# 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 [1] for the bulk, K, and shear moduli μ is given by:

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

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 KS, the bulk modulus of the solid component, and and K1 is Kf, 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, KS = 36.6 GPa, and Kf = 2.2 GPa.

## References

1. Hashin, Z, and Shtrikman, S, 1963, A variational approach to the elastic behavior of multiphase minerals. Journal of the Mechanics and Physics of Solids, 11 (2), 127-140. DOI:10.1016/0022-5096(63)90060-7