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

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.
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, 29 (2008), 1.

[2]

T. Aiki and N. Kenmochi, Some models for shape memory alloys,, in, 17 (2002), 144.

[3]

V. Barbu, "Optimal Control of Variational Inequalities,", Pitman, (1984).

[4]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Springer Verlag, (1996).

[5]

J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis,", Springer-Verlag, (1980).

show all references

References:
[1]

T. Aiki, A mathematical model for a valve made of a spring of a shape memory alloy,, in, 29 (2008), 1.

[2]

T. Aiki and N. Kenmochi, Some models for shape memory alloys,, in, 17 (2002), 144.

[3]

V. Barbu, "Optimal Control of Variational Inequalities,", Pitman, (1984).

[4]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Springer Verlag, (1996).

[5]

J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis,", Springer-Verlag, (1980).

[1]

Diego Grandi, Ulisse Stefanelli. The Souza-Auricchio model for shape-memory alloys. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 723-747. doi: 10.3934/dcdss.2015.8.723

[2]

Tomáš Roubíček. Modelling of thermodynamics of martensitic transformation in shape-memory alloys. Conference Publications, 2007, 2007 (Special) : 892-902. doi: 10.3934/proc.2007.2007.892

[3]

Michel Frémond, Elisabetta Rocca. A model for shape memory alloys with the possibility of voids. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1633-1659. doi: 10.3934/dcds.2010.27.1633

[4]

Peiying Chen. Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations. Electronic Research Announcements, 2017, 24: 38-52. doi: 10.3934/era.2017.24.005

[5]

Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026

[6]

Robert Jensen, Andrzej Świech. Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE. Communications on Pure & Applied Analysis, 2005, 4 (1) : 199-207. doi: 10.3934/cpaa.2005.4.187

[7]

Yen-Lin Wu, Zhi-You Chen, Jann-Long Chern, Y. Kabeya. Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 949-960. doi: 10.3934/cpaa.2014.13.949

[8]

Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. Thermal control of the Souza-Auricchio model for shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 369-386. doi: 10.3934/dcdss.2013.6.369

[9]

Linxiang Wang, Roderick Melnik. Dynamics of shape memory alloys patches with mechanically induced transformations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1237-1252. doi: 10.3934/dcds.2006.15.1237

[10]

Shuji Yoshikawa, Irena Pawłow, Wojciech M. Zajączkowski. A quasilinear thermoviscoelastic system for shape memory alloys with temperature dependent specific heat. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1093-1115. doi: 10.3934/cpaa.2009.8.1093

[11]

Alessia Berti, Claudio Giorgi, Elena Vuk. Free energies and pseudo-elastic transitions for shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 293-316. doi: 10.3934/dcdss.2013.6.293

[12]

Ferdinando Auricchio, Elena Bonetti. A new "flexible" 3D macroscopic model for shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 277-291. doi: 10.3934/dcdss.2013.6.277

[13]

Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73

[14]

Igor Chueshov, Irena Lasiecka. Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 777-809. doi: 10.3934/dcds.2006.15.777

[15]

Toyohiko Aiki. On the existence of a weak solution to a free boundary problem for a model of a shape memory alloy spring. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 1-13. doi: 10.3934/dcdss.2012.5.1

[16]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017

[17]

Chiara Corsato, Colette De Coster, Pierpaolo Omari. Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape. Conference Publications, 2015, 2015 (special) : 297-303. doi: 10.3934/proc.2015.0297

[18]

Xia Huang. Stable weak solutions of weighted nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 293-305. doi: 10.3934/cpaa.2014.13.293

[19]

Takashi Suzuki, Shuji Yoshikawa. Stability of the steady state for multi-dimensional thermoelastic systems of shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 209-217. doi: 10.3934/dcdss.2012.5.209

[20]

Ken Shirakawa. Asymptotic stability for dynamical systems associated with the one-dimensional Frémond model of shape memory alloys. Conference Publications, 2003, 2003 (Special) : 798-808. doi: 10.3934/proc.2003.2003.798

2017 Impact Factor: 0.561

Metrics

  • PDF downloads (2)
  • HTML views (0)
  • Cited by (1)

[Back to Top]