Birch-Murnaghan

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This “finite strain EoS” is derived (Birch, 1947) from the assumption that the strain energy of a solid undergoing compression can be expressed as a Taylor series in the finite Eulerian strain, . Expansion to 4th-order in the strain yields an EoS:



The normalised pressure for the Birch-Murnaghan EoS is defined as (Stacey, Brennan & Irvine, 1981):



This allows the Birch-Murnaghan EoS to be expressed as a simple polynomial:



If this EoS is truncated at 2nd-order in the energy, then the coefficient of fE must be identical to zero, which requires that K'0T has the fixed value of 4 (higher-order terms are ignored). The 3rd-order truncation, in which the coefficient of is set to zero yields a three-parameter EoS (with V0T, K0T and K'0T ) with an implied value of K''0T  given by (Anderson, 1995):        


                       

The expressions for the bulk modulus and its first derivative for the 3rd-order Birch-Murnaghan EoS are therefore (Angel, 2000b):





These are equivalent to the expressions given by Birch (1986) in his appendix 1, and by Anderson (1995) in his equations (6.52) to (6.55), except for a typographical error of K' for K'' in his equation (6.53). The expressions given by Stacey et al. (1981) are correct except that for K' which is truncated at fE rather than after the fE2 which is required for the expression to be exact. Expressions for the 2nd-order Birch-Murnaghan EoS can be obtained by setting K'0T = 4 in all of the above.