Homogenizing the viscoelasticity problem with long-term memory

Шумилова, В. В.

doi:10.1134/s0001434613090125

Cited by 3 publications

(4 citation statements)

References 7 publications

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“…Next, using the basic properties of two-scale convergence and repeating the same arguments as in [5,11,12], we can show that the homogenized problem that corresponds to problem (5) and which is constructed for ε → 0 has the form…”

Section: Effective Acoustic Equationsmentioning

confidence: 88%

“…To construct the homogenized problem, we will use the method proposed in [5,11,12]. This method was developed for the homogenization of acoustics equations in two-phase dissipative media with periodic microstructure.…”

Section: Effective Acoustic Equationsmentioning

confidence: 99%

“…It is easy to check that Z kh (y) = Z hk (y) and V kh (y) = V hk (y), so that we need only to find Z kh (y) and V kh (y) for k h. Solving problems (12) for k h, we obtain…”

Section: Solutions Of Auxiliary Cell Problemsmentioning

confidence: 99%

“…Now, after defining Z kh (y) and V kh (y) in ( 11), we can find parameters M kh . It follows from ( 8), (12), and ( 13) that M kh satisfies the system Applying the inverse Laplace transform to (11), we get…”

Section: Solutions Of Auxiliary Cell Problemsmentioning

confidence: 99%

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Effective Acoustic Equations for a Layered Material Described by the Fractional Kelvin-Voigt Model

Шамаев

Шумилова

2021

Journal of Siberian Federal University. Mathematics &Amp; Physics

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The paper is devoted to the construction of effective acoustic equations for a two-phase layered viscoelastic material described by the Kelvin–Voigt model with fractional time derivatives. For this purpose, the theory of two-scale convergence and the Laplace transform with respect to time are used. It is shown that the effective equations are partial integro-differential equations with fractional time derivatives and fractional exponential convolution kernels. In order to find the coefficients and the convolution kernels of these equations, several auxiliary cell problems are formulated and solved

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Section: Effective Acoustic Equationsmentioning

confidence: 88%

Section: Effective Acoustic Equationsmentioning

confidence: 99%

“…It is easy to check that Z kh (y) = Z hk (y) and V kh (y) = V hk (y), so that we need only to find Z kh (y) and V kh (y) for k h. Solving problems (12) for k h, we obtain…”

Section: Solutions Of Auxiliary Cell Problemsmentioning

confidence: 99%

Section: Solutions Of Auxiliary Cell Problemsmentioning

confidence: 99%

See 2 more Smart Citations