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 and 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), 477-504. doi: 10.1051/m2an/2010063.

[2]

O. V. Besov, V. P. Il'in and S. M. Nikol'skič, Integral Representations of Functions and Imbedding Theorems, Scripta Series in Mathematics, Halsted Press (John Wiley & Sons), New York-Toronto, Ont.-London, 1978 (Vol. I), 1979 (Vol. II). Russian version Nauka, Moscow, 1975.

[3]

E. Bonetti and G. Bonfanti, Well-posedness results for a model of damage in thermoviscoelastic materials, Ann. I. H. Poincaré, 25 (2008), 1187-1208. doi: 10.1016/j.anihpc.2007.05.009.

[4]

E. Bonetti and G. Schimperna, Local existence for Frémond's model of damage in elastic materials, Continuum Mech. Therm., 16 (2004), 319-335. doi: 10.1007/s00161-003-0152-2.

[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), 91-116. doi: 10.1016/j.jde.2005.04.015.

[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, Physica B: Condensed Matter, 435 (2014), 1-3.

[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), 705-737.

[8]

M. Brokate and A. M. Khludnev, Existence of solutions in the Prandtl-Reuss theory of elastoplastic plates, Adv. Math. Sci Appl., 10 (2000), 399-415.

[9]

P. Drábek, P. Krejčí and P. Takáč, Nonlinear Differential Equations, Research Notes in Mathematics, 404, Chapman & Hall/CRC, Boca Raton, FL, 1999.

[10]

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.

[11]

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.

[12]

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.

[13]

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

[14]

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.

[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), 651-672. doi: 10.1007/s00205-012-0517-4.

[16]

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

[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), 183-236. doi: 10.1007/s00205-005-0400-7.

[18]

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.

[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), 2-15.

[20]

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

[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), 565-616. doi: 10.1142/S021820251250056X.

[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, Physica B: Condensed Matter, 435 (2014), 31-33.

[23]

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

[24]

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.

[25]

P. Krejčí and J. Sprekels, Clamped elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators, Discrete Contin. Dyn. Syst.-S, 1 (2008), 283-292. doi: 10.3934/dcdss.2008.1.283.

[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), 31-48.

[27]

M. Liero and A. Mielke, An evolutionary elastoplastic plate model derived via Gamma convergence, Math. Models Meth. Appl. Sci. (M3AS), 21 (2011), 1961-1986. doi: 10.1142/S0218202511005611.

[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) 437-457. doi: 10.1007/s00030-011-0137-y.

[29]

A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear inelasticity, Math. Models Meth. Appl. Sci. (M3AS), 16 (2006), 177-209. doi: 10.1142/S021820250600111X.

[30]

A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA, Nonlinear Differ. Equ. Appl., 11 (2004), 151-189. doi: 10.1007/s00030-003-1052-7.

[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), 369-384. doi: 10.1016/S0997-7538(03)00044-5.

[32]

D. Percivale, Perfectly plastic plates: a variational definition, J. Reine Angew. Math., 411 (1990), 39-50. doi: 10.1515/crll.1990.411.39.

[33]

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

[34]

E. Rocca and R. Rossi, A degenerating PDE system for phase transitions and damage, Math. Models Methods Appl. Sci., 24 (2014), 1265-1341. doi: 10.1142/S021820251450002X.

[35]

R. Rossi and T. Roubíček, Adhesive contact delaminating at mixed mode, its thermodynamics and analysis, Interfaces Free Bound, 14 (2013), 1-37. doi: 10.4171/IFB/293.

[36]

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

[37]

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.

[38]

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.

[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), 88-112. doi: 10.1002/zamm.200900243.

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]

O. V. Besov, V. P. Il'in and S. M. Nikol'skič, Integral Representations of Functions and Imbedding Theorems, Scripta Series in Mathematics, Halsted Press (John Wiley & Sons), New York-Toronto, Ont.-London, 1978 (Vol. I), 1979 (Vol. II). Russian version Nauka, Moscow, 1975.

[3]

E. Bonetti and G. Bonfanti, Well-posedness results for a model of damage in thermoviscoelastic materials, Ann. I. H. Poincaré, 25 (2008), 1187-1208. doi: 10.1016/j.anihpc.2007.05.009.

[4]

E. Bonetti and G. Schimperna, Local existence for Frémond's model of damage in elastic materials, Continuum Mech. Therm., 16 (2004), 319-335. doi: 10.1007/s00161-003-0152-2.

[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), 91-116. doi: 10.1016/j.jde.2005.04.015.

[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, Physica B: Condensed Matter, 435 (2014), 1-3.

[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), 705-737.

[8]

M. Brokate and A. M. Khludnev, Existence of solutions in the Prandtl-Reuss theory of elastoplastic plates, Adv. Math. Sci Appl., 10 (2000), 399-415.

[9]

P. Drábek, P. Krejčí and P. Takáč, Nonlinear Differential Equations, Research Notes in Mathematics, 404, Chapman & Hall/CRC, Boca Raton, FL, 1999.

[10]

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.

[11]

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.

[12]

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.

[13]

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

[14]

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.

[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), 651-672. doi: 10.1007/s00205-012-0517-4.

[16]

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

[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), 183-236. doi: 10.1007/s00205-005-0400-7.

[18]

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.

[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), 2-15.

[20]

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

[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), 565-616. doi: 10.1142/S021820251250056X.

[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, Physica B: Condensed Matter, 435 (2014), 31-33.

[23]

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

[24]

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.

[25]

P. Krejčí and J. Sprekels, Clamped elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators, Discrete Contin. Dyn. Syst.-S, 1 (2008), 283-292. doi: 10.3934/dcdss.2008.1.283.

[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), 31-48.

[27]

M. Liero and A. Mielke, An evolutionary elastoplastic plate model derived via Gamma convergence, Math. Models Meth. Appl. Sci. (M3AS), 21 (2011), 1961-1986. doi: 10.1142/S0218202511005611.

[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) 437-457. doi: 10.1007/s00030-011-0137-y.

[29]

A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear inelasticity, Math. Models Meth. Appl. Sci. (M3AS), 16 (2006), 177-209. doi: 10.1142/S021820250600111X.

[30]

A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA, Nonlinear Differ. Equ. Appl., 11 (2004), 151-189. doi: 10.1007/s00030-003-1052-7.

[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), 369-384. doi: 10.1016/S0997-7538(03)00044-5.

[32]

D. Percivale, Perfectly plastic plates: a variational definition, J. Reine Angew. Math., 411 (1990), 39-50. doi: 10.1515/crll.1990.411.39.

[33]

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

[34]

E. Rocca and R. Rossi, A degenerating PDE system for phase transitions and damage, Math. Models Methods Appl. Sci., 24 (2014), 1265-1341. doi: 10.1142/S021820251450002X.

[35]

R. Rossi and T. Roubíček, Adhesive contact delaminating at mixed mode, its thermodynamics and analysis, Interfaces Free Bound, 14 (2013), 1-37. doi: 10.4171/IFB/293.

[36]

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

[37]

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.

[38]

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.

[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), 88-112. doi: 10.1002/zamm.200900243.

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