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June  2015, 35(6): 2465-2495. doi: 10.3934/dcds.2015.35.2465

A new phase field model for material fatigue in an oscillating elastoplastic beam

1. 

Dipartimento di Matematica ed Informatica “U. Dini”, viale Morgagni 67/a, I-50134 Firenze, Italy

2. 

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

3. 

Institute of Mathematics, Czech Academy of Sciences, Žitná 25, CZ-11567 Praha 1

Received  December 2013 Revised  April 2014 Published  December 2014

We pursue the study of fatigue accumulation in an oscillating elastoplastic beam under the additional hypothesis that the material can partially recover by the effect of melting. The full system consists of the momentum and energy balance equations, an evolution equation for the fatigue rate, and a differential inclusion for the phase dynamics. The main result consists in proving the existence and uniqueness of a strong solution.
Citation: Michela Eleuteri, Jana Kopfová, Pavel Krejčí. A new phase field model for material fatigue in an oscillating elastoplastic beam. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2465-2495. doi: 10.3934/dcds.2015.35.2465
References:
[1]

S. Bosia, M. Eleuteri, J. Kopfová and P. Krejčí, Fatigue and phase transition in a oscillating plate, "Proceedings of the 9th International Symposium on Hysteresis Modeling and Micromagnetics'', Physica B: Condensed Matter, 435 (2014), 1-3.

[2]

M. Brokate, K. Dreßler and P. Krejčí, Rainflow counting and energy dissipation for hysteresis models in elastoplasticity, Euro. J. Mech. A/Solids, 15 (1996), 705-737.

[3]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Appl. Math. Sci. Vol. 121, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.

[4]

C. M. Dafermos, Global smooth solutions to the initial-boundary value problem for the equations of one-dimensional thermoviscoelasticity, SIAM J. Math. Anal., 13 (1982), 397-408. doi: 10.1137/0513029.

[5]

M. Eleuteri, J. Kopfová and P. Krejčí, A thermodynamic model for material fatigue under cyclic loading, "Proceedings of the 8th International Symposium on Hysteresis Modeling and Micromagnetics'', Physica B: Condensed Matter, 407, (2012), 1415-1416.

[6]

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.

[7]

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.

[8]

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.

[9]

A. Flatten, Lokale und Nicht-lokale Modellierung und Simulation thermomechanischer Lokalisierung mit Schädigung für metallische Werkstoffe unter Hochgeschwindigkeitsbeanspruchungen, BAM-Dissertationsreihe, Berlin 2008.

[10]

A. Yu. Ishlinskii, Some applications of statistical methods to describing deformations of bodies, Izv. Akad. Nauk SSSR, Otd. Tekh. Nauk., 9 (1944), 583-590 (In Russian).

[11]

J. Kopfová and P. Sander, Non-isothermal cycling fatigue in an oscillating elastoplastic beam with phase transition, "Proceedings of the 9th International Symposium on Hysteresis Modeling and Micromagnetics'', Physica B: Condensed Matter, 435 (2014), 31-33.

[12]

M. A. Krasnosel'skii and A. V. Pokrovskii, Systems with Hysteresis, Springer-Verlag, Berlin - Heidelberg, 1989. doi: 10.1007/978-3-642-61302-9.

[13]

P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakuto Intern. Ser. Math. Sci. Appl., Vol. 8, Gakkotōsho, Tokyo, 1996.

[14]

P. Krejčí and J. Sprekels, Hysteresis operators in phase-field models of Penrose-Fife type, Appl. Math., 43 (1998), 207-222. doi: 10.1023/A:1023276524286.

[15]

P. Krejčí and J. Sprekels, On a system of nonlinear PDE's with temperature-dependent hysteresis in one-dimensional thermoplasticity, J. Math. Anal. Appl., 209 (1997), 25-46. doi: 10.1006/jmaa.1997.5304.

[16]

P. Krejčí and J. Sprekels, Elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators, Math. Methods Appl. Sci., 30 (2007), 2371-2393. doi: 10.1002/mma.892.

[17]

J. Lemaitre and J.-L. Chaboche, Mechanics of Solid Materials, Cambridge University Press, 1990.

[18]

L. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper, Z. Ang. Math. Mech., 8 (1928), 85-106.

show all references

References:
[1]

S. Bosia, M. Eleuteri, J. Kopfová and P. Krejčí, Fatigue and phase transition in a oscillating plate, "Proceedings of the 9th International Symposium on Hysteresis Modeling and Micromagnetics'', Physica B: Condensed Matter, 435 (2014), 1-3.

[2]

M. Brokate, K. Dreßler and P. Krejčí, Rainflow counting and energy dissipation for hysteresis models in elastoplasticity, Euro. J. Mech. A/Solids, 15 (1996), 705-737.

[3]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Appl. Math. Sci. Vol. 121, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.

[4]

C. M. Dafermos, Global smooth solutions to the initial-boundary value problem for the equations of one-dimensional thermoviscoelasticity, SIAM J. Math. Anal., 13 (1982), 397-408. doi: 10.1137/0513029.

[5]

M. Eleuteri, J. Kopfová and P. Krejčí, A thermodynamic model for material fatigue under cyclic loading, "Proceedings of the 8th International Symposium on Hysteresis Modeling and Micromagnetics'', Physica B: Condensed Matter, 407, (2012), 1415-1416.

[6]

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.

[7]

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.

[8]

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.

[9]

A. Flatten, Lokale und Nicht-lokale Modellierung und Simulation thermomechanischer Lokalisierung mit Schädigung für metallische Werkstoffe unter Hochgeschwindigkeitsbeanspruchungen, BAM-Dissertationsreihe, Berlin 2008.

[10]

A. Yu. Ishlinskii, Some applications of statistical methods to describing deformations of bodies, Izv. Akad. Nauk SSSR, Otd. Tekh. Nauk., 9 (1944), 583-590 (In Russian).

[11]

J. Kopfová and P. Sander, Non-isothermal cycling fatigue in an oscillating elastoplastic beam with phase transition, "Proceedings of the 9th International Symposium on Hysteresis Modeling and Micromagnetics'', Physica B: Condensed Matter, 435 (2014), 31-33.

[12]

M. A. Krasnosel'skii and A. V. Pokrovskii, Systems with Hysteresis, Springer-Verlag, Berlin - Heidelberg, 1989. doi: 10.1007/978-3-642-61302-9.

[13]

P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakuto Intern. Ser. Math. Sci. Appl., Vol. 8, Gakkotōsho, Tokyo, 1996.

[14]

P. Krejčí and J. Sprekels, Hysteresis operators in phase-field models of Penrose-Fife type, Appl. Math., 43 (1998), 207-222. doi: 10.1023/A:1023276524286.

[15]

P. Krejčí and J. Sprekels, On a system of nonlinear PDE's with temperature-dependent hysteresis in one-dimensional thermoplasticity, J. Math. Anal. Appl., 209 (1997), 25-46. doi: 10.1006/jmaa.1997.5304.

[16]

P. Krejčí and J. Sprekels, Elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators, Math. Methods Appl. Sci., 30 (2007), 2371-2393. doi: 10.1002/mma.892.

[17]

J. Lemaitre and J.-L. Chaboche, Mechanics of Solid Materials, Cambridge University Press, 1990.

[18]

L. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper, Z. Ang. Math. Mech., 8 (1928), 85-106.

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