We have applied a homogenization theory, which is based on the Fourier formalism, to calculate the effective parameters of phononic crystals having liquid inclusions embedded in a solid host matrix. The theory provides explicit formulas for determining all the components of the effective mass density and stiffness tensors, which are valid in the long wavelength limit for arbitrary Bravais lattice and any form of the inclusions inside the unit cell. In the previous work, it was shown that rectangular two-dimensional lattices of water-filled holes in an elastic host matrix exhibit solid-like behavior with strongly anisotropic mass density in the low-frequency limit. Such metamaterials were called metasolids. In the present work, we analyze the metasolid behavior of liquid-solid three-dimensional phononic crystals. In particular we have analyzed the effect of the type of Bravais lattice and form of the liquid inclusions on the anisotropy of the effective mass density. In the analysis we have considered different solid host materials (Al, Si, and ribbon) with isolated inclusions of water. We have established that the anisotropy of the effective mass density is considerably strong when the homogenized phononic crystals do not possess inversion symmetry because of the inclusion shape. Our results could be useful for designing metamaterials with predetermined elastic properties.
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