April  2013, 6(4): 909-923. doi: 10.3934/dcdss.2013.6.909

Fatigue accumulation in an oscillating plate

1. 

Dipartimento di Matematica, Universit degli Studi di Milano

2. 

via Saldini 50

3. 

20133 Milano.

4. 

Mathematical Institute of the Silesian University

5. 

Na Rybn?ku 1

6. 

746 01 Opava

7. 

Institute of Mathematics, Czech Academy of Sciences

8. 

?itn 25

9. 

11567 Praha 1

Received  October 2011 Revised  February 2012 Published  December 2012

A thermodynamic model for fatigue accumulation in an oscillating elastoplastic Kirchhoffplate based on the hypothesis that the fatigue accumulation rate is proportional tothe dissipation rate, is derived for the case that both the elastic and the plasticmaterial characteristics change with increasing fatigue. We prove the existence ofa unique solution in the whole time interval before a singularity (material failure) occursunder the simplifying hypothesis that the temperature history is a priori given.
Citation: 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
References:
[1]

M. Brokate, C. Carstensen and J. Valdman, A quasi-static boundary value problem in multi-surface elastoplasticity. I. Analysis,, Math. Methods Appl. Sci., 27 (2004), 1697. doi: 10.1002/mma.524.

[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.

[3]

M. Brokate and A. M. Khludnev, Existence of solutions in the Prandtl-Reuss theory of elastoplastic plates,, Adv. Math. Sci Appl., 10 (2000), 399.

[4]

M. Brokate, P. Krejčí and H. Schnabel, On uniqueness in evolution quasivariational inequalities,, J. of Convex Analysis, 11 (2004), 111.

[5]

P. Drábek, P. Krejčí and P. Takáč, "Nonlinear Differential Equations,", Research Notes in Mathematics, 404 (1999).

[6]

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

[7]

M. Eleuteri, J. Kopfová and P. Krejčí, Non-isothermal cyclic fatigue in an oscillating elastoplastic beam,, Submitted., ().

[8]

G. Friesecke, R. D. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence,, Arch. Ration. Mech. Anal., 180 (2006), 183. doi: 10.1007/s00205-005-0400-7.

[9]

R. Guenther, P. Krejčí and J. Sprekels, Small strain oscillations of an elastoplastic Kirchhoff plate,, Z. Angew. Math. Mech., 88 (2008), 199. doi: 10.1002/zamm.200700111.

[10]

A. L. Gurson, Continuum theory of ductile rupture by void nucleation and growth: Part I - yield criteria and flow rules for porous ductile media,, J. Engng. Mater. Technol., 99 (1977), 2.

[11]

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

[12]

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

[13]

M. Kuczma, P. Litewka, J. Rakowski and J. R. Whiteman, A variational inequality approach to an elastoplastic plate-foundation system,, Foundations of Civil and Environmental Engineering, 5 (2004), 31.

[14]

M. Liero and A. Mielke, An evolutionary elastoplastic plate model derived via Gamma convergence,, WIAS Preprint No. 1583, (2010).

[15]

M. Liero and T. Roche, Rigorous derivation of a plate theory in linear elastoplasticity via Gamma convergence,, WIAS Preprint No. 1636, (2011).

[16]

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

[17]

A. Mielke and F. Theil, On rate-independent hysteresis models,, NoDEA, 11 (2004), 151. doi: 10.1007/s00030-003-1052-7.

[18]

O. Millet, A. Cimetiere and A. Hamdouni, An asymptotic elastic-plastic plate model for moderate displacements and strong strain hardening,, Eur. J. Mech. A Solids, 22 (2003), 369. doi: 10.1016/S0997-7538(03)00044-5.

[19]

D. Percivale, Perfectly plastic plates: A variational definition,, J. Reine Angew. Math., 411 (1990), 39. doi: 10.1515/crll.1990.411.39.

[20]

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

show all references

References:
[1]

M. Brokate, C. Carstensen and J. Valdman, A quasi-static boundary value problem in multi-surface elastoplasticity. I. Analysis,, Math. Methods Appl. Sci., 27 (2004), 1697. doi: 10.1002/mma.524.

[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.

[3]

M. Brokate and A. M. Khludnev, Existence of solutions in the Prandtl-Reuss theory of elastoplastic plates,, Adv. Math. Sci Appl., 10 (2000), 399.

[4]

M. Brokate, P. Krejčí and H. Schnabel, On uniqueness in evolution quasivariational inequalities,, J. of Convex Analysis, 11 (2004), 111.

[5]

P. Drábek, P. Krejčí and P. Takáč, "Nonlinear Differential Equations,", Research Notes in Mathematics, 404 (1999).

[6]

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

[7]

M. Eleuteri, J. Kopfová and P. Krejčí, Non-isothermal cyclic fatigue in an oscillating elastoplastic beam,, Submitted., ().

[8]

G. Friesecke, R. D. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence,, Arch. Ration. Mech. Anal., 180 (2006), 183. doi: 10.1007/s00205-005-0400-7.

[9]

R. Guenther, P. Krejčí and J. Sprekels, Small strain oscillations of an elastoplastic Kirchhoff plate,, Z. Angew. Math. Mech., 88 (2008), 199. doi: 10.1002/zamm.200700111.

[10]

A. L. Gurson, Continuum theory of ductile rupture by void nucleation and growth: Part I - yield criteria and flow rules for porous ductile media,, J. Engng. Mater. Technol., 99 (1977), 2.

[11]

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

[12]

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

[13]

M. Kuczma, P. Litewka, J. Rakowski and J. R. Whiteman, A variational inequality approach to an elastoplastic plate-foundation system,, Foundations of Civil and Environmental Engineering, 5 (2004), 31.

[14]

M. Liero and A. Mielke, An evolutionary elastoplastic plate model derived via Gamma convergence,, WIAS Preprint No. 1583, (2010).

[15]

M. Liero and T. Roche, Rigorous derivation of a plate theory in linear elastoplasticity via Gamma convergence,, WIAS Preprint No. 1636, (2011).

[16]

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

[17]

A. Mielke and F. Theil, On rate-independent hysteresis models,, NoDEA, 11 (2004), 151. doi: 10.1007/s00030-003-1052-7.

[18]

O. Millet, A. Cimetiere and A. Hamdouni, An asymptotic elastic-plastic plate model for moderate displacements and strong strain hardening,, Eur. J. Mech. A Solids, 22 (2003), 369. doi: 10.1016/S0997-7538(03)00044-5.

[19]

D. Percivale, Perfectly plastic plates: A variational definition,, J. Reine Angew. Math., 411 (1990), 39. doi: 10.1515/crll.1990.411.39.

[20]

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

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