The Elastic Constants of a Solid containing Spherical Holes

The effective bulk and shear moduli are calculated by a self-consistent method due to Fröhlich and Sack. The bulk modulus k is determined by applying a hydrostatic pressure, and the shear modulus μ by applying a simple homogeneous shear stress, to a large sphere. Each hole is surrounded by a spherical shell of real material, and the reaction of the rest of the material is estimated by replacing it by equivalent homogeneous material For consistency, both the density and the displacement of the outer spherical boundary must be the same whether the hole and its surrounding shell are replaced by equivalent material or not. The effective elastic constants calculated from these conditions are 1/k = 1/k₀ρ + 3(1 - ρ)/4μ₀ρ + O[(1 - ρ)³], (μ₀ - μ)/μ₀ = 5(1 - ρ)(3k₀ + 4μ₀)/(9k₀ + 8μ₀) + O[(1 - ρ)²], where k₀ and μ₀ refer to the real material and ρ is the density of the actual material relative to that of the real material, in the next approximation k depends on the standard deviation of the volumes of the holes.

The dilatation due to a distribution of pressures in the holes is p(1/k - 1/k₀), where p is the mean obtained when the pressure in each hole has a weight proportional to the volume of the hole. By using the hydrodynamic analogue of the elastic problem, the theory is briefly applied to the theory of sintering, and used to discuss the effective viscosity of a liquid containing small air bubbles.