Multi-scale formulation for heterogeneous materials with fixed and evolving microstructures

This work presents a class of multi-scale mathematical and computational formulations as well as homogenization and localization numerical procedures for multiscale modeling of (1) materials with fixed microstructures, (2) problems with evolving microstructures such as stressed grain growth processes in polycrystalline materials, (3) microstructurel effect under large deformation, (4) problems with local instability such as wrinkling formation, (5) eigenvalue problems.; Wavelet-based computational methods are introduced for multi-scale modeling and homogenization of materials with fixed microstructures. Two wavelet-based methods, the wavelet Galerkin method and wavelet projection method, are presented. The wavelet Galerkin multi-scale homogenization method is more adequate for microstructures with regular shape, whereas the multi-scale wavelet projection method is capable of homogenizing microstructures with arbitrary geometry. Boundary conditions issues in the homogenization process are also addressed. In the case where the microstructures evolve, such as the grain growth process in polycrystalline materials, the proposed wavelet-based multi-scale methods become ineffective in dealing with the moving material interfaces. Alternatively, a multi-scale variational formulation based on asymptotic expansion and principle of virtual power, in conjunction with a double-grid numerical method, is then proposed for modeling and homogenization of stressed grain growth.; The asymptotic expansion method for solving multi-scale problem is further extended to large deformation multi-scale analysis with geometric and material nonlinearities considered. This method is formulated based on a proposed consistent homogenization which yields a symmetric tangent operator. This approach corrected the major deficiency in the conventional asymptotic expansion method that results in an asymmetric tangent operator. This nonlinear multi-scale formulation is applied to prediction of wrinkling formation resulting from the local instability. It is shown that the surface wrinkles and distortions can be captured from the local short-wavelength deformation modes at the microstructure level. In the last part of this work, a multi-scale formulation for solving eigenvalue problem is developed. The approach composes of an asymptotic expansion predictor and an iterative corrector based on an inverse power method enhanced with Rayleigh quotient corrector to achieve good accuracy and high convergent rate.