September  2014, 19(7): 2091-2109. doi: 10.3934/dcdsb.2014.19.2091

Fatigue accumulation in a thermo-visco-elastoplastic plate

1. 

Dipartimento di Matematica e Informatica "U. Dini", Università degli Studi di Firenze, viale Morgagni 67/a, 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, 11567 Praha 1

Received  April 2013 Revised  August 2013 Published  August 2014

We consider a thermodynamic model for fatigue accumulation in an oscillating elastoplastic Kirchhoff plate based on the hypothesis that the fatigue accumulation rate is proportional to the plastic part of the dissipation rate. For the full model with periodic boundary conditions we prove existence of a solution in the whole time interval.
Citation: 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
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]

O. V. Besov, V. P. Il'in and S. M. Nikol'skič, Integral Representations of Functions and Imbedding Theorems,, Scripta Series in Mathematics, (1978). Google Scholar

[3]

E. Bonetti and G. Bonfanti, Well-posedness results for a model of damage in thermoviscoelastic materials,, Ann. I. H. Poincaré, 25 (2008), 1187. doi: 10.1016/j.anihpc.2007.05.009. Google Scholar

[4]

E. Bonetti and G. Schimperna, Local existence for Frémond's model of damage in elastic materials,, Continuum Mech. Therm., 16 (2004), 319. doi: 10.1007/s00161-003-0152-2. Google Scholar

[5]

E. Bonetti, G. Schimperna and A. Segatti, On a doubly nonlinear model for the evolution of damaging in viscoelastic materials,, J. Diff. Equ., 218 (2005), 91. doi: 10.1016/j.jde.2005.04.015. Google Scholar

[6]

S. Bosia, M. Eleuteri, J. Kopfová and P. Krejčí, Fatigue and phase in a oscillating plate,, Proceedings of the 9th International Symposium on Hysteresis Modeling and Micromagnetics, 435 (2014), 1. Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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. Google Scholar

[11]

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

[12]

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. Google Scholar

[13]

M. Eleuteri, J. Kopfová and P. Krejčí, A new phase-field model for material fatigue in oscillating elastoplastic beam,, Discrete Cont. Dynam. Syst., (2014). Google Scholar

[14]

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. Google Scholar

[15]

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

[16]

A. Flatten, Lokale und Nicht-Lokale Modellierung und Simulation Thermomechanischer Lokalisierung mit Schädigung Für metallische Werkstoffe unter Hochgeschwindigkeitsbeanspruchungen,, BAM-Dissertationsreihe, (2008). Google Scholar

[17]

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. Google Scholar

[18]

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

[19]

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. Google Scholar

[20]

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

[21]

D. Knees, R. Rossi and C. Zanini, A vanishing viscosity approach to a rate-independent damage model,, Math. Models Meth. Appl. Sci. (M3AS), 23 (2013), 565. doi: 10.1142/S021820251250056X. Google Scholar

[22]

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, 435 (2014), 31. Google Scholar

[23]

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

[24]

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

[25]

P. Krejčí and J. Sprekels, Clamped elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators,, Discrete Contin. Dyn. Syst.-S, 1 (2008), 283. doi: 10.3934/dcdss.2008.1.283. Google Scholar

[26]

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. Google Scholar

[27]

M. Liero and A. Mielke, An evolutionary elastoplastic plate model derived via Gamma convergence,, Math. Models Meth. Appl. Sci. (M3AS), 21 (2011), 1961. doi: 10.1142/S0218202511005611. Google Scholar

[28]

M. Liero and T. Roche, Rigorous derivation of a plate theory in linear elastoplasticity via Gamma convergence,, Nonlinear Differential Equations and Applications NoDEA, 19 (2012), 437. doi: 10.1007/s00030-011-0137-y. Google Scholar

[29]

A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear inelasticity,, Math. Models Meth. Appl. Sci. (M3AS), 16 (2006), 177. doi: 10.1142/S021820250600111X. Google Scholar

[30]

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

[31]

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. Google Scholar

[32]

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

[33]

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

[34]

E. Rocca and R. Rossi, A degenerating PDE system for phase transitions and damage,, Math. Models Methods Appl. Sci., 24 (2014), 1265. doi: 10.1142/S021820251450002X. Google Scholar

[35]

R. Rossi and T. Roubíček, Adhesive contact delaminating at mixed mode, its thermodynamics and analysis,, Interfaces Free Bound, 14 (2013), 1. doi: 10.4171/IFB/293. Google Scholar

[36]

R. Rossi and M. Thomas, From an adhesive to a brittle delamination model in thermo-visco-elasticity,, ESAIM COCV, (2014). Google Scholar

[37]

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

[38]

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

[39]

M. Thomas and A. Mielke, Damage of nonlinearly elastic materials at small strain Existence and regularity results,, ZAMM - Z. Angew. Math. Mech., 90 (2010), 88. doi: 10.1002/zamm.200900243. 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]

O. V. Besov, V. P. Il'in and S. M. Nikol'skič, Integral Representations of Functions and Imbedding Theorems,, Scripta Series in Mathematics, (1978). Google Scholar

[3]

E. Bonetti and G. Bonfanti, Well-posedness results for a model of damage in thermoviscoelastic materials,, Ann. I. H. Poincaré, 25 (2008), 1187. doi: 10.1016/j.anihpc.2007.05.009. Google Scholar

[4]

E. Bonetti and G. Schimperna, Local existence for Frémond's model of damage in elastic materials,, Continuum Mech. Therm., 16 (2004), 319. doi: 10.1007/s00161-003-0152-2. Google Scholar

[5]

E. Bonetti, G. Schimperna and A. Segatti, On a doubly nonlinear model for the evolution of damaging in viscoelastic materials,, J. Diff. Equ., 218 (2005), 91. doi: 10.1016/j.jde.2005.04.015. Google Scholar

[6]

S. Bosia, M. Eleuteri, J. Kopfová and P. Krejčí, Fatigue and phase in a oscillating plate,, Proceedings of the 9th International Symposium on Hysteresis Modeling and Micromagnetics, 435 (2014), 1. Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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. Google Scholar

[11]

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

[12]

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. Google Scholar

[13]

M. Eleuteri, J. Kopfová and P. Krejčí, A new phase-field model for material fatigue in oscillating elastoplastic beam,, Discrete Cont. Dynam. Syst., (2014). Google Scholar

[14]

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. Google Scholar

[15]

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

[16]

A. Flatten, Lokale und Nicht-Lokale Modellierung und Simulation Thermomechanischer Lokalisierung mit Schädigung Für metallische Werkstoffe unter Hochgeschwindigkeitsbeanspruchungen,, BAM-Dissertationsreihe, (2008). Google Scholar

[17]

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. Google Scholar

[18]

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

[19]

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. Google Scholar

[20]

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

[21]

D. Knees, R. Rossi and C. Zanini, A vanishing viscosity approach to a rate-independent damage model,, Math. Models Meth. Appl. Sci. (M3AS), 23 (2013), 565. doi: 10.1142/S021820251250056X. Google Scholar

[22]

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, 435 (2014), 31. Google Scholar

[23]

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

[24]

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

[25]

P. Krejčí and J. Sprekels, Clamped elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators,, Discrete Contin. Dyn. Syst.-S, 1 (2008), 283. doi: 10.3934/dcdss.2008.1.283. Google Scholar

[26]

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. Google Scholar

[27]

M. Liero and A. Mielke, An evolutionary elastoplastic plate model derived via Gamma convergence,, Math. Models Meth. Appl. Sci. (M3AS), 21 (2011), 1961. doi: 10.1142/S0218202511005611. Google Scholar

[28]

M. Liero and T. Roche, Rigorous derivation of a plate theory in linear elastoplasticity via Gamma convergence,, Nonlinear Differential Equations and Applications NoDEA, 19 (2012), 437. doi: 10.1007/s00030-011-0137-y. Google Scholar

[29]

A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear inelasticity,, Math. Models Meth. Appl. Sci. (M3AS), 16 (2006), 177. doi: 10.1142/S021820250600111X. Google Scholar

[30]

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

[31]

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. Google Scholar

[32]

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

[33]

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

[34]

E. Rocca and R. Rossi, A degenerating PDE system for phase transitions and damage,, Math. Models Methods Appl. Sci., 24 (2014), 1265. doi: 10.1142/S021820251450002X. Google Scholar

[35]

R. Rossi and T. Roubíček, Adhesive contact delaminating at mixed mode, its thermodynamics and analysis,, Interfaces Free Bound, 14 (2013), 1. doi: 10.4171/IFB/293. Google Scholar

[36]

R. Rossi and M. Thomas, From an adhesive to a brittle delamination model in thermo-visco-elasticity,, ESAIM COCV, (2014). Google Scholar

[37]

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

[38]

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

[39]

M. Thomas and A. Mielke, Damage of nonlinearly elastic materials at small strain Existence and regularity results,, ZAMM - Z. Angew. Math. Mech., 90 (2010), 88. doi: 10.1002/zamm.200900243. Google Scholar

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