Effective acoustic equations for a layered material described by the fractional Kelvin–Voigt model

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.