Vibrations of an infinite viscoelastic layer with a dissipative memory

TLDR

The study applies a two‑degree‑of‑freedom elastic‑energy dissipation model with memory to determine the surface response spectrum of a layer given the bottom spectrum. The surface reaction is obtained analytically with the Laplace‑transform method, and the interior response is derived in time and Fourier space. The amplification function at the surface depends on the viscosity coefficient and a fractional power of frequency, and the peak amplitude at resonant frequency is strongly influenced by the dissipation mechanism, indicating that knowledge of its parameters can aid many physical and engineering problems.

Abstract

A model of elastic-energy dissipation based on a memory mechanism with two degrees of freedom is applied to the problem of the determination of the response spectrum at the surface of a layer when the spectrum is given at the bottom. The reaction of the surface of the layer is obtained directly with the Laplace-transform method. With the Fourier method, the amplification function is also found at the surface of the layer; it depends on the first power of the coefficient of viscosity—the Q−1 is proportional to a fractional power of the frequency. The reaction inside of the layer in the case of an infinite layer has also been obtained as function of time and in the Fourier space. It is verified that the amplification function (namely, the peak amplitude response of the free surface of the viscoelastic layer at a “resonant frequency”) depends strongly on the dissipation mechanism; a complete knowledge of the parameters of this mechanism could be of great help in the solution of many physical and engineering problems, especially in the cases when the characteristics of the layer can be varied.