62 (2011), 1013. doi: 10.1007/s00033-011-0160-1.

[17]

M. Fabrizio and A. Morro, Electromagnetism of Continuous Media,, Oxford University Press, (2003). doi: 10.1093/acprof:oso/9780198527008.001.0001.

[18]

M. Frémond, Matériaux à mémoire de forme,, C.R. Acad. Sci. Paris Ser. II, 304 (1987), 239.

[19]

M. Frémond, Non-smooth Thermomechanics,, Springer-Verlag, (2002). doi: 10.1007/978-3-662-04800-9.

[20]

C. Giorgi, Continuum thermodynamics and phase-field models,, Milan J. Math., 77 (2009), 67. doi: 10.1007/s00032-009-0101-z.

[21]

N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group,, Addison-Wesley, (1992).

[22]

D. Grandi, M. Maraldi and L. Molari, A macroscale phase-field model for shape memory alloys with non-isothermal effects: Influence of strain rate and environmental conditions on the mechanical response,, Internat. J. Engrg. Sci., 50 (2012), 31.

[23]

L. D. Landau and V. L. Ginzburg, On the theory of superconductivity,, in Collected papers of L. D. Landau (ed. D. ter Haar), (1965).

[24]

L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics,, Pergamon, (1980).

[25]

L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media,, $2^{nd}$ edition, (1984).

[26]

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.

[27]

S. Miyazaki, Development and characterization of shape memory alloys,, in Shape Memory Alloys, (1996), 69. doi: 10.1007/978-3-7091-4348-3_2.

[28]

A. H. Morrish, The Physical Principles of Magnetism,, Wiley-IEEE Press, (2001). doi: 10.1109/9780470546581.

[29]

A. Visintin, Differential Models of Hysteresis,, Series: Applied Mathematical Sciences, (1994). doi: 10.1007/978-3-662-11557-2.

[30]

C. M. Wayman, Shape memory and related phenomena,, Progress in Materials Science, 36 (1992), 203. doi: 10.1016/0079-6425(92)90009-V.

[31]

J. C. Willems, Dissipative dynamical systems - Part I: General theory,, Arch. Rational Mech. Anal., 45 (1972), 321. doi: 10.1007/BF00276493.

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