The effective Poisson ratio of three-dimensional two-phase composites is discussed from the theoretical point of view. It is shown that – in contrast to porous materials – the Voigt-Reuss and Hashin-Shtrikman (or Walpole) bounds for the bulk and shear modulus impose certain restrictions beyond the general condition that the Poisson ratio must be in the range between – 1 and + 0.5. Here, the conditions for auxetic behavior (occurrence of negative Poisson ratios) are investigated. It is found, that a two-phase composite of non-auxetic phases (i.e. phases with phase bulk moduli higher than the phase shear moduli) may be auxetic, when the phase contrast (ratio of the phase bulk and shear moduli) is sufficiently high. In particular, it is shown that even composites with two phases of the same Poisson ratio (i.e. the same bulk-to-shear modulus ratio) can have a negative effective Poisson ratio (i.e. exhibit auxetic behavior) when the phase contrast is higher than 3.8 for anisotropic materials or higher than 8.6 for isotropic materials. For a given set of phase bulk and shear moduli the minimum Poisson ratio of the composite and the corresponding composition can be predicted, and this information may be exploited in materials design to produce auxetic materials. Examples of actually or potentially auxetic material combinations are given, together with the minimum Poisson ratio that can be achieved in these systems. The special case of porous materials (extreme case with infinite phase contrast) is briefly discussed as well, together with the microstructural features to achieve auxetic behavior (reentrant-corner, node-fibril and rotating-hinge microstructures). On the other hand, when the phase contrast is small, such as in densely sintered ceramics (e.g. pore-free alumina-zirconia composites), auxetic behavior principally cannot be achieved, and in these cases the effective Poisson ratio is well predicted by simple relations, including the mixture-rule approximation.

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1. Effective properties of ceramic matrix composites