Hashin–Shtrikman bounds

The Hashin-Shtrikman bounds are the tightest bounds possible from range of composite moduli for a two-phase material. Specifying the volume fraction of the constituent moduli allows the calculation of rigorous upper and lower bounds for the elastic moduli of any composite material. The so-called Hashin-Shtrikman bounds for the bulk, K, and shear moduli μ is given by:

$K_{\mathrm {HS} }^{\pm }=K_{2}+{\frac {\phi }{(K_{1}-K_{2})^{-1}+{(1-\phi )(K_{2}+{\frac {4}{3}}\mu _{2})^{-1}}}}$

$\mu _{\mathrm {HS} }^{\pm }=\mu _{2}+{\frac {\phi }{(\mu _{1}-\mu _{2})^{-1}+{\frac {2(1-\phi )(K_{2}+2\mu _{2})}{5\mu _{2}(K_{2}+{\frac {4}{3}}\mu _{2})}}}}$

The upper bound is computed when K₂ > K₁. The lower bound is computed by interchanging the indices in the equations.

For the case of a solid-fluid mixture, K₂ is K_S, the bulk modulus of the solid component, and and K₁ is K_f, the bulk modulus of the fluid component.

Visual representation

Bounds on the effective elastic properties are completely independent of grain texture or fabric.

Example

Quartz-Brine mixture: Quartz with solid mineral modulus, K_S = 36.6 GPa, and K_f = 2.2 GPa.

External links

• Theoretical formulae — NIST page on the subject