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

The ability to determine probabilistic characteristics of response quantities in structural mechanics (e.g. displacements, stresses) as well as effective material properties is restricted due to lack of information on the probabilistic characteristics of the uncertain system parameters. The concept of the variability response function (VRF) has been proposed as a means to systematically capture the effect of the stochastic spectral characteristics of uncertain system parameters modeled by homogeneous random fields on the uncertain structural response. The key property of the VRF in its classical sense is its independence from the marginal probability distribution function (PDF) and the spectral density function (SDF) of the uncertain system parameters (it depends only on the deterministic structural configuration and boundary conditions). Proofs have been provided for the existence of VRFs for linear and some nonlinear statically determinate beams. For statically indeterminate structures, the Monte Carlo based generalized variability response function (GVRF) methodology has been proposed recently as a generalization of the VRF concept to indeterminate linear and some nonlinear beams. The methodology computes GVRFs, which are analogous to VRFs for statically determinate structures, and evaluates their dependence (or lack thereof) on the PDF and SDF of the random field, thereby providing an estimate of the accuracy of the GVRF. In this paper, the GVRF methodology is extended to problems involving two-dimensional, linear continua whose stochasticity is characterized by statistically homogeneous random fields. After detailing the GVRF methodology for two-dimensional random fields, two numerical examples are provided: GVRFs are computed for the displacement response and for the effective compliance of linear plane stress systems.

}, keywords = {Effective properties, Random fields, Uncertainty quantification, Variability response functions}, doi = {10.1016/j.cma.2014.01.013}, author = {Teferra Kirubel and Arwade, Sanjay R. and Deodatis, George} } @article {11203, title = {Stochastic variability of effective properties via the generalized variability response function}, journal = {Computers and Structures}, volume = {110-111}, year = {2012}, month = {11/2012}, pages = {107-115}, chapter = {107}, abstract = {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.