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Article Contents

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

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

 Citation:

•  [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. [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. [3] V. Barbu, "Optimal Control of Variational Inequalities," Pitman, Boston, 1984. [4] M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Springer Verlag, Berlin, 1996. [5] J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis," Springer-Verlag, NY, Berlin, 1980.