MESO-SCALE HOMOGENIZATION METHODS FOR MOVING WINDOW ANALYSIS

Abstract

Computational methods are developed to capture the macro-scale behavior of a heterogeneous composite material when the macro-scale is significantly larger than the scale of the heterogeneity. In this case, it may be computationally costly to model the material microstructure directly in a finite element analysis. Often a representative volume element (RVE) can be used with success; however, there are two main disadvantages of an RVE approach. First, when nonlinear behavior or damage initiation is considered, too much information is lost regarding the local variation in the microstructure. Second, an RVE approach effectively eliminates randomness in the meso-scale material representation, which is needed to statistically quantify structural reliability. As an alternative to the RVE approach, a moving windowing homogenization approach has been developed. This captures a degree of small scale material heterogeneity and randomness, and is suited to a priori implementation in a reduced-mesh large scale finite element analysis. However, non-unique solutions may exist for the homogenization of a window (or statistical volume element, SVE) that is below the scale of an RVE. At a scale smaller than the RVE, the inverse of the compliance tensor obtained by a statically uniform boundary condition (SUBC) test does not equal the stiffness tensor obtained by a kinematically uniform boundary condition (KUBC) test; these are lower and upper bonds on the effective behavior of the element, respectively. Mixed uniform boundary conditions and periodic boundary conditions, such as those used in the Generalized Method of Cells (GMC) predict apparent properties between the KUBC and SUBC upper and lower bounds. This work explores the assumption that accurately capturing effective behavior of a composite at the meso-scale and accurately predicting macro-scale behavior in a moving window analysis are equivalent considerations. A composite beam model with randomly varying inclusions is considered, and the deflection at midspan is used as a metric for macro-scale behavior. Homogenization methods including those described above are implemented. Results are compared to determine not only how well each homogenization technique approximates effective behavior, but also by how each predicts the macro-scale behavior of a structure in a finite element analysis.