November  2013, 33(11&12): 5273-5292. doi: 10.3934/dcds.2013.33.5273

A thermo piezoelectric model: Exponential decay of the total energy

1. 

National Laboratory of Scientific Computation, LNCC/MCT, Av. Getulio Vargas 333, Quitandinha, Petrópolis, RJ, 25651-070, Brazil

2. 

National Laboratory of Scientific Computation, LNCC/MCT, Av. Getulio Vargas 333, Quitandinha, Petrópolis, RJ, CEP 25651-070, Brazil

Received  October 2011 Revised  April 2012 Published  May 2013

We consider a linear evolution model describing a piezoelectric phenomenon under thermal effects as suggested by R. Mindlin [13] and W. Nowacki [16]. We prove the equivalence between exponential decay of the total energy and an observability inequality for an anisotropic elastic wave system. Our strategy is to use a decoupling method to reduce the problem to an equivalent observability inequality for an anisotropic elastic wave system and assume a condition which guarantees that the corresponding elliptic operator has no eigenfunctions with null divergence.
Citation: Gustavo Alberto Perla Menzala, Julian Moises Sejje Suárez. A thermo piezoelectric model: Exponential decay of the total energy. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5273-5292. doi: 10.3934/dcds.2013.33.5273
References:
[1]

K. Ammari and S. Nicaise, Stabilization of a piezoelectric system, Asymptotic Analysis, 73 (2011), 125-146.

[2]

I. Babuska, Error bounds for finite element method, Numerishe Mathematik, 16 (1971), 322-333.

[3]

P. G. Ciarlet, "Mathematical Elasticity, Vols I and II," North-Holland, Amsterdam, I, 1988, II, 1997.

[4]

C. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal., 29 (1968), 241-271. doi: 10.1007/BF00276727.

[5]

H. Funakubo, "Ed. Shape Memory Alloys," Translated from the Japanese by J. B. Kennedy, Gordon and Breach Science Publishers, New York, 1984.

[6]

D. Henry, O. Lopes and A. Perissinotto, On the essential spectrum of a semigroup of thermoelasticity, Nonlinear Anal. TMA, 21 (1993), 65-75. doi: 10.1016/0362-546X(93)90178-U.

[7]

D. Iessan, On some theorems in Thermopiezoelectricity, J. Thermal Stresses, 12 (1989), 209-223. doi: 10.1080/01495738908961962.

[8]

B. Kapitonov, B. Miara and G. Perla Menzala, Stabilization of a layered Piezoelectric 3-D body by boundary dissipation, ESAIM, Control Optimization and Calculus of Variations, 12 (2006), 198-215. doi: 10.1051/cocv:2005028.

[9]

B. Kapitonov, B. Miara and G. Perla Menzala, Boundary observation and exact control of a quasi electrostatic piezoelectric system in multilayered media, SIAM, J. Control and Optim., 46 (2007), 1080-1097. doi: 10.1137/050629884.

[10]

G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Rational Mech. Anal., 148 (1999), 179-231. doi: 10.1007/s002050050160.

[11]

J. L. Lions, "Contrôlabilité Exacte, Stabilization et Perturbations de Systèmes Distribués," Tome 1, Contrôlabilité Exacte, Masson 1988.

[12]

B. Miara and M. Lima, Energy decay in piezoelectric systems, Applicable Analysis, 88 (2009), 947-960. doi: 10.1080/00036810903042166.

[13]

R. D. Mindlin, Equations of high frequency vibrations of thermopiezoelectric crystal plates, International Journal of Solid Structures, 10 (1974), 625-637. doi: 10.1016/0020-7683(74)90047-X.

[14]

I. Müller, Six lectures in shape memory, Centre Recherches Mathématiques, CRM, Proceedings and Lectures Notes, 13 (1998).

[15]

S. Nicaise, Stability and controllability of the electromagneto-elastic system, Post. Math., 60 (2003), 73-80.

[16]

W. Nowacki, Some general theorems of thermopiezoelectricity, J. Thermal Stresses, 1 (1978), 171-182. doi: 10.1080/01495737808926940.

[17]

J. M. Sejje Suárez, Modelagem de fenômenos termopiezoelétricos: Analise assintótica e Simulação Numérica, Tese de Doutorado (2011) Laboratório Nacional de Computação Cientifica (LNCC-MCT), Brasil. (in Portuguese)

[18]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed. Applicata, (IV), CXLVI (1987), 65-96. doi: 10.1007/BF01762360.

[19]

R. C. Smith, "Smart Material Systems. Model Development," SIAM, Frontiers in Applied Mathematics, 2005. doi: 10.1137/1.9780898717471.

[20]

A. V. Srinivasan and D. M. McFarland, "Smart Structures: Analysis and Design," Cambridge University Press, Cambridge, UK, 2001.

[21]

K. Uchino, "Piezoelectric Actuators and Ultrasonic Motors," Kluwer Academic Publishers, Boston, 1997. doi: 10.1007/978-1-4613-1463-9.

[22]

E. Zuazua, Controllability of the linear system of thermoelasticity, J. Math. Pures Appl., 74 (1995), 291-315.

show all references

References:
[1]

K. Ammari and S. Nicaise, Stabilization of a piezoelectric system, Asymptotic Analysis, 73 (2011), 125-146.

[2]

I. Babuska, Error bounds for finite element method, Numerishe Mathematik, 16 (1971), 322-333.

[3]

P. G. Ciarlet, "Mathematical Elasticity, Vols I and II," North-Holland, Amsterdam, I, 1988, II, 1997.

[4]

C. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal., 29 (1968), 241-271. doi: 10.1007/BF00276727.

[5]

H. Funakubo, "Ed. Shape Memory Alloys," Translated from the Japanese by J. B. Kennedy, Gordon and Breach Science Publishers, New York, 1984.

[6]

D. Henry, O. Lopes and A. Perissinotto, On the essential spectrum of a semigroup of thermoelasticity, Nonlinear Anal. TMA, 21 (1993), 65-75. doi: 10.1016/0362-546X(93)90178-U.

[7]

D. Iessan, On some theorems in Thermopiezoelectricity, J. Thermal Stresses, 12 (1989), 209-223. doi: 10.1080/01495738908961962.

[8]

B. Kapitonov, B. Miara and G. Perla Menzala, Stabilization of a layered Piezoelectric 3-D body by boundary dissipation, ESAIM, Control Optimization and Calculus of Variations, 12 (2006), 198-215. doi: 10.1051/cocv:2005028.

[9]

B. Kapitonov, B. Miara and G. Perla Menzala, Boundary observation and exact control of a quasi electrostatic piezoelectric system in multilayered media, SIAM, J. Control and Optim., 46 (2007), 1080-1097. doi: 10.1137/050629884.

[10]

G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Rational Mech. Anal., 148 (1999), 179-231. doi: 10.1007/s002050050160.

[11]

J. L. Lions, "Contrôlabilité Exacte, Stabilization et Perturbations de Systèmes Distribués," Tome 1, Contrôlabilité Exacte, Masson 1988.

[12]

B. Miara and M. Lima, Energy decay in piezoelectric systems, Applicable Analysis, 88 (2009), 947-960. doi: 10.1080/00036810903042166.

[13]

R. D. Mindlin, Equations of high frequency vibrations of thermopiezoelectric crystal plates, International Journal of Solid Structures, 10 (1974), 625-637. doi: 10.1016/0020-7683(74)90047-X.

[14]

I. Müller, Six lectures in shape memory, Centre Recherches Mathématiques, CRM, Proceedings and Lectures Notes, 13 (1998).

[15]

S. Nicaise, Stability and controllability of the electromagneto-elastic system, Post. Math., 60 (2003), 73-80.

[16]

W. Nowacki, Some general theorems of thermopiezoelectricity, J. Thermal Stresses, 1 (1978), 171-182. doi: 10.1080/01495737808926940.

[17]

J. M. Sejje Suárez, Modelagem de fenômenos termopiezoelétricos: Analise assintótica e Simulação Numérica, Tese de Doutorado (2011) Laboratório Nacional de Computação Cientifica (LNCC-MCT), Brasil. (in Portuguese)

[18]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed. Applicata, (IV), CXLVI (1987), 65-96. doi: 10.1007/BF01762360.

[19]

R. C. Smith, "Smart Material Systems. Model Development," SIAM, Frontiers in Applied Mathematics, 2005. doi: 10.1137/1.9780898717471.

[20]

A. V. Srinivasan and D. M. McFarland, "Smart Structures: Analysis and Design," Cambridge University Press, Cambridge, UK, 2001.

[21]

K. Uchino, "Piezoelectric Actuators and Ultrasonic Motors," Kluwer Academic Publishers, Boston, 1997. doi: 10.1007/978-1-4613-1463-9.

[22]

E. Zuazua, Controllability of the linear system of thermoelasticity, J. Math. Pures Appl., 74 (1995), 291-315.

[1]

Mirelson M. Freitas, Anderson J. A. Ramos, Manoel J. Dos Santos, Eraldo R. N. Fonseca. Attractors and pullback dynamics for non-autonomous piezoelectric system with magnetic and thermal effects. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3745-3765. doi: 10.3934/cpaa.2021129

[2]

Michela Eleuteri, Luca Lussardi. Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials. Evolution Equations and Control Theory, 2014, 3 (3) : 411-427. doi: 10.3934/eect.2014.3.411

[3]

Mirelson M. Freitas, Anderson J. A. Ramos, Manoel J. Dos Santos, Jamille L.L. Almeida. Dynamics of piezoelectric beams with magnetic effects and delay term. Evolution Equations and Control Theory, 2022, 11 (2) : 583-603. doi: 10.3934/eect.2021015

[4]

Sandra Carillo. Materials with memory: Free energies & solution exponential decay. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1235-1248. doi: 10.3934/cpaa.2010.9.1235

[5]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184

[6]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196

[7]

Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. A rate-independent model for permanent inelastic effects in shape memory materials. Networks and Heterogeneous Media, 2011, 6 (1) : 145-165. doi: 10.3934/nhm.2011.6.145

[8]

Haolei Wang, Lei Zhang. Energy minimization and preconditioning in the simulation of athermal granular materials in two dimensions. Electronic Research Archive, 2020, 28 (1) : 405-421. doi: 10.3934/era.2020023

[9]

Tomasz Komorowski, Stefano Olla, Marielle Simon. Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities. Kinetic and Related Models, 2018, 11 (3) : 615-645. doi: 10.3934/krm.2018026

[10]

Willy Sarlet, Tom Mestdag. Compatibility aspects of the method of phase synchronization for decoupling linear second-order differential equations. Journal of Geometric Mechanics, 2022, 14 (1) : 91-104. doi: 10.3934/jgm.2021019

[11]

Mohammed Aassila. On energy decay rate for linear damped systems. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 851-864. doi: 10.3934/dcds.2002.8.851

[12]

Rachid Assel, Mohamed Ghazel. Energy decay for the damped wave equation on an unbounded network. Evolution Equations and Control Theory, 2018, 7 (3) : 335-351. doi: 10.3934/eect.2018017

[13]

Bopeng Rao. Optimal energy decay rate in a damped Rayleigh beam. Discrete and Continuous Dynamical Systems, 1998, 4 (4) : 721-734. doi: 10.3934/dcds.1998.4.721

[14]

Gustavo Alberto Perla Menzala, Julian Moises Sejje Suárez. On the one-dimensional version of the dynamical Marguerre-Vlasov system with thermal effects. Conference Publications, 2009, 2009 (Special) : 536-547. doi: 10.3934/proc.2009.2009.536

[15]

Irena Lasiecka, To Fu Ma, Rodrigo Nunes Monteiro. Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1037-1072. doi: 10.3934/dcdsb.2018141

[16]

Gilbert Peralta. Uniform exponential stability of a fluid-plate interaction model due to thermal effects. Evolution Equations and Control Theory, 2020, 9 (1) : 39-60. doi: 10.3934/eect.2020016

[17]

Asim Aziz, Wasim Jamshed, Yasir Ali, Moniba Shams. Heat transfer and entropy analysis of Maxwell hybrid nanofluid including effects of inclined magnetic field, Joule heating and thermal radiation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2667-2690. doi: 10.3934/dcdss.2020142

[18]

Brenton LeMesurier. Modeling thermal effects on nonlinear wave motion in biopolymers by a stochastic discrete nonlinear Schrödinger equation with phase damping. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 317-327. doi: 10.3934/dcdss.2008.1.317

[19]

Xiaoxiao Li, Yingjing Shi, Rui Li, Shida Cao. Energy management method for an unpowered landing. Journal of Industrial and Management Optimization, 2022, 18 (2) : 825-841. doi: 10.3934/jimo.2020180

[20]

Zhuangyi Liu, Ramón Quintanilla. Energy decay rate of a mixed type II and type III thermoelastic system. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1433-1444. doi: 10.3934/dcdsb.2010.14.1433

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (0)

[Back to Top]