The Elastic Constants of Materials Loaded with Non‐Rigid Fillers

摘要

General expressions are given for the modulus of rigidity, λ¯2, and compressibility, k¯, of a medium of Lame''s constants, λ1 and λ2, loaded with a volume fraction, Φ, of a filler of constants λ1′ and λ2′. To terms in the first power of Φ λ¯2=λ¯21+ 15(λ2′-λ2)(λ1-2λ2) 2λ2′(3λ1+8λ2)+λ2(9λ1+14λ2) φk¯=k+ 3+4λ2k′ 3+4λ2k (k′-k)φ. If the ``filler'''' is a gas at a pressure, p, in a nearly incompressible medium, these give Young''s modulus Ē=E1- E 9p+4E φ. Modulus of rigidity λ¯2=λ2(1- 5 3 φ). Compressibility k¯=k+ 3 3p+4λ2 φ. Poisson''s ratio σ¯=σ1- 3E 9p+4E φ, where Φ is the volume loading, p is the pressure within the spherical cavities in the defo-nrmed state. Barred symbols refer to the properties of the loaded material; unbarred, to the medium alone. Expressions for the displacements and stresses within the medium and the particles, neglecting interactions between particles, are also given.