| Heterogeneous materials are present everywhere in the physical reality, and their importance appears to be continuously growing as the technology evolves. Several advantages such as light weight, high strength, high thermal performance, corrosion resistance and weak thermal expansion prompt heterogeneous materials widely being used in various fields. Hence, it is thus evident that a true comprehension of the overall macroscopic property in heterogeneous materials cannot leave aside the detailed random micro-mechanical analysis, whereby the random micro-scale properties and their correlation as well as random morphology of microstructure should be fully considered. In this work, random homogenization analysis of heterogeneous materials under infinitesimal deformation is addressed in the context of elasticity where the uncertainty in microstructure is fully and completely considered, and random effective quantities as well as their numerical characteristics together with the correlation among these effective quantities under four types of boundary conditions are then sought.1. The background and status of homogenization for the heterogeneous materials are firstly presented, and the importance and necessity of this work as well as the problems to be solved are then proposed. Moreover, the definition and classification as well as the application of heterogeneous materials are simply introduced.2. For the homogenization of heterogeneous materials, a RVE should be firstly abstracted from heterogeneous materials. Since the stress and strain field will be solved by multi-scale finite element method to obtain the macroscopic properties, the boundary conditions should be applied to the RVE, and the effective properties are formulated based on the multi-scale finite element method.3. It is not perfect only to analyze the effective properties of heterogeneous materials with definite microstructure, and it is necessary to consider the random microstructure of the materials during the homogenization. In this work, random homogenization analysis of heterogeneous materials under infinitesimal deformation is addressed in the context of elasticity by combining the multi-scale finite element with Monte-carlo method.4. Based on the multi-scale finite element method and Monte-carlo method, the homogenization of a two-phase heterogeneous materials in two dimension is used as the numerical example. The generation and discretization of RVE with randomly distributing particles identified by a numerical convergence scheme are firstly given, and different types of boundary conditions for homogenization are then employed on the RVE where the randomness of morphology and material properties of constituents as well as the correlation of material properties are accounted for. The numerical characteristics of random effective quantities are then derived, and impacts of different correlative, randomness cases, mismatch ration and random morphology on random effective quantities are finally investigated and illustrated, the simulation of the correlation among random variables is realized by MCM here, Moreover, random physical fields including random deformation and stress as well as strain energy distribution within an RVE under different boundary conditions are revealed as well. |
