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.

Plain explanations are given in the lecture page about equations of state in Tomo Katsura's Website.

In more detail,

- Finite strain
- 2nd-order Birch-Murnaghan equation of state
- 3rd-order Birch-Murnaghan equation of state

Full discussion is of course given in this publicaion

Katsura, T. & Tange, Y., A simple derivation of the Birch–Murnaghan equations
of state (EOSs) and comparison with EOSs derived from other definitions of finite strain. *Minerals* 9, 745, 2019. doi.org/10.3390/min9120745