The Viscoelasticity of the Disperse System of Spherical Shell Structures

Abstract

The bulk modulus κ^* and the rigidity μ^* of dilute dispersions of spherical shell structures have been calculated by using the following equations:
κ^*=Q-4cμQ'/Q+3cQ' and μ^*/μ=Γ-3/2cΓ'/Γ+cΓ'
where κ and μ are the bulk modulus and the rigidity of the medium, c is the volume concentration of the structures composed of the shell region with κ_m, μ_m and the inside region with κ', μ', Q=(3κ'+4μ_m)(3κ_m+4μ)+12A³(κ'-κ_m)(μ_m-μ), Q'=(κ-κ_m)(3κ'+4μ_m)-A³(κ'-κ_m)(3κ+4μ_m), Γ=Γ_Aμ_mμ_m²+Γ_Bμ_mμ+Γ_Cμμ², Γ'=Γ_A'μ_mμ_m²+Γ_B'μ_mμ+Γ_C'μμ², Γ_A=-38μ₂μ₄+(450-672AA²)A³+μ₁μ₄+400A⁷μ₁μ₃-1824A¹⁰μ₁μ₁²=-Γ_A', Γ_B=-89μ₂μ₄+(150-336AA²)A³μ₁μ₄+200A⁷μ₁μ₃+3648A¹⁰μ₁μ₁², Γ_B'=-6μ₂μ₄+(500-1344AA²)A³μ₁μ₄+800A⁷μ₁μ₃-608A¹⁰μ₁μ₁², Γ_C=-48μ₂μ₄-(600-1008AA²)A³μ₁μ₄-600A⁷μ₁μ₃-1824A¹⁰μ₁μ₁²=3/2Γ_C', μ₁=μ_m-μ', μ₂=3μ_m+2μ', μ₃=18μ_m+19μ', μ₄=16μ_m+19μ', A=a'/a, a and a' are the inner and the outer radii of the shell, respectively.