# 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:

The upper bound is computed when *K*_{2} > *K*_{1}. The lower bound is computed by interchanging the indices in the equations.

For the case of a solid-fluid mixture, *K*_{2} is *K*_{S}, the bulk modulus of the solid component, and and *K*_{1} 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.

## References

- ↑ 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

## External links

- Theoretical formulae — NIST page on the subject