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Free vibrations in space of the single mode for the Kirchhoff string
Non-isothermal cyclic fatigue in an oscillating elastoplastic beam
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 |
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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