Finite strain equation of states

Finite strain vs. Birch-Murnaghan

Explicit expression for Birch-Murnaghan equation of states

The explicit expressions for third-order Birch-Murnaghan (BM3) equation of states (EoS) are

$E(V)=E^{0}+\tfrac{9}{2} V^{0} \kappa^{0} f^{2}\left[1+f\left(\kappa^{\prime 0}-4\right)\right]$

$P(V)=-\left(\frac{\partial E}{\partial V}\right)_{S}=3 \kappa^{0} f(1+2 f)^{\frac{5}{2}}\left[1+\tfrac{3}{2}\left(\kappa^{\prime 0}-4\right)\right]$

where the Eulerian strain $f$ is given by

$f \equiv \tfrac{1}{2}\left[\left(V^{0} / V\right)^{\frac{2}{3}}-1\right] \,.$

The effect of using of an arbitrary $V_\mathrm{ref}$ instead of $V_0$

Since we have $f=\frac{1}{2}[(V_0/V)^{2/3}-1]$ and we assume $f' \equiv\frac{1}{2}[(V_\mathrm{ref}/V)^{2/3}-1]$ , we have

$\frac{2f + 1}{2f'+1} = \frac{(V_0/V)^{2/3}}{(V_\mathrm{ref}/V)^{2/3}} = \left(\frac{V_0}{V_\mathrm{ref}}\right)^{2/3}$

So we have

$f = (2f'+1)\left(\frac{V_0}{V_\mathrm{ref}}\right)^{2/3} - 1 = pf' + q$

Since $E(V) = a f^3 + bf^2 + d$ , we can fit with $f'$

$E(V) = a(pf' + q)^3 + b(pf'+q)^2 + d = a'f'^3 + b'f'^2 + c'f' + d'$

So with an arbitrary $V_\mathrm{ref}$ , the fitting result is equivalent.

Analytical derivatives to obtain $P(V)$

Not easy to do, because

$P = -\frac{\partial E}{\partial V} = -\frac{\partial E}{\partial f'} \frac{\partial f'}{\partial V} = -(3af'^2 + 2bf' + c)\frac{\partial f'}{\partial V}$

where

$\frac{\partial f'}{\partial V} = \frac{\partial}{\partial V} \left[\left(\frac{V_\mathrm{ref}}{V}\right)^{2/3}-1\right] = -\frac{2}{3} \left(\frac{V_\text{ref}^{2/3}}{V^{5/3}}\right) = -\frac{2}{3} \left(\frac{V_\mathrm{ref}}{V}\right)^{2/3} V^{-1} = -\tfrac{2}{3} (2f'+1)V^{-1} \,.$

But the expression for $PV$ is a 3^rd-order polynomial vs. $f'$ ,

$PV = \tfrac{2}{3} (2f' + 1) (3af'^2 + 2bf' + c) \,.$

Gibbs free energy or enthalpy

Usually, these equation of states are for internal energy ( $U$ ), Helmholtz free energy ( $F$ ) as a function of volume $V$ . It is also possible to think about the form of enthalpy ( $H$ ), Gibbs free energy ( $G$ ) as a function of strain.

Because

$\begin{split} G &= F + PV \,,\\ H &= U + PV \,, \end{split}$

assuming $T = 0$ , from the above statement, $PV$ is a 3^rd-order polynomial of $f'$ , it is obvious that $G$ and $H$ will be 3^rd order polynomials of $f'$ too.

References

Birch–Murnaghan equation of state - Wikipedia