April  2011, 4(2): 239-246. doi: 10.3934/dcdss.2011.4.239

The position of the joint of shape memory alloy and bias springs

1. 

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

Received  January 2009 Revised  June 2009 Published  November 2010

In our previous work we proposed the mathematical model for a device made of the standard spring and the shape memory alloy spring. The model was given by the system of partial differential equations with the dynamic boundary condition. Also, we have proved the existence and the uniquess theorems for the model. The purpose of this paper is to improve the existence theorem.
Citation: Toyohiko Aiki. The position of the joint of shape memory alloy and bias springs. Discrete and Continuous Dynamical Systems - S, 2011, 4 (2) : 239-246. doi: 10.3934/dcdss.2011.4.239
References:
[1]

T. Aiki, Multi-dimensional Stefan problems with dynamic boundary conditions, Applicable Analysis, Applicable Analysis, 56 (1995), 71-94. doi: 10.1080/00036819508840311.

[2]

T. Aiki, Multi-dimensional two-phase Stefan problems with nonlinear dynamic boundary conditions, in "Nonlinear Analysis and Applications," Gakuto International Series, Mathematical Sciences and Applications, Vol. 7, (1996), 1-25.

[3]

T. Aiki, A mathematical model for a valve made of a spring of a shape memory alloy, in "Nonlinear Phenomena with Energy Dissipation, Mathematical Analysis, Modeling and Simulation," Gakuto International Series, Mathematical Sciences and Applications, Vol. 29, (2008), 1-18.

[4]

K. T. Andrews, K. L. Kuttler and M. Shillor, Second order evolution equations with dynamic boundary conditions, J. Math. Anal. Appl., 197 (1996), 781-795. doi: 10.1006/jmaa.1996.0053.

[5]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Appl. Math. Sci., 121, Springer-Verlag, New York, 1996.

[6]

R. E. Langer, A problem in diffusion or in the flow of heat for a solid in contact with fluid, Tôhoku Math. J. Ser. 1, 35 (1932), 260-275.

[7]

J. L. Lions, "Quelques Methods de Resolutions des Problems aux Limites Non Lineares," Dunod, Paris, 1969.

[8]

N. Sato and T. Aiki, Phase field equations with constraints under nonlinear dynamic boundary conditions, Commun. Appl. Anal., 5 (2001), 215-234.

show all references

References:
[1]

T. Aiki, Multi-dimensional Stefan problems with dynamic boundary conditions, Applicable Analysis, Applicable Analysis, 56 (1995), 71-94. doi: 10.1080/00036819508840311.

[2]

T. Aiki, Multi-dimensional two-phase Stefan problems with nonlinear dynamic boundary conditions, in "Nonlinear Analysis and Applications," Gakuto International Series, Mathematical Sciences and Applications, Vol. 7, (1996), 1-25.

[3]

T. Aiki, A mathematical model for a valve made of a spring of a shape memory alloy, in "Nonlinear Phenomena with Energy Dissipation, Mathematical Analysis, Modeling and Simulation," Gakuto International Series, Mathematical Sciences and Applications, Vol. 29, (2008), 1-18.

[4]

K. T. Andrews, K. L. Kuttler and M. Shillor, Second order evolution equations with dynamic boundary conditions, J. Math. Anal. Appl., 197 (1996), 781-795. doi: 10.1006/jmaa.1996.0053.

[5]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Appl. Math. Sci., 121, Springer-Verlag, New York, 1996.

[6]

R. E. Langer, A problem in diffusion or in the flow of heat for a solid in contact with fluid, Tôhoku Math. J. Ser. 1, 35 (1932), 260-275.

[7]

J. L. Lions, "Quelques Methods de Resolutions des Problems aux Limites Non Lineares," Dunod, Paris, 1969.

[8]

N. Sato and T. Aiki, Phase field equations with constraints under nonlinear dynamic boundary conditions, Commun. Appl. Anal., 5 (2001), 215-234.

[1]

Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10

[2]

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

[3]

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

[4]

Donatella Danielli, Marianne Korten. On the pointwise jump condition at the free boundary in the 1-phase Stefan problem. Communications on Pure and Applied Analysis, 2005, 4 (2) : 357-366. doi: 10.3934/cpaa.2005.4.357

[5]

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

[6]

Jiayue Zheng, Shangbin Cui. Bifurcation analysis of a tumor-model free boundary problem with a nonlinear boundary condition. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4397-4410. doi: 10.3934/dcdsb.2020103

[7]

Xilu Wang, Xiaoliang Cheng. Continuous dependence and optimal control of a dynamic elastic-viscoplastic contact problem with non-monotone boundary conditions. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2021064

[8]

Wenzhen Gan, Peng Zhou. A revisit to the diffusive logistic model with free boundary condition. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 837-847. doi: 10.3934/dcdsb.2016.21.837

[9]

Jesús Ildefonso Díaz, L. Tello. On a climate model with a dynamic nonlinear diffusive boundary condition. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 253-262. doi: 10.3934/dcdss.2008.1.253

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

Barbora Benešová, Miroslav Frost, Lukáš Kadeřávek, Tomáš Roubíček, Petr Sedlák. An experimentally-fitted thermodynamical constitutive model for polycrystalline shape memory alloys. Discrete and Continuous Dynamical Systems - S, 2021, 14 (11) : 3925-3952. doi: 10.3934/dcdss.2020459

[17]

Yang Zhang. A free boundary problem of the cancer invasion. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1323-1343. doi: 10.3934/dcdsb.2021092

[18]

Hayk Mikayelyan, Henrik Shahgholian. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1431-1443. doi: 10.3934/cpaa.2013.12.1431

[19]

Qi An, Chuncheng Wang, Hao Wang. Analysis of a spatial memory model with nonlocal maturation delay and hostile boundary condition. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5845-5868. doi: 10.3934/dcds.2020249

[20]

Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure and Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]