Estimation of effective moduli of viscoelastic materials with periodic microstructure using a homogenization method

Abstract

A systematic way of obtaining the effective viscoelastic moduli in time and frequency domain is developed for viscoelastic composites with periodic microstructures. The problem of estimating the effective moduli is formulated using the asymptotic homogenization method. With the effective moduli in time domain, effective complex moduli in frequency domain are obtained by using Fourier transformation. The memory effects of viscoelastic composites due to homogenization are shown in general form and a sufficient condition for the effects to disappear is discussed. It is shown that memory effects disappear if the relaxation moduli are separable in space and time. Several numerical examples are presented to illustrate and verify present approach and to discuss the memory effects and inverse Laplace transforms. The effective relaxation moduli are computed in Laplace transform domain and are numerically inverse-transformed into time domain. The least-square fitting in Laplace transform domain is employed based on the Prony series representations of the relaxation moduli. The effect of number of fitting terms on effective moduli is shown and the scheme for the selection of the fitting terms is proposed. The validity of the approach for the estimation of the effective moduli in general viscoelastic composites is demonstrated by comparing the estimated results with those of other approaches through numerical examples. In order to observe the memory effects, viscoelastic composites with the circular inclusion in the viscoelastic matrix, whose moduli are represented by the mechanical analogy models, are treated. Effective moduli for composite of two different viscoelastic materials are computed. Results showed that due to the presence of memory effects, additional terms were required for the accurate inverse Laplace transformation in the Prony series approximation of the homogenized relaxation moduli in Laplace transform domain. It is also shown in the numerical examples th...