Poisson ratio in composites of auxetics

Mean-field theory of elastic moduli of a two-phase disordered composite with ellipsoidal inclusions is reviewed together with an indication as to how interactions among inclusions may be taken into account. In the mean-field approximation, the effective Poisson ratio ${\sigma }_e$ in composites with auxetic inclusions of various shapes such as discs, spheres, blades, needles, and disks is studied analytically and numerically. It is shown that phase properties such as inclusion volume or area fraction $\phi$ and matrix and inclusion Poisson ratios $({\sigma }_m$ and $\sigma )$ and Young’s moduli ${(E}_m$ and $E)$ have a marked effect on ${\sigma }_e.$ The earlier theoretical findings of the existence of auxeticity windows and the widening effect of inclusion-inclusion interactions on the window for $\delta {=E/E}_m$ are reconfirmed for composites of auxetic spheres in both two and three dimensions, with new auxeticity windows discovered for the other inclusion shapes. For a composite with $\sigma =-0.8,$ ${\sigma }_m=0.25,$ and $\phi =0.4,$ it is found that the sphere is the most ${\sigma }_e$-lowering or negative-${\sigma }_e$-producing inclusion shape for $\delta$ around $1/2,$ while disklike inclusions yield a most negative ${\sigma }_e$ for $\delta$ greater than $1.$