The Elastic Constants of Materials Loaded with Non-Rigid Fillers

Abstract

General expressions are given for the modulus of rigidity, ${\mathrm{\lambda ̄}}_2$ , and compressibility, k̄, 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 $$\begin{array}{c}{\mathrm{\lambda ̄}}_2{\mathrm{=\lambda ̄}}_2\left(\text{1+}\frac{{\text{15(λ}}_2{\mathrm{\prime -\lambda }}_2{\mathrm{)(\lambda }}_1{\text{−2λ}}_2)}{{\text{2λ}}_2{\text{′(3λ}}_1{\text{+8λ}}_2{\mathrm{)+\lambda }}_2{\text{(9λ}}_1{\text{+14λ}}_2)}\phi \right)\\ \mathrm{k̄=k+}\frac{{\text{3+4λ}}_2\mathrm{k\prime }}{{\text{3+4λ}}_{2}k}\mathrm{(k\prime -k)\phi .}\end{array}$$ If the ``filler'' is a gas at a pressure, p, in a nearly incompressible medium, these give $$\begin{array}{cc}\mathrm{Young\text{'}s\; modulus}& \mathrm{Ē=E}\left(\text{1−}\frac{E}{\text{9p+4E}}\phi \right).\\ \mathrm{Modulus\; of\; rigidity}& {\mathrm{\lambda ̄}}_2{\mathrm{=\lambda }}_2\text{(1−}\frac{5}{3}\mathrm{\phi ).}\\ \mathrm{Compressibility}& \mathrm{k̄=k+}\frac{3}{{\text{3p+4λ}}_2}\mathrm{\phi .}\\ \mathrm{Poisson\text{'}s\; ratio}& \mathrm{\sigma ̄=\sigma }\left(\text{1−}\frac{\text{3E}}{\text{9p+4E}}\phi \right),\end{array}$$ where φ is the volume loading, p is the pressure within the spherical cavities in the deformed 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.