# Hashin–Shtrikman bounds

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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:

$K_\mathrm{HS}^{\pm}=K_2+\frac{\phi }{( K_1-K_2 )^{-1}+{ ( 1-\phi ) ( K_2 + \frac43 \mu_2 )^{-1}}}$
$\mu_\mathrm{HS}^{\pm}=\mu_2+\frac{\phi }{( \mu_1-\mu_2 )^{-1}+\frac{2( 1-\phi )( K_{2}+2\mu _{2} )}{5\mu _{2}( K_{2} + \frac43\mu_{2} )}}$

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