The computation of apparent material properties for a random heterogeneous material requires the assumption of a solution field on a finite domain over which the apparent properties are to be computed. In this paper the assumed solution field is taken to be that defined by the shape functions that underpin the finite element method and it is shown that the variance of the apparent properties calculated using the shape functions to define the solution field can be expressed in terms of a variability response function (VRF) that is independent of the marginal distribution and spectral density function of the underlying random heterogeneous material property field. The variance of apparent material properties can be an important consideration in problems where the domain over which the apparent properties are computed is smaller than the representative volume element and the approach introduced here provides an efficient means of calculating that variance and performing sensitivity studies with respect to the characteristics of the material property field. The approach is illustrated using examples involving heat transfer problems and finite elements with linear and nonlinear shape functions and in one and two dimensions. Features of the VRF are described, including dependency on shape and scale of the finite element and the order of the shape functions

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\ [1], 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\ [19]. In this paper, this methodology is extended to establish estimates of the VRF for effective properties of statically indeterminate beams.