Statistics of Residual Stress in Random Microstructures: Mean-Field Estimates and Full-Field Validations

Abstract

Fluctuations of local fields are crucial for the prediction of failure in random composites across different scales as well as estimating the inelastic behaviour of it. This can be quantified statistically through second moments of the local fields, which can be quickly estimated using mean field homogenization (MFH). However, the exact fluctuation field can be estimated using full-field methods though it comes at the cost of intensive computational resources and limited scalability to complex microstructures. In this work, MFH is used to estimate the statistical variation of the field quantities and then cross-verified with full-field methods for a linear-thermoelastic homogenization problem. An analytical expression to calculate the second moments of the local fields for a linear thermo-elastic problem using MFH is obtained based on the Hill-Mandel condition. The expressions fundamentally rely on the solution of linear elastic problem which in turn depends on the derivatives of Hill's polarization tensor. Solution of this derivative term has been analytically and semianalytically derived in previous work [1]. The statistical distribution of residual stress tensor components and equivalent stress in particulate and unidirectional fibrous composites, arising purely due to differential thermal expansion, is computed and compared with full-field homogenization. Full-field simulations indicated a non-Gaussian distribution of stress components, whereas Weibull-like distributions for equivalent residual stress. Nevertheless, the assumed Gaussian distribution in mean-field estimates captures the essential features.