Hashin–Shtrikman Based FE Analysis of the Elastic Behaviour of Finite Random Composite Bodies

Abstract

This paper demonstrates the effect of non-local interactions in an edge-cracked body made of composite material, in the case that the scale of the body is not large in comparison with the scale of the microstructure, as can occur, for example, in laboratory testing of concrete specimens. Not only are there significant boundary layers in the vicinity of all of the specimen boundaries, but the small size of the body permits their interaction, to alter the value of the mean field globally relative to the value that would be predicted by the use of “ordinary” homogenization, under conditions of “displacement control”. Such an effect would be present even for an uncracked body but the presence of the crack induces strong gradients in the mean field which significantly enhance this effect. The mean strain component $\langle {e_{22}}\rangle$ tends to become concentrated around the line ahead of the crack in comparison with the prediction of homogenization; there is, in addition, an enhancement of the stress and strain concentrations in the harder phase. The method of analysis is a finite-element implementation of a variational formulation related to the Hashin–Shtrikman variational principle, developed in the context of non-local effective response by the present authors. The present calculations are considered to be more accurate than some previously reported (Luciano and Willis, Journal of the Mechanics and Physics of Solids 53, 1505–1522, 2005) and the conclusions differ correspondingly.