A second-order multiscale approach for viscoelastic analysis of statistically inhomogeneous materials

摘要

An effective second-order multiscale approach is developed in this work to discuss viscoelastic properties of statistically inhomogeneous materials. In these materials, the sophisticated microscale structure of inclusions, including related shape, orientation, size, spatial distribution, volume fraction and so on, results in varying of the macroscale properties. At first, the Laplace transform is applied to the linear viscoelastic problems, and expected relaxation modulus in Laplace domain for the materials is given. Also, the second-order multiscale formulas for evaluating the viscoelastic problems of statistically inhomogeneous materials are derived. Next, the stochastic multiscale algorithm is proposed, and the expected relaxation moduli in time domain are obtained by the least-square and inverse Laplace transform. The salient features of the proposed approach are the asymptotic high-order homogenizations that do not require high-order continuity of the coarse-scale (or macroscale) solutions and can handle the random materials with complicated microstructure at a fraction of computational cost. Finally, some examples for the composites are computed by the effective algorithm, and compared with the data by the theoretical models and experimental results. The comparison illustrates that the stochastic multiscale model is efficient for determining the viscoelastic properties of the materials and shows their potential application in practical engineering calculation.