Third-order bounds on the elastic moduli of metal-matrix composites

Elastic behaviour of a model two-phase material

“…The Voigt and Reuss bounds are relatively wide apart, and modifications have accordingly been proposed by many researchers. [1][2][3][4][5][6][7][8][9][10][11] Among these modifications, the Hashin and Shtrikman model (H-S model) has received wide attention. [11][12][13][14] Hashin and Shtrikman treated the system containing one particulate phase and one continuous matrix phase.…”

“…3 Most models can predict the elastic properties of two-phase materials simply by knowing the elastic properties of each constituent. 1,2,[4][5][6][7][8][9][10][11] Many experimental studies have also been carried out to determine the elastic properties of two phases materials. [12][13][14] The predictions made by the theoretical models can always match some experimental data well.…”

“…[1][2][3][4][5][6][7][8][9][10][11][12][13][14] Many theoretical models have been developed to describe and predict the elastic properties of two-phase materials. [1][2][3][4][5][6][7][8][9][10][11] These models all start with the assumption that the two-phase materials are homogeneous on a scale much larger than the size of inclusions, and that the displacements and tractions at the interface between the two phases are continuous. Since the dimension of the specimen considered is much larger than the inclusion size, the shape of the inclusion would make relatively little difference.…”

The boundary-due terms in the Green operator of inclusion patterns from distant to contact and to connected situations using radon transforms: Illustration for spheroid alignments in isotropic media

“…cannot be correctly approached if the inclusion interactions are ignored, except where concentrations are ''dilute enough" (Willis and Acton, 1976;Christensen, 1979). For ''high" concentrations, the isolated (one site) inclusion approximation has been significantly improved by using either the solution for the pair interaction problem between two ellipsoids (Berveiller et al, 1987;Kouris and Tsuchida, 1991;Anttreter and Fisher, 1996) or statistical estimates that are based on correlation functions of rank two and more (Davis, 1991;Helsing, 1993;Ponte Castaneda and Willis, 1995;Kanaun, 2003). However, when inclusions are not homogeneously or randomly distributed within a matrix, the estimate of the cluster or pattern effects needs to approach some mean, or effective, interaction contribution in the arrangement (De Bartolo and Hillberry, 1998;Estevez et al, 1995Estevez et al, , 1999.…”