Exponential stability of viscoelastic structure with second sound

Authors

https://doi.org/10.61383/ejam.20242250

Keywords:

Viscoelastic structure, Cattaneo’s law, existence of solution, exponential decay

Abstract

This manuscript deals with a thermo-viscoelastic system describing the vibrations of a flexible structure. We study this structure's stabilization problem when subjected to Kelvin-Voigt damping and the second sound. Cattaneo-Vernotte's law governs the thermal effect, eliminating the physical paradox of equal speed of waves in the classical thermoelastic theory. Semigroup theory proves the solution's existence and uniqueness. Exponential stability is proved by the energy method.

References

R. A. Adams, Sobolev Spaces, Academic Press, New York, (1975).

M. Alves, P. Gamboa, G. Gorain, A. Rambaud, O. Vera, Asymptotic behavior of a flexible structure with Cattaneo type of thermal effect, Indag. Math. 27 (2016), 821–834, https://doi.org/10.1016/j.indag.2016.03.001. DOI: https://doi.org/10.1016/j.indag.2016.03.001

J. L. Auriault, Cattaneo-Vernotte equation versus Fourier thermoelastic hyperbolic heat equation, Int. J. Eng. Sci. 101 (2016), 45–49. https://doi.org/10.1016/j.ijengsci.2015.12.002. DOI: https://doi.org/10.1016/j.ijengsci.2015.12.002

H. T. Banks, R. C. Smith, Y. Wang, Smart Material Structures: Modeling, Estimation, and Control, Wiley, London, (1996).

D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature Appl. Mech. Rev. 51 (1998), 705–728, https://doi.org/10.1115/1.3098984. DOI: https://doi.org/10.1115/1.3098984

J. H. Choi, S.H. Yoon, S. G. Park, S. H. Choi, Analytical solution of the Cattaneo-Vernotte equation (non-Fourier heat conduction), J. Korean Soc. of Marine Engineering 40 (2016), 389–396, https://doi.org/10.5916/jkosme.2016.40.5.389. DOI: https://doi.org/10.5916/jkosme.2016.40.5.389

W. A. Day, Heat Conduction within Linear Thermoelasticity, Springer Science & Business Media, New York, (2013).

R. D´ıaz, J. Mu ˜noz, C. Mart´ınez, O. Vera, Exponential stability of a flexible structure with history and thermal effect, Appl. Math. 65 (2020), 407–420, https://doi.org/10.21136/AM.2020.0117-19. DOI: https://doi.org/10.21136/AM.2020.0117-19

Y. C. Fung, Foundations of Solid Mechanics, Prentice-Hall, New Delhi, (1968).

G. C. Gorain, Exponential stabilization of longitudinal vibrations of an inhomogeneous beam, J. Math. Sci. 198 (2014), 245–251, https://doi.org/10.1007/s10958-014-1787-1. DOI: https://doi.org/10.1007/s10958-014-1787-1

G. C. Gorain, C.A. Raposo, O. Vera, Stabilization of non-homogeneous viscoelastic waves coupled with heat conduction due to Cattaneo-Vernotte, Ricerche Mat. 72 (2023), 359–377, https://doi.org/10.1007/s11587-022-00722-4. DOI: https://doi.org/10.1007/s11587-022-00722-4

M. Houasni, S. Zitouni, Flexible structure with second sound and past history - Existence, uniqueness and exponential stability, Math. Eng. Sci. Aerosp. 13 (2022), 251–262.

M. Houasni, S. Zitouni, A. Djebabla, On the exponential stability of a flexible structure in thermo-elasticity with micro-temperature effects, Sigma J. Eng. Nat. Sci. 39 (2021), 157–164, https://doi.org/10.14744/sigma.2021.00015. DOI: https://doi.org/10.14744/sigma.2021.00015

G. Li, Y. Luan, J Yu, F. Jiang, Well-posedness and exponential stability of a flexible structure with second sound and time delay , Applicable Analysis 98 (2019), 2903–2915, https://doi.org/10.1080/00036811.2018.1478081. DOI: https://doi.org/10.1080/00036811.2018.1478081

W. Liu, M. Chen, Well-posedness and exponential decay for a porous thermoelastic system with second sound and a time-varying delay term in the internal feedback, Continuum Mechanics and Thermodynamics 29 (2017), 741–746, https://doi.org/10.1007/s00161-017-0556-z. DOI: https://doi.org/10.1007/s00161-017-0556-z

K. Liu, Z. Liu, Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping, SIAM J. Control Optim. 36 (1998), 1086–1098, https://doi.org/10.1137/S0363012996310703. DOI: https://doi.org/10.1137/S0363012996310703

Z. Liu, S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall, London, (1999).

R. Ma, K. Bi, H. Hao, Inerter-based structural vibration control: A state-of-the-art review, Eng. Struct. 243 (2021), 112655, https://doi.org/10.1016/j.engstruct.2021.112655. DOI: https://doi.org/10.1016/j.engstruct.2021.112655

Y. Meng, Y. Wang, Stability for a nonlinear coupled system of elasticity and thermoelasticity with second sound, Analysis and Applications 13 (2021), 45–75, https://doi.org/10.1142/S0219530514500171. DOI: https://doi.org/10.1142/S0219530514500171

S. Misra, M. Alves, G. Gorain, O. Vera, Stability of the vibrations of an inhomogeneous flexible structure with thermal effect Int. J. Dyn. Control, 3 (2015), 354–362, https://doi.org/10.1007/s40435-014-0113-6. DOI: https://doi.org/10.1007/s40435-014-0113-6

B. Mochnacki, M. Paruch, Cattaneo-Vernotte equation. Identification of relaxation time using evolutionary algorithms, J. Appl. Math. and Comp. Mech. 12 (2013), 97–102, https://doi.org/10.17512/jamcm.2013.4.12. DOI: https://doi.org/10.17512/jamcm.2013.4.12

R. Racke, Thermoelasticity with second sound - Exponential stability in linear and non-linear 1-d, Math. Methods Appl. Sci. 25 (2002), 409–441, https://doi.org/10.1002/mma.298. DOI: https://doi.org/10.1002/mma.298

A. J. A. Ramos, A. D. S. Campelo, D. S. Almeida J ´unior, M. M. Freitas, R. C. Barbosa Numerical exponential decay of thermoelastic waves connected in parallel, Adv. Comput. Math. 49 (2023), https://doi.org/10.1007/s10444-023-10027-1. DOI: https://doi.org/10.1007/s10444-023-10027-1

P. ˇSkvorc, H. Kozmar, Wind energy harnessing on tall buildings in urban environments, Renewable and Sustainable Energy Reviews, 152 (2021), 111662, https://doi.org/10.1016/j.rser.2021.111662. DOI: https://doi.org/10.1016/j.rser.2021.111662

V. C. Zannini, T. Q. M ´endez, A. J. A. Ramos, Optimal stability for laminated beams with Kelvin–Voigt damping and Fourier’s law Asymptotic Analysis 137 (2023), 1–29, https://doi.org/10.3233/ASY-231883. DOI: https://doi.org/10.3233/ASY-231883

M. Zhang, Global existence and exponential stability of solutions for thermodiffusion equations of type III, Zeitschrift f ¨ur angewandte Mathematik und Physik 74 (2023), https://doi.org/10.1007/s00033-023-02006-9. DOI: https://doi.org/10.1007/s00033-023-02006-9

H. Zuo, K. Bi, H. Hao, A state-of-the-art review on the vibration mitigation of wind turbines, Renew. Sust. Energ. Rev. 121 (2020), 109710, https://doi.org/10.1016/j.rser.2020.109710. DOI: https://doi.org/10.1016/j.rser.2020.109710

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Published

2024 Jun 10

How to Cite

[1]
G. C. Gorain, Carlos, O. Vera, and S. Ram, “Exponential stability of viscoelastic structure with second sound”, Electron. J. Appl. Math., vol. 2, no. 2, pp. 10–21, Jun. 2024.

Issue

Vol. 2 No. 2 (2024)

Section

Research Article

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Copyright (c) 2024 Ganesh C. Gorain; Carlos Raposo (Corresponding Author); Octavio Vera, Srikanta Ram

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
 
 
Received 2024 Feb 04
Accepted 2024 Jun 02
Published 2024 Jun 10

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