American Institute of Mathematical Sciences

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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 & 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.  doi: 10.1051/m2an/2010063.  Google Scholar

[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.   Google Scholar

[3]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Appl. Math. Sci., 121 (1996).  doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[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.  doi: 10.1137/0513029.  Google Scholar

[5]

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

[6]

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.  Google Scholar

[7]

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

[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.  doi: 10.1007/s00205-012-0517-4.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  doi: 10.1016/0013-7944(85)90052-9.  Google Scholar

[12]

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

[13]

P. Krejčí, "Hysteresis, Convexity and Dissipation in Hyperbolic Equations,", Gakuto Intern. Ser. Math. Sci. Appl., 8 (1996).   Google Scholar

[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.  doi: 10.1006/jmaa.1997.5304.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

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.  doi: 10.1051/m2an/2010063.  Google Scholar

[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.   Google Scholar

[3]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Appl. Math. Sci., 121 (1996).  doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[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.  doi: 10.1137/0513029.  Google Scholar

[5]

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

[6]

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.  Google Scholar

[7]

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

[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.  doi: 10.1007/s00205-012-0517-4.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  doi: 10.1016/0013-7944(85)90052-9.  Google Scholar

[12]

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

[13]

P. Krejčí, "Hysteresis, Convexity and Dissipation in Hyperbolic Equations,", Gakuto Intern. Ser. Math. Sci. Appl., 8 (1996).   Google Scholar

[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.  doi: 10.1006/jmaa.1997.5304.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

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