Averaging the acoustics equations for a viscoelastic material with channels filled with a viscous compressible fluid

“…The second class of problems contains averaging problems in multiphase media, where one of the phases is an elastic (or viscoelastic) medium and another one is a viscous (compressible or incompressible) fluid (see [21,22]). The averaging problem is to construct an effective (averaged) model of the two-phase medium such that some switchings of one or another phase interlace rapidly under a change of the space variables.…”

Section: Introductionmentioning

confidence: 99%

Analysis of Operator Models Arising in Problems of Hereditary Mechanics

2014

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The correct solvability of initial-value problems for integrodifferential equations with unbounded operator coefficients in Hilbert spaces is determined. Numerous problems of hereditary mechanics and thermal physics have motivated the study of such equations. Models taking into account a Kelvin-Voight friction are considered. A spectral analysis of the operator functions, which are the symbols of integro-differential equations considered in this paper, is made.

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“…where ρ is the density of the viscoelastic medium, u(x, t) is displacement of the point with abscissa x at time t at the equilibrium state, α and β depend on properties of the viscoelastic medium, g(t) is the convolution kernel, the prime denotes the derivative with respect to x. Equations of the form (1) also arise as a result of homogenization of the acoustic equations for periodic combined media of two viscous fluids [4,5] or of porous or viscoelastic material and a viscous liquid occupying pores [6]- [8].…”

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confidence: 99%

On the spectrum of an integro-differential equation arising in viscoelasticity theory

Шамаев

Шумилова

2012

J Math Sci

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We study the spectral properties of a boundary value problem for an integro-differential equation arising in viscoelasticity theory. We prove that the spectrum of the boundary value problem is either completely real or contains, together with the real part, only finitely many complex eigenvalues. Bibliography: 8 titles.The spectral properties of boundary value problems for integro-differential equations were studied in many works (cf., for example, [1]-[3]). The interest in the study of spectral properties of such problems is caused by numerous applications of integro-differential equations to mechanics and physics, in particular, in the visoelasticity theory and the theory of heat transfer. For example, the dynamics of a one-dimensional viscoelastic medium with long-time memory is described by the following integro-differential equation:where ρ is the density of the viscoelastic medium, u(x, t) is displacement of the point with abscissa x at time t at the equilibrium state, α and β depend on properties of the viscoelastic medium, g(t) is the convolution kernel, the prime denotes the derivative with respect to x. Equations of the form (1) also arise as a result of homogenization of the acoustic equations for periodic combined media of two viscous fluids [4,5] or of porous or viscoelastic material and a viscous liquid occupying pores [6]-[8].This paper is devoted to the spectral analysis of Equation (1) with the homogeneous initial and boundary conditions u(0, t) = u(l, t) = 0, t > 0, u(x, 0) =u(x, 0) = 0, x ∈ (0, l).

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Averaging the acoustics equations for a viscoelastic material with channels filled with a viscous compressible fluid

Cited by 13 publications

References 8 publications

Homogenization of Acoustic Equations for a Partially Perforated Elastic Material with Slightly Viscous Fluid

Homogenization of Acoustic Equations for a Partially Perforated Elastic Material with Slightly Viscous Fluid

Analysis of Operator Models Arising in Problems of Hereditary Mechanics

On the spectrum of an integro-differential equation arising in viscoelasticity theory

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