The asymptotic behavior of the linear transmission problem in viscoelasticity

Alves, Margareth S.; Rivera, Jaime Muñoz; Sepúlveda, Mauricio; Villagrán, Octavio Vera; Garay, Maria Zegarra

doi:10.1002/mana.201200319

Cited by 47 publications

(44 citation statements)

References 21 publications

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“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php decay rate of 1 t 2 ln t as an application of a theory on semigroup and recently by Alves et al [1] with the complete characterization of polynomial stability result by Borichev and Tomilov [2]. Thus, we are able to obtain the best decay rate for our model, which is also the best decay rate for the two-piece model.…”

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confidence: 50%

The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin--Voigt Dissipation

Alves¹,

Rivera²,

Sepúlveda³

et al. 2014

SIAM J. Appl. Math.

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In this paper we consider the transmission problem of a material composed of three components; one of them is a Kelvin-Voigt viscoelastic material, the second is an elastic material (no dissipation), and the third is an elastic material inserted with a frictional damping mechanism. The main result of this paper is that the rate of decay will depend on the position of each component. When the viscoelastic component is not in the middle of the material, then there exists exponential stability of the solution. Instead, when the viscoelastic part is in the middle of the material, there is not exponential stability. In this case we show that the corresponding solution decays polynomially as 1/t 2 . Moreover we show that the rate of decay is optimal over the domain of the infinitesimal generator. Finally, using a second order scheme that ensures the decay of energy (Newmark-β method), we give some numerical examples which demonstrate this asymptotic behavior.

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confidence: 50%

The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin--Voigt Dissipation

Alves¹,

Rivera²,

Sepúlveda³

et al. 2014

SIAM J. Appl. Math.

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“…(2) We find here the particular cases studied in [23,24,1,17,22]. Note that concerning the result of polynomial stability in [1,17] the authors proved that the 1 t 2 decay rate of solution is optimal when the damping coefficient is a characteristic function.…”

Section: Now By (327) We Obtain After Integrating By Parts Thatmentioning

confidence: 84%

Stability of the wave equations on a tree with local Kelvin–Voigt damping

2019

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In this paper we study the stability problem of a tree of elastic strings with local Kelvin-Voigt damping on some of the edges. Under the compatibility condition of displacement and strain and continuity condition of damping coefficients at the vertices of the tree, exponential/polynomial stability are proved. Our results generalizes the cases of single elastic string with local Kelvin-Voigt damping in [21,24,5].2010 Mathematics Subject Classification. 35B35, 35B40, 93D20.

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“…With this, the fully discrete energy obtained by the −Newmark method is decreasing, and we expect that its asymptotic behavior be a reflection of the continuous case (see Krenk 22 and also 23,24 ).…”

Section: Equation Of Motion and Time Discretizationmentioning

confidence: 89%

Energy decay to Timoshenko system with indefinite damping

Fatori

Saito

Sepúlveda

et al. 2019

Math Methods in App Sciences

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We consider the classical Timoshenko system for vibrations of thin rods. The system has an indefinite damping mechanism, ie, it has a damping function a = a(x) possibly changing sign, present only in the equation for the vertical displacement. We shall prove that exponential stability depends on conditions regarding of the indefinite damping function a and a nice relationship between the coefficient of the system. Finally, we give some numerical result to verify our analytical results. KEYWORDS exponential stability, finite difference, indefinite damping, Timoshenko system MSC CLASSIFICATION35L53; 35B40; 93D20; 65N06 Here, t denotes the time variable, x is the distance until the beam's centerline in equilibrium, the function = (t, x) denotes the vertical displacement of the beam's centerline, and the function = (t, x) denotes the rotation of the vertical fibers in the beam. Moreover, the coefficients 1 , 2 , b, and k denote positive constants, and they depend on the density of the mass material, the area of the cross-section, the second moment of the cross-section area, the Young's model, the modulus of rigidity, and the shear factor.The system (1)-(2) is conservative. So, if we want to search about asymptotic behavior, we must add a damping term. In this direction, the main types of dissipative mechanisms considered are frictional, thermal, viscoelastic and their combinations.Recently, researches have shown that the exponential stability of the Timoshenko system is achieved regardless of any specific relations between the coefficients when there are a dissipative mechanism in both equations. We refer the reader to, eg, previous studies 1-4 and the reference therein.Math Meth Appl Sci. 2020;43:225-241.wileyonlinelibrary.com/journal/mma

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The asymptotic behavior of the linear transmission problem in viscoelasticity

Cited by 47 publications

References 21 publications

The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin--Voigt Dissipation

The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin--Voigt Dissipation

Stability of the wave equations on a tree with local Kelvin–Voigt damping

Energy decay to Timoshenko system with indefinite damping

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