Stochastic variability of effective properties via the generalized variability response function

Homogenization of randomly heterogeneous material properties into effective properties is an essential procedure in facilitating the analysis of a wide range of mechanics problems. Although formulas exist to calculate deterministic effective properties for structures larger than the representative volume element (RVE), no general method other than Monte Carlo simulation exists to evaluate the variability of these effective properties for structures smaller than the RVE. In a recent paper , a method was proposed for evaluating the stochastic variability of effective properties by incorporating the variability response function (VRF) concept. Subsequently, the existence of the VRF for effective properties for linear, statically determinate structures was formally proven. The concept of the VRF has been proposed as a means to systematically capture the effect of the spectral characteristics of uncertain system parameters modeled by homogeneous stochastic fields on the uncertain structural response. Although the existence of VRFs can be formally proven only for statically determinate structures, a Monte Carlo-based methodology has been proposed recently as a generalization of the VRF concept . In this paper, this methodology is extended to establish estimates of the VRF for effective properties of statically indeterminate beams.