Mean field homogenization schemes using averages of stress, strain and strain energy density and applications

Abstract

The average of stress (AS) and average of strain (AN) were often used in the development of conventional mean-field homogenization (MFH) schemes, but their correlation with the average of strain energy density (AE) has rarely been discussed. In this work, we examined the correlations of AE with AS and AN, and found that AE can be reduced to the product of AS or AN under uniform boundary stress or linear surface displacement associated with a uniform strain field. By introducing a reference matrix and using any two among AE, AS and AN, three MFH schemes were developed: NS (using AN and AS); NE (using AN and AE); and SE (using AS and AE). Following the concept of the Mori-Tanaka's (MT) method, three MFH schemes obtained above were reduced to MT-NS, MT-NE and MT-SE, respectively. Effective elastic properties of composites consisting of isotropic elastic matrices and spherical isotropic elastic inclusions were predicted using these schemes and compared with experimental results. The comparison revealed that MT-SE and MT-NS can reasonably replicate the experimental results in the case of moderate particle volume fractions and ratios between the Young's moduli of matrix and the reinforcement particles, demonstrating the validity of the MFH schemes. This work is significant because it not only clarifies the correlation of AE with AN and AS, but also provides more options for the development of MFH schemes, which can enrich the theory and methods for the analysis of the effective properties of composites.