| Title:
|
Asymptotic behaviour for a phase-field model with hysteresis in one-dimensional thermo-visco-plasticity (English) |
| Author:
|
Klein, Olaf |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
49 |
| Issue:
|
4 |
| Year:
|
2004 |
| Pages:
|
309-341 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The asymptotic behaviour for $t \rightarrow \infty $ of the solutions to a one-dimensional model for thermo-visco-plastic behaviour is investigated in this paper. The model consists of a coupled system of nonlinear partial differential equations, representing the equation of motion, the balance of the internal energy, and a phase evolution equation, determining the evolution of a phase variable. The phase evolution equation can be used to deal with relaxation processes. Rate-independent hysteresis effects in the strain-stress law and also in the phase evolution equation are described by using the mathematical theory of hysteresis operators. (English) |
| Keyword:
|
phase-field system |
| Keyword:
|
phase transition |
| Keyword:
|
hysteresis operator |
| Keyword:
|
thermo-visco-plasticity |
| Keyword:
|
asymptotic behaviour |
| MSC:
|
34C55 |
| MSC:
|
35B40 |
| MSC:
|
35K60 |
| MSC:
|
47J40 |
| MSC:
|
74K05 |
| MSC:
|
74N30 |
| idZBL:
|
Zbl 1099.74051 |
| idMR:
|
MR2076488 |
| DOI:
|
10.1007/s10492-004-6402-1 |
| . |
| Date available:
|
2009-09-22T18:18:22Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134571 |
| . |
| Reference:
|
[1] G. Andrews: On the existence of solutions to the equation $u_{tt} = u_{xxt} + \sigma (u_x)_x$.J. Differential Equations 35 (1980), 200–231. Zbl 0415.35018, MR 0561978, 10.1016/0022-0396(80)90040-6 |
| Reference:
|
[2] G. Bourbon, P. Vacher, C. Lexcellent: Comportement thermomécanique d’un alliage polycristallin à mémoire de forme Cu-Al-Ni.Phys. Stat. Sol. (A) 125 (1991), 179–190. 10.1002/pssa.2211250115 |
| Reference:
|
[3] M. Brokate, J. Sprekels: Hysteresis and Phase Transitions.Springer-Verlag, New York, 1996. MR 1411908 |
| Reference:
|
[4] : Shape Memory Materials and their Applications. Vol. 394–395 of Materials Science Forum.Y. Y. Chu, L. C. Zhao (eds.), Trans Tech Publications, Switzerland, 2002. |
| Reference:
|
[5] C. M. Dafermos, L. Hsiao: Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity.Nonlinear Anal. 6 (1982), 434–454. MR 0661710 |
| Reference:
|
[6] F. Falk: Model free energy, mechanics and thermodynamics of shape memory alloys.Acta Met. 28 (1980), 1773–1780. 10.1016/0001-6160(80)90030-9 |
| Reference:
|
[7] F. Falk: Ginzburg-Landau theory of static domain walls in shape-memory alloys.Z. Physik B—Condensed Matter 51 (1983), 177–185. 10.1007/BF01308772 |
| Reference:
|
[8] F. Falk: Pseudoelastic stress-strain curves of polycrystalline shape memory alloys calculated from single crystal data.Internat. J. Engrg. Sci. 27 (1989), 277–284. 10.1016/0020-7225(89)90115-8 |
| Reference:
|
[9] M. Frémond: Non-Smooth Thermomechanics.Springer-Verlag, Berlin, 2002. Zbl 0990.80001, MR 1885252 |
| Reference:
|
[10] M. Frémond, S. Miyazaki: Shape Memory Alloys, CISM Courses and Lectures, Vol. 351.Springer-Verlag, , 1996. |
| Reference:
|
[11] G. Gilardi, P. Krejčí, J. Sprekels: Hysteresis in phase-field models with thermal memory.Math. Methods Appl. Sci. 23 (2000), 909–922. MR 1765906, 10.1002/1099-1476(20000710)23:10<909::AID-MMA142>3.0.CO;2-E |
| Reference:
|
[12] T. Jurke, O. Klein: Existence results for a phase-field model in one-dimensional thermo-visco-plasticity involving unbounded hysteresis operators.In preparation. |
| Reference:
|
[13] O. Klein, P. Krejčí: Outwards pointing hysteresis operators and and asymptotic behaviour of evolution equations.Nonlinear Anal. Real World Appl. 4 (2003), 755–785. MR 1978561 |
| Reference:
|
[14] M. Krasnosel’skii, A. Pokrovskii: Systems with Hysteresis.Springer-Verlag, Heidelberg, 1989; Russian edition: Nauka, Moscow, 1983. MR 0742931 |
| Reference:
|
[15] P. Krejčí: Hysteresis, Convexity and Dissipation in Hyperbolic Equations.Gakuto Internat. Ser. Math. Sci. Appl. Vol. 8, Gakkōtosho, Tokyo, 1996. MR 2466538 |
| Reference:
|
[16] P. Krejčí, J. Sprekels: 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:
|
[17] P. Krejčí, J. Sprekels: Hysteresis operators in phase-field models of Penrose-Fife type.Appl. Math. 43 (1998), 207–222. MR 1620620, 10.1023/A:1023276524286 |
| Reference:
|
[18] P. Krejčí, J. Sprekels: Temperature-dependent hysteresis in one-dimensional thermovisco-elastoplasticity.Appl. Math. 43 (1998), 173–205. MR 1620624, 10.1023/A:1023224507448 |
| Reference:
|
[19] P. Krejčí, J. Sprekels: A hysteresis approach to phase-field models.Nonlinear Anal. Ser. A 39 (2000), 569–586. MR 1727271, 10.1016/S0362-546X(98)00222-3 |
| Reference:
|
[20] P. Krejčí, J. Sprekels: Phase-field models with hysteresis.J. Math. Anal. Appl. 252 (2000), 198–219. MR 1797852 |
| Reference:
|
[21] P. Krejčí, J. Sprekels: Phase-field systems and vector hysteresis operators.In: Free Boundary Problems: Theory and Applications, II (Chiba, 1999), Gakkōtosho, Tokyo, 2000, pp. 295–310. MR 1794360 |
| Reference:
|
[22] P. Krejčí, J. Sprekels: On a class of multi-dimensional Prandtl-Ishlinskii operators.Physica B 306 (2001), 185–190. 10.1016/S0921-4526(01)01001-8 |
| Reference:
|
[23] P. Krejčí, J. Sprekels: Phase-field systems for multi-dimensional Prandtl-Ishlinskii operators with non-polyhedral characteristics.Math. Methods Appl. Sci. 25 (2002), 309–325. MR 1875705, 10.1002/mma.288 |
| Reference:
|
[24] P. Krejčí, J. Sprekels, and U. Stefanelli: One-dimensional thermo-visco-plastic processes with hysteresis and phase transitions.Adv. Math. Sci. Appl. 13 (2003), 695–712. MR 2029939 |
| Reference:
|
[25] P. Krejčí, J. Sprekels, and U. Stefanelli: Phase-field models with hysteresis in one-dimensional thermoviscoplasticity.SIAM J. Math. Anal. 34 (2002), 409–434. MR 1951781, 10.1137/S0036141001387604 |
| Reference:
|
[26] P. Krejčí, J. Sprekels, and S. Zheng: Existence and asymptotic behaviour in phase-field models with hysteresis.In: Lectures on Applied Mathematics (Munich, 1999), Springer, Berlin, 2000, pp. 77–88. MR 1767764 |
| Reference:
|
[27] P. Krejčí, J. Sprekels, and S. Zheng: Asymptotic behaviour for a phase-field system with hysteresis.J. Differential Equations 175 (2001), 88–107. MR 1849225, 10.1006/jdeq.2001.3950 |
| Reference:
|
[28] P. Krejčí: Resonance in Preisach systems.Appl. Math. 45 (2000), 439–468. MR 1800964, 10.1023/A:1022333500777 |
| Reference:
|
[29] I. Müller: Grundzüge der Thermodynamik, 3. ed.Springer-Verlag, Berlin-New York, 2001. |
| Reference:
|
[30] I. Müller, K. Wilmanski: A model for phase transition in pseudoelastic bodies.Il Nuovo Cimento 57B (1980), 283–318. |
| Reference:
|
[31] : Space Memory Materials, first paperback.K. Otsuka, C. Wayman (eds.), Cambridge University Press, Cambridge, 1999. |
| Reference:
|
[32] R. L. Pego: Phase transitions in onedimensional nonlinear viscoelasticity: Admissibility and stability.Arch. Ration. Mech. Anal. 97 (1987), 353–394. MR 0865845, 10.1007/BF00280411 |
| Reference:
|
[33] R. Racke, S. Zheng: Global existence and asymptotic behavior in nonlinear thermoviscoelasticity.J. Differential Equations 134 (1997), 46–67. MR 1429091, 10.1006/jdeq.1996.3216 |
| Reference:
|
[34] W. Shen, S. Zheng: On the coupled Cahn-Hilliard equations.Comm. Partial Differential Equations 18 (1993), 701–727. MR 1214877, 10.1080/03605309308820946 |
| Reference:
|
[35] J. Sprekels, S. Zheng, P. Zhu: Asymptotic behavior of the solutions to a Landau-Ginzburg system with viscosity for martensitic phase transitions in shape memory alloys.SIAM J. Math. Anal. 29 (1998), 69–84 (electronic). MR 1617175, 10.1137/S0036141096297698 |
| Reference:
|
[36] A. Visintin: Differential Models of Hysteresis.Springer-Verlag, Berlin, 1994. Zbl 0820.35004, MR 1329094 |
| Reference:
|
[37] A. Visintin: Models of Phase Transitions. Progress in Nonlinear Differential Equations and Their Applications, Vol. 28.Birkhäuser-Verlag, Boston, 1996. MR 1423808 |
| Reference:
|
[38] K. Wilmanski: Symmetric model of stress-strain hysteresis loops in shape memory alloys.Internat J. Engrg. Sci. 31 (1993), 1121–1138. Zbl 0774.73016, 10.1016/0020-7225(93)90086-A |
| Reference:
|
[39] S. Zheng: Nonlinear Parabolic Equations and Hyperbolic—Parabolic Coupled Systems. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 76.Longman, New York, 1995. MR 1375458 |
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