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February  2012, 5(1): 15-28. doi: 10.3934/dcdss.2012.5.15

On a one-dimensional shape-memory alloy model in its fast-temperature-activation limit

Toyohiko Aiki 1, , Martijn Anthonissen 2, and  Adrian Muntean 3,

1. 

Department of Mathematics, Faculty of Education, Gifu University, Yanagido 1-1, Gifu, 501-1193
2. 

Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, Netherlands
3. 

Department of Mathematics and Computer Science, Eindhoven University of Technology, Institute of Complex Molecular Systems (ICMS), P.O. Box 513, 5600 MB Eindhoven

Received  May 2009 Revised  November 2009 Published  February 2011

We study a one-dimensional model describing the motion of a shape-memory alloy spring at a small characteristic time scale, called here fast-temperature-activation limit. At this level, the standard Falk's model reduces to a nonlinear elliptic partial differential equation (PDE) with Newton boundary condition. We show existence and uniqueness of a bounded weak solution and approximate this numerically. Interestingly, in spite of the nonlinearity of the model, the approximate solution exhibits nearly a linear profile. Finally, we extend the reduced model to the simplest PDE system for shape memory alloys that can capture oscillations and then damp out these oscillations numerically. The numerical results for both limiting cases show excellent agreement. The graphs show that the valve opens in an instant, which is realistic behavior of the free boundary.
Keywords: modeling, nonlinear elliptic equation., existence and uniqueness of weak solutions, Shape-memory alloys.
Mathematics Subject Classification: Primary: 35J66; Secondary: 65N06, 74D9.
Citation: Toyohiko Aiki, Martijn Anthonissen, Adrian Muntean. On a one-dimensional shape-memory alloy model in its fast-temperature-activation limit. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 15-28. doi: 10.3934/dcdss.2012.5.15
References:
[1]

T. Aiki, A mathematical model for a valve made of a spring of a shape memory alloy, in "Mathematical Aspects of Modeling Structure Formation Phenomena,'' GAKUTO Internat. Ser. Math. Sci. Appl., 29 (2008), 1-18.  Google Scholar

[2]

T. Aiki and N. Kenmochi, Some models for shape memory alloys, in "Mathematical Aspects of Modeling Structure Formation Phenomena,'' GAKUTO Internat. Ser. Math. Sci. Appl., 17 (2002), 144-162.  Google Scholar

[3]

V. Barbu, "Optimal Control of Variational Inequalities," Pitman, Boston, 1984.  Google Scholar

[4]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Springer Verlag, Berlin, 1996.  Google Scholar

[5]

J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis," Springer-Verlag, NY, Berlin, 1980.  Google Scholar

show all references

References:
[1]

T. Aiki, A mathematical model for a valve made of a spring of a shape memory alloy, in "Mathematical Aspects of Modeling Structure Formation Phenomena,'' GAKUTO Internat. Ser. Math. Sci. Appl., 29 (2008), 1-18.  Google Scholar

[2]

T. Aiki and N. Kenmochi, Some models for shape memory alloys, in "Mathematical Aspects of Modeling Structure Formation Phenomena,'' GAKUTO Internat. Ser. Math. Sci. Appl., 17 (2002), 144-162.  Google Scholar

[3]

V. Barbu, "Optimal Control of Variational Inequalities," Pitman, Boston, 1984.  Google Scholar

[4]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Springer Verlag, Berlin, 1996.  Google Scholar

[5]

J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis," Springer-Verlag, NY, Berlin, 1980.  Google Scholar

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