Abstract
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.
Highlights
Density distributions are one of the most fundamental properties to describe planetary interiors.Materials within planetary interiors are under high pressure and, have higher densities than under ambient conditions
Poirier [3] plainly derived the Eulerian finite strain and Birch–Murnaghan equation of state” (EOS) starting from the elastic anisotropy for matter compressed uniformly, there is no need to use a tensor or any special reason infinitesimal length squared with tensors in the three-dimensional space
Another question is why is a change in squared length considered; why not other powers of length? Murnaghan’s [1] argument leaves one to reflect that the use of squared length was based on Pythagorean theorem but there is no physical reason
Summary
Density distributions are one of the most fundamental properties to describe planetary interiors. Birch [2] extended the theory of Murnaghan [1] to derive a prototype of the EOS, which is referred to as the Birch–Murnaghan EOS They proposed an EOS using tensors in three dimensions, finite strain. They proposed an EOS using tensors in three dimensions, Poirier [3] plainly derived the Eulerian finite strain and Birch–Murnaghan EOS starting from the but other simpler arguments are available. Poirier [3] plainly derived the Eulerian finite strain and Birch–Murnaghan EOS starting from the elastic anisotropy for matter compressed uniformly, there is no need to use a tensor or any special reason infinitesimal length squared with tensors in the three-dimensional space. A partial derivative of a quantity Y with respect to some parameter X at constant temperature is, referred to as a X derivative of Y
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