February  2012, 5(1): 209-217. doi: 10.3934/dcdss.2012.5.209

Stability of the steady state for multi-dimensional thermoelastic systems of shape memory alloys

1. 

Division of Mathematical Science, Department of System Innovation, Graduate School of Engineering Science, Osaka University, 1-3 Machikane-yama, Toyonaka, Osaka, 560-8531, Japan

2. 

Department of Engineering for Production and Environment, Graduate School of Science and Engineering, Ehime University, 3 Bunkyo-cho, Matsuyama, Ehime 790-8577, Japan

Received  April 2009 Revised  November 2009 Published  February 2011

This paper studies a dynamical stability of the steady state for some thermoelastic and thermoviscoelastic systems in multi-dimensional space domain. More general nonlinear term can be taken here than the one in [6] which studied the stability for the one-dimensional system called the Falk model system. We also give applications to thermoviscoelastic systems treated in [8] and [9].
Citation: 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
References:
[1]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Appl. Math. Sci., 121 (1996). Google Scholar

[2]

F. Falk and P. Konopka, Three-dimensional Landau theory describing the martensitic phase transformation of shape memory alloys,, Journal of Physics: Condensed Matter, 2 (1990), 61. doi: 10.1088/0953-8984/2/1/005. Google Scholar

[3]

I. Pawłow, Three-dimensional model of thermomechanical evolution of shape memory materials,, Control Cybernet., 29 (2000), 341. Google Scholar

[4]

T. Suzuki, "Mean Field Theories and Dual Variation,", Atlantis Press, (2008). Google Scholar

[5]

T. Suzuki and S. Tasaki, Stationary solutions to the Falk system on shape memory alloys,, in preparation., (). Google Scholar

[6]

T. Suzuki and S. Yoshikawa, Stability of the steady state for the Falk model system of shape memory alloys,, Math. Methods Appl. Sci., 30 (2007), 2233. doi: 10.1002/mma.889. Google Scholar

[7]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North-Holland Mathematical Library 18, 18 (1978). Google Scholar

[8]

S. Yoshikawa, I. Pawłow and W. M. Zajączkowski, Quasilinear thermoelasticity system arising in shape memory materials,, SIAM J. Math. Anal., 38 (2007), 1733. doi: 10.1137/060653159. Google Scholar

[9]

S. Yoshikawa, I. Pawłow and W. M. Zajązkowski, A quasilinear thermoviscoelastic system for shape memory alloys with temperature dependent specific heat, , Commun. Pure Appl. Anal., 8 (2009), 1093. doi: 10.3934/cpaa.2009.8.1093. Google Scholar

show all references

References:
[1]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Appl. Math. Sci., 121 (1996). Google Scholar

[2]

F. Falk and P. Konopka, Three-dimensional Landau theory describing the martensitic phase transformation of shape memory alloys,, Journal of Physics: Condensed Matter, 2 (1990), 61. doi: 10.1088/0953-8984/2/1/005. Google Scholar

[3]

I. Pawłow, Three-dimensional model of thermomechanical evolution of shape memory materials,, Control Cybernet., 29 (2000), 341. Google Scholar

[4]

T. Suzuki, "Mean Field Theories and Dual Variation,", Atlantis Press, (2008). Google Scholar

[5]

T. Suzuki and S. Tasaki, Stationary solutions to the Falk system on shape memory alloys,, in preparation., (). Google Scholar

[6]

T. Suzuki and S. Yoshikawa, Stability of the steady state for the Falk model system of shape memory alloys,, Math. Methods Appl. Sci., 30 (2007), 2233. doi: 10.1002/mma.889. Google Scholar

[7]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North-Holland Mathematical Library 18, 18 (1978). Google Scholar

[8]

S. Yoshikawa, I. Pawłow and W. M. Zajączkowski, Quasilinear thermoelasticity system arising in shape memory materials,, SIAM J. Math. Anal., 38 (2007), 1733. doi: 10.1137/060653159. Google Scholar

[9]

S. Yoshikawa, I. Pawłow and W. M. Zajązkowski, A quasilinear thermoviscoelastic system for shape memory alloys with temperature dependent specific heat, , Commun. Pure Appl. Anal., 8 (2009), 1093. doi: 10.3934/cpaa.2009.8.1093. Google Scholar

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