Multiscale material modeling using variational principles and random matrix theory

The focus of this dissertation is to estimate the material model for an elastic macro-scale structure from micro-mechanical analyses performed on meso-scale heterogeneous microstructures. The general viewpoint adopted here is: to model the macro-scale material parameters as random so as to account for uncertainty due to limited availability of high-fidelity microstructural data and modeling inaccuracy. The relevant probability distribution is characterized based on a limited number of deterministic micro-mechanical analyses. The material uncertainty is then propagated through the macro-scale boundary value problem via Monte Carlo analysis. More specifically, in this work, this general framework is first used to model the linear elastic macro-scale constitutive matrix as a Random Matrix. The parameters of the appropriate probability density functions are determined by using variational principles on response data from micro-mechanical analyses performed on heterogeneous microstructural specimens with randomly dispersed micro-cracks. Such micro-cracks are not discernible to the naked eye and are difficult to model by conventional continuum damage mechanics. Therefore, to address this challenge, an optimization scheme is developed that predicts both the location of micro-cracks and the weakened macro-scale elasticity matrix from a single set of strain observation on a macro-scale structure. This inverse analysis framework makes significant use of the statistical information on macroscale response (generated by propagating the material uncertainty) as well as the deterministic information in the form of positive definite bounds on the elasticity matrix (generated from deterministic micro-mechanical analyses). The variational principles are further extended theoretically to encompass finite deformation nonlinear elastic microstructures and a corresponding macro-scale material uncertainty modeling framework is proposed and numerically demonstrated. Finally, bounds on appropriate moduli matrices are theoretically established using variational principles when the meso-scale continuum is governed by Skew Symmetric Couple Stress theory.