Generalized Variational Principles for Uncertainty Quantification of Boundary Value Problems of Random Heterogeneous Materials

Asymptotic theories of classical micromechanics are built on a fundamental assumption of large separation of scales. For random heterogeneous materials the scale-decoupling assumption however is inapplicable in many circumstances from conventional failure problems to novel small-scale engineering systems. Development of new theories for scale-coupling mechanics and uncertainty quantification is considered to have significant impacts on diverse disciplines. Scale-coupling effects become crucial when size of boundary value problems (BVPs) dynamic wavelength is comparable to the characteristic length of heterogeneity or when local heterogeneity becomes crucial due to extreme sensitivity of local instabilities. Stochasticity, vanishing in deterministic homogenization, resurfaces amid multiscale interactions. Multiscale stochastic modeling is expected to play an increasingly important role in simulation and prediction of material failure and novel multiscale systems such as microelectronic-mechanical systems and metamaterials. In computational mechanics a prevalent issue is, while a fine mesh is desired to achieve high accuracy, a certain mesh size threshold exists below which deterministic finite elements become questionable. This work starts investigation of the scale-coupling problems by first looking at uncertainty of material responses due to randomness or incomplete information of microstructures. The classical variational principles are generalized from scale-decoupling problems to scale-coupling BVPs, which provides upper and lower variational bounds for probabilistic prediction of material responses. It is expected that the developed generalized variational principles will lead to novel computational methods for uncertainty quantification of random heterogeneous materials.