A Verified Non-Linear Regression Model for Elastic Stiffness Estimates of Finite Composite Domains Considering Combined Effects of Volume Fractions, Shapes, Orientations, Locations, and Number of Multiple Inclusions

Abstract

A non-linear regression model using SAS/STAT (JMP® software; Proc regression module) is developed for estimating the elastic stiffness of finite composite domains considering the combined effects of volume fractions, shapes, orientations, inclusion locations, and number of multiple inclusions. These estimates are compared to numerical solutions that utilized another developed homogenization methodology by the authors (dubbed the generalized stiffness formulation, GSF) to numerically determine the elastic stiffness tensor of a composite domain having multiple inclusions with various combinations of geometric attributes. For each inclusion, these considered variables represent the inclusions’ combined attributes of volume fraction, aspect ratio, orientation, number of inclusions, and their locations. The GSF methodology’s solutions were compared against literature-reported solutions of simple cases according to such well-known techniques as Mori-Tanaka and generalized self-consistent type methods. In these test cases, the effect of only one variable was considered at a time: volume fraction, aspect ratio, or orientation (omitting the number and locations of inclusions). For experimental corroboration of the numerical solutions, testing (uniaxial compression) was performed on test cases of 3D printed test cubes. The regression equation returns estimates of the composite’s ratio of normalized longitudinal modulus (E11) to that of the matrix modulus (Em) or E11/Em when considering any combination of all of the aforementioned inclusions’ variables. All parameters were statistically analyzed with the parameters retained are only those deemed statistically significant (p-values less than 0.05). Values returned by the regression stiffness formulation solutions were compared against values returned by the GSF formulation numerical and against the experimentally found stiffness values. Results show good agreement between the regression model estimates as compared with both numerical and experimental results.