<i>Mathematical Analysis and Numerical Methods for Science and Technology</i>

“…Furthermore, the connectedness condition implies the positive definiteness of the tensors α and D(λ), λ > 0. The positive definiteness of the last tensor is, in turn, sufficient for the existence and uniqueness for the homogenized problem (2.9) [1,11]. At the same time, according to the example of a layered medium considered below, the connectedness of Y 2 and E 2 is not necessary for the tensor D(λ), λ > 0, to be positive definite.…”

Section: Homogenized Initial-boundary Value Problemmentioning

confidence: 97%

Homogenization of Equations of Dynamics of a Medium Consisting of Viscoelastic Material with Memory and Incompressible Kelvin–Voigt Fluid

Шамаев

Шумилова

2023

J Math Sci

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We consider the initial-boundary value problem describing the dynamics of a two-phase medium with periodic structure. The phases of the medium are an isotropic viscoelastic material with memory and an incompressible viscoelastic Kelvin-Voigt viscoelastic material. We derive the corresponding homogenized problem describing the dynamics of a homogenous viscoelastic medium with memory. We find explicit analytic expressions for coefficients and convolution kernels of the homogenized equations corresponding to the case of a layered medium. Bibliography: 12 titles.Homogenization of the equations of dynamics of a periodic two-phase medium consisting of a solid (elastic or viscoelastic) material and a viscous Newton fluid was studied, for example, in [1]- [6]. According to the results of the cited works, the homogenized equations are partial and integro-differential equations of the third order. Their coefficients and convolution kernels are found by solving auxiliary stationary and evolution periodic problems in the unit cube. If the solid phase is an elastic material, then the evolution periodic problems are formulated for a system of differential equations, whereas if the solid phase is a viscoelastic material with memory, then we deal with a system of differential and integro-differential equations.In this paper, we consider the homogenization problem for the initial-boundary value problem describing the motion of a two-phase medium with periodic structure. One phase of the medium consists of an isotropic viscoelastic material with memory, whereas the second phase is an incompressible viscoelastic Kelvin-Voigt fluid [7]. Using the Laplace transform method and the results of [4]-[6], we formulate a homogenized problem as an initial-boundary value problem * To whom the correspondence should be addressed.

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“…The proof is based on the standard Galerkin approach, see, e.g., [9,10]. We first establish the existence of a weak solution.…”

Section: Proof Of Theorem 31mentioning

confidence: 99%

Electromagnetic wave propagation in media consisting of dispersive metamaterials

Nguyên

Vinoles

2018

Comptes Rendus. Mathématique

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Mathematical Analysis and Numerical Methods for Science and Technology

Abstract: Articles you may be interested in (I.) applications of mathematical methods and models in sciences and engineering AIP Conf.

Cited by 175 publications

References 0 publications

The Euler-Bernoulli equation with distributional coefficients and forces

The Euler-Bernoulli equation with distributional coefficients and forces

Homogenization of Equations of Dynamics of a Medium Consisting of Viscoelastic Material with Memory and Incompressible Kelvin–Voigt Fluid

Electromagnetic wave propagation in media consisting of dispersive metamaterials

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