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

Toyohiko Aiki; Martijn Anthonissen; Adrian Muntean

doi:10.3934/dcdss.2012.5.15

Article Contents

2012, Volume 5, Issue 1: 15-28. Doi: 10.3934/dcdss.2012.5.15

This issue Previous Article On the existence of a weak solution to a free boundary problem for a model of a shape memory alloy spring Next Article Modelling phase transitions via Young measures

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

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
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: April 30, 2009

Revised: October 31, 2009

Published: January 2012

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Abstract

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:

Shape-memory alloys,
modeling,
existence and uniqueness of weak solutions,
nonlinear elliptic equation.

Mathematics Subject Classification: Primary: 35J66; Secondary: 65N06, 74D99.

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References

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[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.
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