Not to be confused with Birch–Murnaghan equation of state.
The Murnaghan equation of state is a relationship between the volume of a body and the pressure to which it is subjected. This is one of many state equations that have been used in earth sciences and shock physics to model the behavior of matter under conditions of high pressure. It owes its name to Francis D. Murnaghan[1] who proposed it in 1944 to reflect material behavior under a pressure range as wide as possible to reflect an experimentally established fact: the more a solid is compressed, the more difficult it is to compress further.
The Murnaghan equation is derived, under certain assumptions, from the equations of continuum mechanics. It involves two adjustable parameters: the modulus of incompressibility K0 and its first derivative with respect to the pressure, K′0, both measured at ambient pressure. In general, these coefficients are determined by a regression on experimentally obtained values of volume V as a function of the pressure P. These experimental data can be obtained by X-ray diffraction or by shock tests. Regression can also be performed on the values of the energy as a function of the volume obtained from ab-initio and molecular dynamics calculations.
The Murnaghan equation of state is typically expressed as:
If the reduction in volume under compression is low, i.e., for V/V0 greater than about 90%, the Murnaghan equation can model experimental data with satisfactory accuracy. Moreover, unlike many proposed equations of state, it gives an explicit expression of the volume as a function of pressure V(P). But its range of validity is limited and physical interpretation inadequate. However, this equation of state continues to be widely used in models of solid explosives. Of more elaborate equations of state, the most used in earth physics is the Birch–Murnaghan equation of state. In shock physics of metals and alloys, another widely used equation of state is the Mie–Grüneisen equation of state.
^F.D., Murnaghan (1944), "The Compressibility of Media under Extreme Pressures", Proceedings of the National Academy of Sciences of the United States of America, 30 (9): 244–247, Bibcode:1944PNAS...30..244M, doi:10.1073/pnas.30.9.244, PMC 1078704, PMID 16588651
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Global Information![{\displaystyle P(V)={\frac {K_{0}}{K_{0}'}}\left[\left({\frac {V}{V_{0}}}\right)^{-K_{0}'}-1\right]\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d03e1532bb5031e21c996b0a54e6cc212553a340)