Relation between mean stress,thermodynamic, and lithostatic pressure

Pressure is one of the most important parameters to be quantified in geological problems. However, in metamorphic systems the pressure is usually calculated with two different approaches. One pressure calculation is based on petrological phase equilibria and this pressure is often termed thermodynamic pressure. The other calculation is based on continuum mechanics, which provides a mean stress that is commonly used to estimate the thermodynamic pressure. Both thermodynamic pressure calculations can be justified by the accuracy and applicability of the results. Here, we consider systems with low‐differential stress (<1 kbar) and no irreversible volumetric deformation, and refer to them as conventional systems. We investigate the relationship between mean stress and thermodynamic pressure. We discuss the meaning of thermodynamic pressure and its calculation for irreversible processes such as viscous deformation and heat conduction, which exhibit entropy production. Moreover, it is demonstrated that the mean stress for incompressible viscous deformation is essentially equal to the mean stress for the corresponding viscous deformation with elastic compressibility, if the characteristic time of deformation is five times longer than the Maxwell viscoelastic relaxation time that is equal to the ratio of shear viscosity to bulk modulus. For typical lithospheric rocks, this Maxwell time is smaller than c. 10,000 years. Therefore, numerical simulations of long‐term (>10 kyr) geodynamic processes, employing incompressible deformation, provide mean stress values that are close to the mean‐stress value associated with elastic compressibility. Finally, we show that for conventional systems the mean stress is essentially equal to the thermodynamic pressure. However, mean stress and, hence, thermodynamic pressure can be significantly different from the lithostatic pressure.