Mathematical study of an evolution problem describing the thermomechanical process in shape memory alloys

Pierluigi Colli

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1991)

  • Volume: 2, Issue: 1, page 55-64
  • ISSN: 1120-6330

Abstract

top
In this paper we prove existence, uniqueness, and continuous dependence for a one-dimensional time-dependent problem related to a thermo-mechanical model of structural phase transitions in solids. This model assumes the free energy depending on temperature, macroscopic deformation and also on the proportions of the phases. Here we neglect regularizing terms in the momentum balance equation and in the constitutive laws for the phase proportions.

How to cite

top

Colli, Pierluigi. "Mathematical study of an evolution problem describing the thermomechanical process in shape memory alloys." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 2.1 (1991): 55-64. <http://eudml.org/doc/244136>.

@article{Colli1991,
abstract = {In this paper we prove existence, uniqueness, and continuous dependence for a one-dimensional time-dependent problem related to a thermo-mechanical model of structural phase transitions in solids. This model assumes the free energy depending on temperature, macroscopic deformation and also on the proportions of the phases. Here we neglect regularizing terms in the momentum balance equation and in the constitutive laws for the phase proportions.},
author = {Colli, Pierluigi},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Shape memory alloys; Thermo-mechanical model; Initial-boundary value problem; Existence and uniqueness; austenite-martensite phase transition; microscopic scale; contraction mapping principle; regularity},
language = {eng},
month = {3},
number = {1},
pages = {55-64},
publisher = {Accademia Nazionale dei Lincei},
title = {Mathematical study of an evolution problem describing the thermomechanical process in shape memory alloys},
url = {http://eudml.org/doc/244136},
volume = {2},
year = {1991},
}

TY - JOUR
AU - Colli, Pierluigi
TI - Mathematical study of an evolution problem describing the thermomechanical process in shape memory alloys
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1991/3//
PB - Accademia Nazionale dei Lincei
VL - 2
IS - 1
SP - 55
EP - 64
AB - In this paper we prove existence, uniqueness, and continuous dependence for a one-dimensional time-dependent problem related to a thermo-mechanical model of structural phase transitions in solids. This model assumes the free energy depending on temperature, macroscopic deformation and also on the proportions of the phases. Here we neglect regularizing terms in the momentum balance equation and in the constitutive laws for the phase proportions.
LA - eng
KW - Shape memory alloys; Thermo-mechanical model; Initial-boundary value problem; Existence and uniqueness; austenite-martensite phase transition; microscopic scale; contraction mapping principle; regularity
UR - http://eudml.org/doc/244136
ER -

References

top
  1. ACHENBACH, M. - MÜLLER, I., Model for shape memory. J. Physique, C4 Suppl. 12, 43, 1982, 163-167. 
  2. ALT, H. W. - HOFFMANN, K. H. - NIEZGÓDKA, M. - SPREKELS, J., A numerical study of structural phase transitions in shape memory alloys. Preprint n. 90, Institut fur Mathematik, Universitàt Augsburg, 1985. 
  3. S. ANTMAN - J. L. ERICKSEN - D. KINDERLEHRER - I. MÜLLER (eds.), Metastability and incompletely posed problems. Springer-Verlag, Berlin1987. MR870005DOI10.1007/978-1-4613-8704-6
  4. BLANCHARD, D. - FRÉMOND, M. - VISINTIN, A., Phase change with dissipation. In: H. D. BUY - Q. S. NGUYEN (eds.), Thermomechanical coupling in solids. North-Holland, Amsterdam1987, 411-418. Zbl0665.73079MR934936
  5. BREZIS, H., Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam1973. Zbl0252.47055MR348562
  6. COLLI, P., An evolution problem related to shape memory alloys. In: J. F. RODRIGUES (ed.), Mathematical models for phase change problems. International Series of Numerical Mathematics, 88, Birkhäuser Verlag, Basel1989, 75-88. Zbl0692.73004MR1038065
  7. COLLI, P. - FREMONO, M. - VISINTIN, A., Thermo-mechanical evolution of shape memory alloys. Quart. Appl. Math., 48, 1990, 31-47. Zbl0693.73003MR1040232
  8. FALK, F., Martensitic domain boundaries in shape-memory alloys as solitary waves. J. Physique, C4 Suppl. 12, 43, 1982, 3-15. 
  9. FREMONO, M., Matériaux a mémoire de forme. C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre, 304, 1987, 239-244. 
  10. FREMOND, M. - VISINTIN, A., Dissipation dans le changement de phase. Surfusion. Changement de phase irreversible. C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre, 301, 1985, 1265-1268. Zbl0582.73007MR880589
  11. HERGET, G. - MÜLLNER, M. - ECKOLD, G. - TIETZE, H. - ASSMUS, W., Strukturelle phasenübergänge der formgedächtnislegierung NiTi. Jahresbericht 1987 des Inst, fur Kernphysik, Johann Wolfgang Göthe Universität, Frankfurt/Main1987, 77. 
  12. HOFFMANN, K. H. - NIEZGODKA, M., Mathematical models of dynamical martensitic transformations in shape memory alloys. Preprint n. 201, Institut für Mathematik, Universität Augsburg, 1990. 
  13. HOFFMANN, K. H. - NIEZGODKA, M. - ZHENG, S., Existence and uniqueness to an extended model of the dynamical developments in shape memory alloys. Nonlinear Analysis, to appear. Zbl0728.35055
  14. NIEZGODKA, M. - SPREKELS, J., Existence of solutions for a mathematical model of structural phase transitions in shape memory alloys. Math. Methods Appl. Sci., 10, 1988, 197-223. Zbl0668.35013MR949654DOI10.1002/mma.1670100302
  15. SPREKELS, J. - ZHENG, S., Global solutions of a Ginzburg-Landau theory for structural phase transitions in shape memory alloys. Physica D39, 1989, 59-76. Zbl0696.35145MR1021182DOI10.1016/0167-2789(89)90039-0
  16. TIIHONEN, T., A numerical approach to a shape memory model. Preprint n. 98, Institut für Mathematik, Universität Augsburg, 1988. MR1230883

NotesEmbed ?

top

You must be logged in to post comments.