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November  2013, 12(6): 2973-2996. doi: 10.3934/cpaa.2013.12.2973

Non-isothermal cyclic fatigue in an oscillating elastoplastic beam

Michela Eleuteri 1, , Jana Kopfová 2, and  Pavel Krejčí 3,

1. 

Dipartimento di Matematica, Università degli Studi di Milano, via Saldini 50, 20133 Milano.
2. 

Mathematical Institute of the Silesian University, Na Rybníčku 1, 746 01 Opava
3. 

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

Received  July 2012 Revised  February 2013 Published  May 2013

We propose a temperature dependent model for fatigue accumulation in an oscillating elastoplastic beam. The full system consists of the momentum and energy balance equations, and an evolution equation for the fatigue rate. The main modeling hypothesis is that the fatigue accumulation rate is proportional to the dissipation rate. In nontrivial cases, the process develops a thermal singularity in finite time. The main result consists in proving the existence and uniqueness of a strong solution in a time interval depending on the size of the data.
Keywords: blow-up., Elastoplastic beam, material fatigue, thermo-elasto-plasticity, Prandtl-Ishlinskii operator.
Mathematics Subject Classification: Primary: 74R20; Secondary: 74K10, 74H35, 74N3.
Citation: Michela Eleuteri, Jana Kopfová, Pavel Krejčí. Non-isothermal cyclic fatigue in an oscillating elastoplastic beam. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2973-2996. doi: 10.3934/cpaa.2013.12.2973
References:
[1]

S. Bartels and T. Roubíček, Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion, Math. Modelling Numer. Anal., 45 (2011), 477-504. doi: 10.1051/m2an/2010063.

[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), 705-735.

[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, Physica B, Condensed Matter, 407 (2012), 1415-1416. doi: 10.1016/j.physb.2011.10.017.

[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, L. Lussardi and U. Stefanelli, Thermal control of the Souza-Auricchio model for shape memory alloys, Discrete Cont. Dynam. Syst., Ser. S, 6 (2013), 369-386. doi: 10.3934/dcdss.2013.6.369.

[8]

E. Feireisl, M. Frémond, E. Rocca and G. Schimperna, A new approach to non-isothermal models for nematic liquid crystals, Arch. Ration. Mech. Anal., 205 (2012), 651-672. doi: 10.1007/s00205-012-0517-4.

[9]

R. Guenther, P. Krejčí and J. Sprekels, Small strain oscillations of an elastoplastic Kirchhoff plate, Z. Angew. Math. Mech., 88 (2008), 199-217. doi: 10.1002/zamm.200700111.

[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]

G. R. Johnson and W. H. Cook, Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures, Engineering Fracture Mechanics, 21 (1985), 31-48. doi: 10.1016/0013-7944(85)90052-9.

[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, Gakkot\=osho, Tokyo, 1996.

[14]

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.

[15]

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.

[16]

J. Lemaitre and J.-L. Chaboche, "Mechanics of Solid Materials," Cambridge University Press, 1990. doi: 10.1017/CBO9781139167970.

[17]

L. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper, Z. Ang. Math. Mech., 8 (1928), 85-106. doi: 10.1002/zamm.19280080202.

[18]

T. Roubíček, Thermodynamics of rate independent processes in viscous solids at small strains, SIAM J. Math. Anal., 42 (2010), 256-297. doi: 10.1137/080729992.

[19]

T. Roubíček and G. Tomassetti, Thermodynamics of shape-memory alloys under electric current, Zeit. angew. Math. Phys., 61 (2010), 1-20. doi: 10.1007/s00033-009-0007-1.

show all references

References:
[1]

S. Bartels and T. Roubíček, Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion, Math. Modelling Numer. Anal., 45 (2011), 477-504. doi: 10.1051/m2an/2010063.

[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), 705-735.

[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, Physica B, Condensed Matter, 407 (2012), 1415-1416. doi: 10.1016/j.physb.2011.10.017.

[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, L. Lussardi and U. Stefanelli, Thermal control of the Souza-Auricchio model for shape memory alloys, Discrete Cont. Dynam. Syst., Ser. S, 6 (2013), 369-386. doi: 10.3934/dcdss.2013.6.369.

[8]

E. Feireisl, M. Frémond, E. Rocca and G. Schimperna, A new approach to non-isothermal models for nematic liquid crystals, Arch. Ration. Mech. Anal., 205 (2012), 651-672. doi: 10.1007/s00205-012-0517-4.

[9]

R. Guenther, P. Krejčí and J. Sprekels, Small strain oscillations of an elastoplastic Kirchhoff plate, Z. Angew. Math. Mech., 88 (2008), 199-217. doi: 10.1002/zamm.200700111.

[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]

G. R. Johnson and W. H. Cook, Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures, Engineering Fracture Mechanics, 21 (1985), 31-48. doi: 10.1016/0013-7944(85)90052-9.

[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, Gakkot\=osho, Tokyo, 1996.

[14]

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.

[15]

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.

[16]

J. Lemaitre and J.-L. Chaboche, "Mechanics of Solid Materials," Cambridge University Press, 1990. doi: 10.1017/CBO9781139167970.

[17]

L. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper, Z. Ang. Math. Mech., 8 (1928), 85-106. doi: 10.1002/zamm.19280080202.

[18]

T. Roubíček, Thermodynamics of rate independent processes in viscous solids at small strains, SIAM J. Math. Anal., 42 (2010), 256-297. doi: 10.1137/080729992.

[19]

T. Roubíček and G. Tomassetti, Thermodynamics of shape-memory alloys under electric current, Zeit. angew. Math. Phys., 61 (2010), 1-20. doi: 10.1007/s00033-009-0007-1.

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