۱- Introduction

۲- Asymptotic homogenization method for linear viscoelasticity in the frequency domain

۳- Finite element formulation

۴- New implementation algorithm of the AH of viscoelasticity

۵- Flow chart

۶- Numerical examples

۷- Conclusions

References

Abstract

There is a growing demand for methods to estimate the effective viscoelastic response of viscoelastic composites, for their applications in structural vibration and noise control. This paper proposes a novel reformulation and numerical implementation algorithm for the asymptotic homogenization theory for predicting the effective complex moduli of viscoelastic composites in the frequency domain. In the new algorithm, an equivalent harmonic analysis is established and a double-layer elements method is proposed to solve the local problem in the homogenization process. On the basis of the new algorithm, the effective complex moduli can be obtained easily by using commercial software to serve as a black box. Numerous elements and techniques for modeling and analysis available in commercial software can be applied to complicated microstructures without mathematical derivation. The numerical examples presented show the validity of this new implementation algorithm.

Introduction

Increasing demands for controlling vibration and noise in structures have boosted the search for high-performance wave or vibration-absorbing structures and materials. Active or semi-active vibration-control techniques (1-3) and the applications of passive damping (4) are two of the most widely used methods for effectively vibration absorption, over a range of frequencies. Of these methods, passive damping using viscoelastic materials is popular due to its simple implementation and cost-effectiveness. However, very few solid materials can reach the engineering standards necessary for damping applications. Viscoelastic composites, which have desirable damping characteristics and provide design flexibility (5-7), have captured the attention of researchers (8-10). The increasing applications of the damping materials have also driven up demands for accurate estimation of the effective viscoelastic response of composites. Micromechanical method provides overall behavior of the composites from known properties of their constituents through an analysis of a unit-cell model (11, 12), then the heterogeneous structure of the composite is replaced by a homogeneous medium with anisotropic properties. Several methods have been developed over the past few decades to theoretically predict the effective elastic properties of composite materials, such as self-consistent scheme (SCS) (13), generalized self-consistent scheme (GSCS) (14), the Mori–Tanaka method (M-T) (15), representative volume element method (RVE) (16) and asymptotic homogenization method (AH) (17). Of which, SCS, GSCS and M-T method are usually be applied to derive the approximate analytic formula of composite material with simple microstructures. When the geometry configuration of the microstructure is complex or material properties differ greatly between different phases, these methods often exhibit a large error. RVE method and AH method are two widely used numerical methods to determine the effective moduli of heterogeneous materials with complicated microstructures. These two methods all construct the boundary-value problems of the microstructure and then obtain the effective material properties of composite materials based on the characteristic displacement or strain fields.