One of the more-frequently cited and widely-used articles by geoscientists, engineers, and physicists is the classic paper by Hashin and Shtrikman entitled A variational approach to the theory of the elastic behaviour of multiphase materials, published in 1963. The Hashin–Shtrikman (HS) approach yields bounds (lower and upper estimates) for the elastic moduli of a homogeneous, isotropic mixture of different materials, given the elastic moduli and volume fractions of the constituents. The HS bounds on the bulk modulus have been shown to be the best possible — that is, the tightest — given only the elastic moduli and fractions of the constituents, and without specifying anything about the geometry or texture of the mixture. Although a similar proof has not been given for the bounds on the shear modulus, the HS shear modulus bounds are also believed by many authors to similarly be the best possible, and are frequently described as such.
Hashin–Shtrikman bounds can also be constructed for other physical properties, such as conductivity, effective dielectric permittivity, magnetic permeability, thermal conductivity, and so on. This follows from the mathematical equivalence of the equations that describe these phenomena. Those interested are encouraged to read, for example, Chapter 9 of The Rock Physics Handbook by Mavko et al. (2003), where one can also find the Hashin–Shtrikman bound equations.
Why might you care?
The HS bounds on the bulk and shear moduli have been used in a number of geoscience applications. Some authors have used them to describe the elastic moduli of mineral mixtures, towards describing the compressional and shear wave velocities in mixed-mineral sedimentary rocks, such as limestones containing both calcite and dolomite, or sandstones containing quartz and feldspar. Yet other authors have used the bounds to roughly describe the elastic moduli of unconventional resource shales by mixing clay, organic matter, quartz, calcite, and pore fluid. The elastic moduli of mudrocks, consisting of a porous clay matrix mixed with quartz, for example, are actually well described using the equations for just the lower HS bounds, as has been reported by several authors. Indeed, the mudrock geometry — consisting of an elastically weak connected clay matrix in which elastically stronger quartz is embedded — corresponds well to that for which the HS lower bound is not simply a bound but an exact solution (at least for the bulk modulus).
The history of the bounds
Although the HS bounds are, or should be, well known to geoscientists, their history is not. The interested reader is encouraged to get a copy of the historical commentary by Hashin in Citation Classics, 1980 (available at the time of writing from ageo.co/L7CRR5). I offer a brief summary to whet your appetite.
Hashin and Shtrikman were two Israeli engineers who happened to be on leave in the United States at the same time. Both were involved in similar research involving predicting effective properties of composite media. Hashin was a civil engineer working on elastic properties, whereas Shtrikman was an electrical engineer working on electrical and magnetic properties. When they finished writing their seminal paper in 1961 they submitted it for publication to a prestigious American applied mechanics journal. As Hashin recalls, their manuscript was ‘ignominiously rejected [by an] outstanding authority’ who called their work ‘ramblings. ’ Undaunted, the authors submitted the paper to the Journal of Mechanics and Physics of Solids, where it was quickly accepted, and the rest is history.
Never give up when you feel you have a good idea that should be shared with others. The geoscience community is glad Hashin and Shtrikman persevered!
Hashin, Z, and S Shtrikman (1963). A variational approach to the theory of the elastic behaviour of multiphase materials, Journal of Mechanics and Physics of Solids 11, 127–140, DOI 10.1016/0022-5096(63)90060-7.
Hashin, Z (1980). Citation Classics 6, 11 February 1980, 344. Available online at ageo.co/L7CRR5.
Mavko, G, T Mukerji, and J Dvorkin (2003). The Rock Physics Handbook: Tools for Seismic Analysis in Porous Media, Cambridge University Press.