The Randomly Computational Homogenization Of Thermal Properties For Three-dimensional Linear Elasticity Based On Monte-carlo Method

Heterogeneous materials have several advantages such as high specific strength, high specific stiffness and high anti-radiation ability, and they have a wide application and development space in various fields including aerospace, weapons, high-speed trains, civil construction industry and medical, etc. Due to some objective causes, the uncertainty in the effective properties of heterogeneous materials exists inevitably, and the macroscopic properties of heterogeneous materials are closely related to the random micro-scale properties and their correlation as well as random morphology of microstructure. Moreover, the randomness of micro-structural properties and morphology as well as their correlation have a great impact on random effective properties of the composites since a small change in microstructure parameters may result in a greater change in the effective macroscopic properties. Therefore, it is of great importance to study the influences of the uncertainty in the micro-structure on the random effective properties.In this work, computational homogenization of heterogeneous materials under infinitesimal deformation is addressed in the context of linear elasticity when the uncertainty in microstructure is fully considered. The main work consists of two aspects as follows:(1) Based on the multi-scale finite element method and Monte-carlo method, the random homogenization model of heterogeneous materials is established when the randomness of micro-structural morphology and of material properties of constituents as well as the correlation among material properties are accounted for simultaneously. The random effective quantities such as effective moduli and effective elastic parameters as well as effective elastic tensor together with their numerical characteristics under different boundary conditions are then sought. The impacts of micro-structural parameters on random effective quantities are also investigated and illustrated. Finally, the random stress and strain energy distributions within a representative volume element under different boundary conditions are revealed as well. Obviously, it is necessary that the randomness and correlation existing in the microstructure should be fully considered during the solution of the effective mechanical properties of heterogeneous materials.(2) The computational homogenization method and the analytical homogenization method are used when the randomness of micro-structural morphology and of material properties of constituents are considered simultaneously. The random thermal properties and their numerical characteristics are addressed. The approximating degree of the results from two methods to the truth value are analyzed. Moreover, the random heat flux distributions within a representative volume element under different boundary conditions are displayed as well. The numerical example indicates that the results from the numerical solution is closer to the truth value of effective thermal properties. In addition, the randomness of material properties of constituents and of micro-structural morphology have an important influence on the random homogenization results.