A Numerical Study of Parameter Sensitivities in Landau-Ginzburg Models of Phase Transitions in Shape Memory Alloys

Abstract

In this paper we consider a one-dimensional mathematical model for the dynamics of phase transitions in Shape Memory Alloys (SMAs). We employ a finite element scheme to investi gate the dependence of solutions on the parameters that define the Landau-Devonshire potential.

Get full access to this article

View all access options for this article.

References

Achembach, M. and I. Muller. 1982. "A Model for Shape Memory", Journal de Physique, Colloque C4, Supplément au No 12, Tome 43, pp. C4 163-167.

Achembach, M. and I. Müller. 1983. "Creep and Yield in Martensitic Transformations' ; Igenieur-Archiv, 53:73-83.

Achembach, M. and I. Muller. 1985. "Shape Memory as a Thermally Activated Process", in Plasticity Today: Modelling Methods and Applications , A. Sawczuk and G Bianchi, eds., Elsevier Applied Science Pub., pp. 515-531.

Burns, J.A. and R.D. Spies. 1991. "Modelling for Control of Shape Memory Alloys", Proceeding of the 30th IEEE Conference on Decision and Control, Brighton, England, December 1991, pp. 2334-2339.

Falk, F. 1960. "Wie hang die Freie Energie einer Memory-Legierung von der Verzerrung und der Teperatur ab?" ZAMM , 60:T118-T120.

Falk, F. 1980. "Model Free Energy, Mechanics and Thermodynamics of Shape Memory Alloys", Acta Metallurgica , 28:1773-1780.

Falk, F. 1983a. "Ginzburg-Landau Theory and Solitary Waves in Shape Memory Alloys", Z. Physik B-Condensed Matter , 54:177-185.

Falk, F. 1983b. "One Dimensional Model of Shape Memory Alloys ", Arch. Mech. , Warszawa, 35:63-84.

Falk, F. 1984. "Ginzburg-Landau Theory of Static Domain Walls in Shape Memory Alloys", Z. Physik B-Condensed Matter , 51:159-167.

10.

Funakubo, H. 1987. "Shape Memory Alloys", Precision Machinery and Robotics, Vol. 1 , translated from the Japanese by J. B. Kennedy, Gordon and Breach Science Publishers.

11.

Hoffman, K.H. and Z. Songmu. 1986. "Uniqueness for Nonlinear Coupled Equations Arising from Alloy Mechanism", Preprint 118, Institut fur Mathematik, Ausburg.

12.

Hoffman, K.H. and Z. Songmu. 1988. "Uniqueness for Structural Phase Transitions in Shape Memory Alloys", Mathematical Methods in the Applied Sciences, 10:145-151.

13.

Lohman, R. and I. Muller. 1986. "A Model for the Qualitative Description of Martensitic Transformations in Memory Alloys" in Phase Transformations, E. Aifantis and J. Gittus, Elsevier Applied Science Pub., pp. 55-75.

14.

Muller, I. 1979. "A Model for a Body with Shape Memory", Arch. Rational Mechanics Anal. , 70:61-67.

15.

Müller, I. and P. Villagio. 1977. "A Model for an Elastic-Plastic Body ", Arch. Rational Mechanics Anal. , 65(1):25-46.

16.

Müller, I. and K. Wilmanski. 1980. "A Model for Phase Transitions in Pseudoelastic Bodies", Il Nuovo Cimento , 57B(2):283-318.

17.

Niezgodka, M. and J. Sprekels. 1988. "Existence of Solutions for a Mathematical Model of Structural Phase Transitions in Shape Memory Alloys", Mathematical Methods in the Applied Sciences, 10:197-223.

18.

Niezgodka, M. , Z. Songmu and J. Sprekels. 1988. "Global Solutions to a Model of Structural Phase Transitions in Shape Memory Alloys", Journal of Mathematical Analysis and Appliations, 130:39-54.

19.

Songmu, Z. 1989. "Global Solutions to the Thermomechanical Equations with Non-Convex Landau-Ginzburg Free Energy", Journal of Applied Mathematics and Physics (ZAMP), 40:111-127.

20.

Songmu, Z. and J. Sprekels. 1989. "Global Solutions to the Equations of a Ginzburg-Landau Theory for Structural Phase Transitions in Shape Memory Alloys", Physica D. , 39:59-76.

21.

Spies, R.D. 1992. "A State-Space Approach to a One-Dimensional Mathematical Model for the Dynamics of Phase Transitions in Pseudoelastic Materials", Preprint #1054, Institute for Mathematics and Its Applications, University of Minnesota. Submitted to the Journal of Math. Anal. and Applic

22.

. Sprekels, J. 1989. "Global Existence for Thermomechanical Processes with Nonconvex Free Energies of Ginzburg-Landau Form", Journal of Mathematical Analysis and Applications, 141:333-348.

23.

Wilmanski, K. 1983. "Propagation of the Interface in Stress-Induced Austenite-Martensite Transformation", Ingenieur-Archiv ., 53:291-301.