Poirier & Tarantola (1998) developed an EoS based upon the “natural” or “Hencky” measure of linear strain fN which, for hydrostatic compression, may be written as . Note that this definition has been inverted with respect to that used in Angel (2000a) in order to obtain positive values of fN on compression. This yields a pressure-volume relationship expanded to 4th-order in strain of:
which can also be written as in terms of a normalized pressure as:
with and
Truncation of this “Natural strain” EoS at 2nd-order in the strain is obtained by setting a = b = 0 and it implies a value of K'0T = 2, different from that of the 2nd-order Birch-Murnaghan EoS. For truncation at 3rd-order in the strain, a ≠ 0, b = 0, and the implied value of K''0T is given by:
This value for K''0T is normally substantially larger than that implied by the truncation of the 3rd-order Birch-Murnaghan EoS, and this often results in a significantly poorer fit of P-V data. The bulk modulus and its derivatives for all orders of this EoS are: