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:

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

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