Local Field Statistics in Linear Elastic Unidirectional Fibrous Composites

TL;DR

This paper derives explicit analytical formulas for second moments of local tensorial fields in unidirectional fibrous composites, validating mean-field estimates against finite element solutions and highlighting the influence of fiber volume fraction on stress fluctuations.

Contribution

It introduces new analytical expressions for derivatives of Hill's Polarization tensor, improving the estimation of local field fluctuations in fibrous composites.

Findings

Fiber volume fraction significantly affects stress tensor fluctuations.

Aspect ratio has minimal impact on local field fluctuations.

Validated mean-field estimates with exact finite element solutions.

Abstract

Statistical fluctuations of local tensorial fields beyond the mean are relevant to predict localized failure or overall behavior of the inelastic composites. The expression for second moments of the local fields can be established using the Hill-Mandel condition. Complete estimation of statistical fluctuations via second moments is usually ignored despite its significance. In Eshelby-based mean-field approaches, the second moments are evaluated through derivatives of Hill's Polarization tensor $ \left ( \mathbb{P}_{\mathrm{o}} \right )$ using a singular approximation. Typically, semi-analytical procedures using numerical integration are used to evaluate the derivatives of the polarization tensor $\mathbb{P}_{\mathrm{o}}$. Here, new analytically derived explicit expressions are presented for calculating the derivatives, specifically for unidirectional fibrous composites with isotropic phases. Full-field homogenization using finite element is used to compute the statistical distribution of local fields (exact solution) for the class of random fibrous microstructures. The mean-field estimates are validated with the exact solution across different fiber volume fractions and aspect ratios. The results indicate that the fiber volume fraction significantly influences the fluctuation of stress tensor invariants, whereas the aspect ratio has minimal effect.