| Title:
|
Temperature-dependent hysteresis in one-dimensional thermovisco-elastoplasticity (English) |
| Author:
|
Krejčí, Pavel |
| Author:
|
Sprekels, Jürgen |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
43 |
| Issue:
|
3 |
| Year:
|
1998 |
| Pages:
|
173-205 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we develop a thermodynamically consistent description of the uniaxial behavior of thermovisco-elastoplastic materials for which the total stress $\sigma $ contains, in addition to elastic, viscous and thermic contributions, a plastic component $\sigma ^p$ of the form $\sigma ^p(x,t)={\mathcal P}[\varepsilon ,\theta (x,t)](x,t)$. Here $\varepsilon $ and $\theta $ are the fields of strain and absolute temperature, respectively, and $\lbrace {\mathcal P}[\cdot ,\theta ]\rbrace _{\theta > 0}$ denotes a family of (rate-independent) hysteresis operators of Prandtl-Ishlinskii type, parametrized by the absolute temperature. The system of momentum and energy balance equations governing the space-time evolution of the material forms a system of two highly nonlinearly coupled partial differential equations involving partial derivatives of hysteretic nonlinearities at different places. It is shown that an initial-boundary value problem for this system admits a unique global strong solution which depends continuously on the data. (English) |
| Keyword:
|
thermoplasticity |
| Keyword:
|
viscoelasticity |
| Keyword:
|
hysteresis |
| Keyword:
|
Prandtl-Ishlinskii operator |
| Keyword:
|
PDEs with hysteresis |
| Keyword:
|
thermodynamical consistency |
| MSC:
|
35G25 |
| MSC:
|
73B05 |
| MSC:
|
73B30 |
| MSC:
|
73E60 |
| MSC:
|
74N30 |
| idZBL:
|
Zbl 0940.35052 |
| idMR:
|
MR1620624 |
| DOI:
|
10.1023/A:1023224507448 |
| . |
| Date available:
|
2009-09-22T17:57:43Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134384 |
| . |
| Reference:
|
[BS1] Brokate, M., Sprekels, J.: Existence and optimal control of mechanical processes with hysteresis in viscous solids.IMA J. Appl. Math. 43 (1989), 219–229. MR 1042633, 10.1093/imamat/43.3.219 |
| Reference:
|
[BS] Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions.Springer-Verlag, New York, 1996. MR 1411908 |
| Reference:
|
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| Reference:
|
[DH] Dafermos, C. M., Hsiao, L.: Global smooth thermomechanical processes in one-dimensional thermoviscoelasticity.Nonlin. Anal. TMA 6 (1982), 435–454. MR 0661710, 10.1016/0362-546X(82)90058-X |
| Reference:
|
[Is] Ishlinskii, A. Yu.: Some applications of statistical methods to describing deformations of bodies.Izv. AN SSSR, Techn. Ser. 9 (1944), 583–590. |
| Reference:
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[KP] Krasnosel’skii, M. A., Pokrovskii, A. V.: Systems with Hysteresis.Springer-Verlag, Heidelberg, 1989. MR 0987431 |
| Reference:
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| Reference:
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| Reference:
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| Reference:
|
[KS] Krejčí, P., Sprekels, J.: On a system of nonlinear PDEs with temperature-dependent hysteresis in one-dimensional thermoplasticity.J. Math. Anal. Appl. 209 (1997), 25–46. MR 1444509, 10.1006/jmaa.1997.5304 |
| Reference:
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| Reference:
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| Reference:
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| Reference:
|
[P] Prandtl, L.: Ein Gedankenmodell zur kinetischen Theorie der festen Körper.Z. Ang. Math. Mech. 8 (1928), 85–106. 10.1002/zamm.19280080202 |
| Reference:
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[S] Šilhavý, M.: The Mechanics and Thermodynamics of Continuous Media.Springer, Berlin-Heidelberg, 1996. MR 1423807 |
| . |
