Exact second moments of strain for composites with isotropic phases

In micromechanical approaches to failure prediction of inhomogeneous microstructures or homogenization of inelastic materials, local field fluctuations beyond the mean field are relevant, but not easily accessible except for resorting to full-field simulations. However, any analytical estimate of the effective strain energy density combined with the Hill-Mandel condition implies some information about the probability distributions of local fields such as stresses and strains. In this paper, analytical equations for the second moments of local fields are derived for particulate composites with isotropic phases. Explicit expressions are provided using the Mori- Tanaka approach, the Hashin-Shtrikman bounds and the singular approximation. The results are compared to full-field simulations of a random sphere assemblage. As the Hashin-Shtrikman covariance is isotropic, it is not well-suited for predicting the behavior under deviatoric loadings. The singular approximation and Mori-Tanaka methods yield sensible approximations for the second moment of matrix strain.

Automated summary

This paper derives analytical equations for the second moments of local fields in particulate composites with isotropic phases. Using the Mori-Tanaka approach, the Hashin-Shtrikman bounds, and the singular approximation, explicit expressions are provided. The results are compared to full-field simulations, showing that the Hashin-Shtrikman covariance is not suitable for predicting behavior under deviatoric loadings, while the singular approximation and Mori-Tanaka methods yield sensible approximations for the second moment of matrix strain.