Mechanics of deformable solids Homogenization and composites Theorem

Hashin–Shtrikman bounds

Builds on Homogenization Bulk modulus Shear modulus

Statement

Consider a statistically isotropic two-phase composite in $\R^3$ whose phases occupy volume fractions $f_1$ and $f_2 = 1 - f_1$ .

If the phases have conductivities $0 < \sigma_1 \le \sigma_2$ , the effective conductivity $\sigma^*$ satisfies $\sigma_1 + \frac{f_2}{\frac{1}{\sigma_2 - \sigma_1} + \frac{f_1}{3 \sigma_1}} \le \sigma^* \le \sigma_2 + \frac{f_1}{\frac{1}{\sigma_1 - \sigma_2} + \frac{f_2}{3 \sigma_2}}.$
If the phases are isotropic linear elastic with bulk moduli $K_1 \le K_2$ and shear moduli $G_1 \le G_2$ , all positive, the effective bulk modulus $K^*$ satisfies $K_1 + \frac{f_2}{\frac{1}{K_2 - K_1} + \frac{3 f_1}{3 K_1 + 4 G_1}} \le K^* \le K_2 + \frac{f_1}{\frac{1}{K_1 - K_2} + \frac{3 f_2}{3 K_2 + 4 G_2}}.$
Under the same hypotheses, the effective shear modulus $G^*$ satisfies $G_1 + \frac{f_2}{\frac{1}{G_2 - G_1} + \frac{6 f_1 (K_1 + 2 G_1)}{5 G_1 (3 K_1 + 4 G_1)}} \le G^* \le G_2 + \frac{f_1}{\frac{1}{G_1 - G_2} + \frac{6 f_2 (K_2 + 2 G_2)}{5 G_2 (3 K_2 + 4 G_2)}}.$

When $\sigma_1 = \sigma_2$ (or $K_1 = K_2$ , or $G_1 = G_2$ ), the corresponding bounds are read as the common value.

Remarks

The proof uses the Hashin–Shtrikman variational principle. It compares the composite with a homogeneous reference medium and optimizes over a polarization field. Choosing the reference equal to the softer or the stiffer phase gives the lower or the upper bound. The bounds on conductivity and on the bulk modulus are optimal: they are attained by Hashin's assemblage of coated spheres. The stiffer phase is the coating for the upper bound, and the softer phase for the lower bound. They always lie between the Voigt and Reuss bounds and are much closer to each other. For example, with $\sigma_1 = 1$ , $\sigma_2 = 10$ and $f_1 = f_2 = 1/2$ , they give $2.8 \le \sigma^* \le 4.71$ , whereas the harmonic and arithmetic means give $1.82 \le \sigma^* \le 5.5$ , to two decimals. The isotropy hypothesis is needed: a laminate is not isotropic, and its conductivity along the layers is the arithmetic mean, which exceeds the upper bound.

Going deeper

For a student

The bounds are a sharpening of the Voigt and Reuss bounds that uses the isotropy of the composite. They are a standard test for effective medium approximations: a sound approximation must give values between them. The Mori–Tanaka method for spherical inclusions in a matrix reproduces one of the Hashin–Shtrikman bounds for the bulk modulus.

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Tag	`FZGYFV69HK`
URI	`m2://FZGYFV69HK`
Kind	statement, theorem
Writing	W0 stub
MSC	74Q20, 74E30, 78A48
Status	This copy of the site has no build data. Run `m2 status FZGYFV69HK`.
About	Homogenization
Uses	Homogenization, Bulk modulus, Shear modulus
Changed	3 November 2026