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

Thermodynamical consistency - a mystery or?

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.
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).

[2]

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

[3]

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

[4]

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

[5]

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

[6]

P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations,, Mathematical Sciences and Applications, (1996).

[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. doi: 10.1007/s00161-010-0172-7.

[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, (2014), 31.

[9]

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

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).

[2]

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

[3]

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

[4]

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

[5]

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

[6]

P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations,, Mathematical Sciences and Applications, (1996).

[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. doi: 10.1007/s00161-010-0172-7.

[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, (2014), 31.

[9]

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

[1]

Pavel Krejčí. The Preisach hysteresis model: Error bounds for numerical identification and inversion. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 101-119. doi: 10.3934/dcdss.2013.6.101

[2]

Sebastián Ferrer, Francisco Crespo. Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems. Journal of Geometric Mechanics, 2014, 6 (4) : 479-502. doi: 10.3934/jgm.2014.6.479

[3]

Ghendrih Philippe, Hauray Maxime, Anne Nouri. Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution. Kinetic & Related Models, 2009, 2 (4) : 707-725. doi: 10.3934/krm.2009.2.707

[4]

Antonio DeSimone, Natalie Grunewald, Felix Otto. A new model for contact angle hysteresis. Networks & Heterogeneous Media, 2007, 2 (2) : 211-225. doi: 10.3934/nhm.2007.2.211

[5]

Michela Eleuteri, Jana Kopfová, Pavel Krejčí. A new phase field model for material fatigue in an oscillating elastoplastic beam. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2465-2495. doi: 10.3934/dcds.2015.35.2465

[6]

Faker Ben Belgacem. Uniqueness for an ill-posed reaction-dispersion model. Application to organic pollution in stream-waters. Inverse Problems & Imaging, 2012, 6 (2) : 163-181. doi: 10.3934/ipi.2012.6.163

[7]

Shaojun Zhang, Zhong Wan. Polymorphic uncertain nonlinear programming model and algorithm for maximizing the fatigue life of V-belt drive. Journal of Industrial & Management Optimization, 2012, 8 (2) : 493-505. doi: 10.3934/jimo.2012.8.493

[8]

Youssef Amal, Martin Campos Pinto. Global solutions for an age-dependent model of nucleation, growth and ageing with hysteresis. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 517-535. doi: 10.3934/dcdsb.2010.13.517

[9]

D. Lannes. Consistency of the KP approximation. Conference Publications, 2003, 2003 (Special) : 517-525. doi: 10.3934/proc.2003.2003.517

[10]

Imre Csiszar and Paul C. Shields. Consistency of the BIC order estimator. Electronic Research Announcements, 1999, 5: 123-127.

[11]

Michela Eleuteri, Jana Kopfov, Pavel Krej?. Fatigue accumulation in an oscillating plate. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 909-923. doi: 10.3934/dcdss.2013.6.909

[12]

Toufik Bakir, Bernard Bonnard, Jérémy Rouot. A case study of optimal input-output system with sampled-data control: Ding et al. force and fatigue muscular control model. Networks & Heterogeneous Media, 2019, 14 (1) : 79-100. doi: 10.3934/nhm.2019005

[13]

Sebastian Reich, Seoleun Shin. On the consistency of ensemble transform filter formulations. Journal of Computational Dynamics, 2014, 1 (1) : 177-189. doi: 10.3934/jcd.2014.1.177

[14]

Ruikuan Liu, Tian Ma, Shouhong Wang, Jiayan Yang. Thermodynamical potentials of classical and quantum systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1411-1448. doi: 10.3934/dcdsb.2018214

[15]

Alexander Pimenov, Dmitrii I. Rachinskii. Linear stability analysis of systems with Preisach memory. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 997-1018. doi: 10.3934/dcdsb.2009.11.997

[16]

Antonio DeSimone, Martin Kružík. Domain patterns and hysteresis in phase-transforming solids: Analysis and numerical simulations of a sharp interface dissipative model via phase-field approximation. Networks & Heterogeneous Media, 2013, 8 (2) : 481-499. doi: 10.3934/nhm.2013.8.481

[17]

Michela Eleuteri, Jana Kopfová, Pavel Krejčí. Fatigue accumulation in a thermo-visco-elastoplastic plate. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2091-2109. doi: 10.3934/dcdsb.2014.19.2091

[18]

Vincent Pavan. Thermodynamical considerations implying wall/particles scattering kernels. Kinetic & Related Models, 2014, 7 (1) : 133-168. doi: 10.3934/krm.2014.7.133

[19]

Benjamin Couéraud, François Gay-Balmaz. Variational discretization of thermodynamical simple systems on Lie groups. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-28. doi: 10.3934/dcdss.2020064

[20]

Rod Cross, Hugh McNamara, Leonid Kalachev, Alexei Pokrovskii. Hysteresis and post Walrasian economics. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 377-401. doi: 10.3934/dcdsb.2013.18.377

2017 Impact Factor: 0.561

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]