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

摘要

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