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August  2015, 8(4): 757-767. doi: 10.3934/dcdss.2015.8.757

Thermodynamical consistency - a mystery or?

Jana Kopfová 1,

1. 

Mathematical Institute of the Silesian University, Na Rybníčku 1, 746 01 Opava

Received  January 2014 Revised  July 2014 Published  October 2014

The goal of this note is to discuss the basic thermodynamical principles and show how they need to be considered in the process of developing new mathematical models. We give numerous examples: linear elasticity with constant or non-constant temperature, we discuss classical hysteresis models as the play operator, the Preisach operator as well as new models introduced in the last years - the temperature dependent Preisach model, models of magnetostriction and models of an oscillating beam with fatigue.
Keywords: hysteresis, Thermodynamical consistency, fatigue, Preisach model., magnetostriction.
Mathematics Subject Classification: 35Q79, 74A15, 82B3.
Citation: Jana Kopfová. Thermodynamical consistency - a mystery or?. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 757-767. doi: 10.3934/dcdss.2015.8.757
References:
[1]

D. Davino, P. Krejčí and C. Visone, Fully couples modeling of magneto-mechanical hysteresis thorugh thermodynamic compatibility, Smart Materials and Structures, 22 (2013), 095009. Google Scholar

[2]

M. Eleuteri, J. Kopfová and P. Krejčí, A thermodynamic model for material fatigue under cyclic loading, Physica B: Condensed Matter, 407 (2012), 1415-1416. doi: 10.1016/j.physb.2011.10.017.  Google Scholar

[3]

M. Eleuteri, J. Kopfová and P. Krejčí, Fatigue accumulation in an oscillating plate, Discrete Cont. Dynam. Syst., Ser. S, 6 (2013), 909-923. doi: 10.3934/dcdss.2013.6.909.  Google Scholar

[4]

M. Eleuteri, J. Kopfová and P. Krejčí, Non-isothermal cyclic fatigue in an oscillating elastoplastic beam, Comm. Pure Appl. Anal., 12 (2013), 2973-2996. doi: 10.3934/cpaa.2013.12.2973.  Google Scholar

[5]

M. Eleuteri, J. Kopfová and P. Krejčí, Fatigue accumulation in a thermo-visco-elastoplastic plate, Discrete Cont. Dynam. Syst., Ser. B, 19 (2014), 2091-2109. doi: 10.3934/dcdsb.2014.19.2091.  Google Scholar

[6]

P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Mathematical Sciences and Applications, 8, Gakkōtosho Co., Ltd., Tokyo, 1996.  Google Scholar

[7]

J. Kopfová and P. Krejčí, A Preisach type model for temperature driven hysteresis memory erasure in shape memory materials, Continuum Mechanics and Thermodynamics, 23 (2011), 125-137. doi: 10.1007/s00161-010-0172-7.  Google Scholar

[8]

J. Kopfová and P. Sander, Non-isothermal cycling fatigue in an oscillating elastoplastic beam with phase transition, Special HMM issue of Physica B: Condensed Matter, Physica B: Physics of Condensed Matter, (2014), 31-33. Google Scholar

[9]

A. Visintin, Differential Models of Hysteresis, Applied Mathematical Sciences, 111, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-11557-2.  Google Scholar

show all references

References:
[1]

D. Davino, P. Krejčí and C. Visone, Fully couples modeling of magneto-mechanical hysteresis thorugh thermodynamic compatibility, Smart Materials and Structures, 22 (2013), 095009. Google Scholar

[2]

M. Eleuteri, J. Kopfová and P. Krejčí, A thermodynamic model for material fatigue under cyclic loading, Physica B: Condensed Matter, 407 (2012), 1415-1416. doi: 10.1016/j.physb.2011.10.017.  Google Scholar

[3]

M. Eleuteri, J. Kopfová and P. Krejčí, Fatigue accumulation in an oscillating plate, Discrete Cont. Dynam. Syst., Ser. S, 6 (2013), 909-923. doi: 10.3934/dcdss.2013.6.909.  Google Scholar

[4]

M. Eleuteri, J. Kopfová and P. Krejčí, Non-isothermal cyclic fatigue in an oscillating elastoplastic beam, Comm. Pure Appl. Anal., 12 (2013), 2973-2996. doi: 10.3934/cpaa.2013.12.2973.  Google Scholar

[5]

M. Eleuteri, J. Kopfová and P. Krejčí, Fatigue accumulation in a thermo-visco-elastoplastic plate, Discrete Cont. Dynam. Syst., Ser. B, 19 (2014), 2091-2109. doi: 10.3934/dcdsb.2014.19.2091.  Google Scholar

[6]

P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Mathematical Sciences and Applications, 8, Gakkōtosho Co., Ltd., Tokyo, 1996.  Google Scholar

[7]

J. Kopfová and P. Krejčí, A Preisach type model for temperature driven hysteresis memory erasure in shape memory materials, Continuum Mechanics and Thermodynamics, 23 (2011), 125-137. doi: 10.1007/s00161-010-0172-7.  Google Scholar

[8]

J. Kopfová and P. Sander, Non-isothermal cycling fatigue in an oscillating elastoplastic beam with phase transition, Special HMM issue of Physica B: Condensed Matter, Physica B: Physics of Condensed Matter, (2014), 31-33. Google Scholar

[9]

A. Visintin, Differential Models of Hysteresis, Applied Mathematical Sciences, 111, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-11557-2.  Google Scholar

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