A Simple Derivation of the Birch–Murnaghan Equations of State (EOSs) and Comparison with EOSs Derived from Other Definitions of Finite Strain

• Published in: Minerals (Basel) 2019-12, Vol.9 (12), p.745
• Main Authors: Katsura, Tomoo, Tange, Yoshinori
• Format: Article
• Language: English
• Subjects: Bulk modulus; Compression; Differential equations; Dimensions; Equations of state; Free energy; Isotopes; Length; Mathematical analysis; Pressure; Strain
• ISSN: 2075-163X; 2075-163X
• DOI: 10.3390/min9120745

Eulerian finite strain of an elastically isotropic body is defined using the expansion of squared length and the post-compression state as reference. The key to deriving second-, third- and fourth-order Birch–Murnaghan equations-of-state (EOSs) is not requiring a differential to describe the dimensions of a body owing to isotropic, uniform, and finite change in length and, therefore, volume. Truncation of higher orders of finite strain to express the Helmholtz free energy is not equal to ignoring higher-order pressure derivatives of the bulk modulus as zero. To better understand the Eulerian scheme, finite strain is defined by taking the pre-compressed state as the reference and EOSs are derived in both the Lagrangian and Eulerian schemes. In the Lagrangian scheme, pressure increases less significantly upon compression than the Eulerian scheme. Different Eulerian strains are defined by expansion of linear and cubed length and the first- and third-power Eulerian EOSs are derived in these schemes. Fitting analysis of pressure-scale-free data using these equations indicates that the Lagrangian scheme is inappropriate to describe P-V-T relations of MgO, whereas three Eulerian EOSs including the Birch–Murnaghan EOS have equivalent significance.