An approximate calculation on the elastic constants of a solid containing varying volume fractions of cavities

An approximate calculation for the elastic constants of a solid containing varying volume fractions of cavities is presented. The calculation is based on the mechanisms of deformation of solids containing varying volume fractions of cavities namely straining, bending and stretching. The work concentrates first on the deformation of solids containing a small number of cavities and then secondly on the deformation of the solid containing a large fraction of cavities. The analysis represents the unification of two modes of deformation of solids containing cavities in the dilute case with that of the concentrated case. It is found that KfK = (1 − φ)2(1 + ∝φ) + (1 − 2v)[1 −φ2e(1 − φ)2 + (1 − φe)(1 − φ)](1 − 2vfGfG = (1 − φ)2I + βφ + 38eGφ[φ2e(1 − φ)2 + (1 − φe)(1 − φ)], where ∝=12(1 + v)(1 − 2v), β = 2(4 − 5v)7 − 5v, φ is the volume fraction, K is the bulk modulus of the material, Kf is the effective bulk modulus of the material with cavities, G is the shear modulus of the material, Gf is the effective modulus of the material with cavities, e is the elastic modulus of the material, ν is the Poisson ratio of the material, νf Poisson ratio of the cellular material and φe is the fraction of solid in the struts.