Voigt and Reuss

Part 2, Part 2: Bounds and Mixtures

Learning objectives

Write the Voigt (iso-strain) and Reuss (iso-stress) averages and identify each as a rigorous bound
Recognize the Voigt bound as the arithmetic mean and the Reuss bound as the harmonic mean of the phase moduli
Explain why the Reuss average is the exact modulus of a fluid suspension, which is Wood's equation
State what any pore fluid does to the Reuss shear bound and why

The Two Widest Bounds

The two arrangements of the last section have names. The iso-strain limit is the Voigt average, the stiffest a mixture can be: with volume fractions $f_i$ i and phase moduli $M_i$ i, it is the arithmetic mean $M_V = \sum_i f_i M_i$ i. The iso-stress limit is the Reuss average, the softest: it is the harmonic mean $\dfrac{1}{M_R} = \sum_i \dfrac{f_i}{M_i}$ dfracfiMi. The same two formulas apply to the bulk modulus $K$ and to the shear modulus $G$ separately. They are not estimates or fits; they are rigorous bounds, true for every possible arrangement of the given phases, and they are the widest such bounds you can draw knowing only the composition. No real mixture of these ingredients can be stiffer than Voigt or softer than Reuss.

Reading the Envelope

The figure draws both averages against volume fraction for a pair of end-members you choose. At the pure ends the two curves meet, because a one-phase rock has no arrangement freedom. In between they open into an envelope, and the width of that envelope is the honest uncertainty that composition alone leaves on the table. Notice again that the shear envelope is wider than the bulk envelope for the quartz-clay pair, for the same reason as before: the phases differ more in shear stiffness than in bulk stiffness, and the harmonic mean punishes a soft phase harder than the arithmetic mean rewards a stiff one.

The Suspension Is Exactly Reuss

The Reuss bound has a special status that the Voigt bound does not. Consider a suspension: solid grains floating in a fluid, with no grain contacts and therefore no frame. Every phase feels the same pressure, because a fluid transmits pressure equally, so the mixture is exactly iso-stress and its bulk modulus is exactly the Reuss average, not merely bounded by it. That identity is Wood's equation, the correct modulus of a mud or a fresh sediment before it has any rigidity. And because a suspension has no frame to resist shape change, its shear modulus is zero: put any finite fraction of a fluid into the Reuss shear average and the harmonic mean collapses to zero, since one phase with $G = 0$ makes the whole series average vanish. The Reuss bound with a fluid present therefore pins $G$ to zero, which is physically exactly right for a suspension and a warning sign anywhere else.

So the Voigt and Reuss bounds bracket every mixture, and at the soft end the Reuss bound turns into the exact physics of a suspension. But for a real, well-consolidated rock the envelope is uncomfortably wide, especially in shear. The next section asks whether we can do better with a single estimate inside the envelope, and meets the Hill average.

References

Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.
Wood, A. B. (1955). A Textbook of Sound (3rd ed.). G. Bell and Sons.