Computational homogenization for the advanced materials and structures with multiple spatial and temporal scales

This thesis is aimed at exploring a class of multiscale physical processes using asymptotic homogenization method. The emphasis is given to establishing (i) the homogenized descriptions for obtaining the global response fields, and (ii) the global-local inter scale relations among the response fields at various scales. The interactions between multiple spatial and temporal scales are also studied for the multi-physical behavior of composite materials. Four major topics are covered in this research: (1)  Damage evolution in brittle composites. A nonlocal damage theory for obtaining numerical approximation to a boundary value problem describing damage phenomena in brittle composites is developed. The damage evolution is defined on the smallest scale of interest and described within the context of Continuum Damage Mechanics. The mathematical homogenization method based on the asymptotic expansion is generalized to account for damage effects in heterogeneous media. (2) Fatigue in brittle composites. By extending the damage cumulative law defined for the monotonic loading to the case of cyclic loading, the homogenization framework of the two-scale nonlocal damage theory is applied to the fatigue damage in brittle composites. The evolution of fatigue damage is approximated by the first order initial value problem with respect to the number of load cycles. (3)  Multiple temporal scale analysis for the rate-dependent solids under locally periodic loading. For the rate-dependent solids under locally periodic loading, the multiple temporal scales are determined by the material intrinsic time, such as creep time, and the frequency of the external loading. Two rate-dependent material models, including the Maxwell viscoelastic model and the power-law viscoplastic model, are considered. (4) Interactions among multiple spatial scales, multiple temporal scales and multiple physical processes. The coupling of multiple physical processes may introduce additional temporal scale separations which interact with the existing multiple length scales in space and time on a single physical process. A general setting of the space-time asymptotic homogenization process is developed and then applied to the coupled thermo-viscoelastic composites with microscopically periodic mechanical and thermal properties. A rapidly varying spatial and temporal scales are introduced to capture the effects of spatial and temporal fluctuations induced by spatial heterogeneities at diverse time scales. (Abstract shortened by UMI.)...