Relation between mean stress, thermodynamic, and lithostatic pressure

Received: 26 January 2018 | Accepted: 26 July 2018 DOI: 10.1111/jmg.12446 ORIGINAL ARTICLE Relation between mean stress, thermodynamic, and lithostatic pressure Evangelos Moulas1 | Stefan M. Schmalholz1 | Yury Podladchikov1 | Lucie Tajčmanová2 | Dimitrios Kostopoulos3 | Lukas Baumgartner1 1 Institute of Earth Science (ISTE), University of Lausanne, Lausanne, Switzerland 2 Department of Earth Science, ETH Zurich, Zurich, Switzerland 3 Faculty of Geology & Geoenvironment, National & Kapodistrian University of Athens, Athens, Greece Correspondence Evangelos Moulas, Institute of Earth Science (ISTE), University of Lausanne, Lausanne, Switzerland. Email: evangelos.moulas@unil.ch Funding information University of Lausanne; ERC, Grant/ Award Number: 335577 Abstract 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 incom- Handling Editor: Doug Robinson pressible 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. KEYWORDS geodynamic models, lithostatic pressure, local thermodynamic equilibrium, mean stress, thermodynamic pressure 1 | INTRODUCTION Metamorphic rocks are a unique window to conditions prevailing in the earth's crust and mantle. Thermodynamic pressure and temperature (P–T) information is culled from J Metamorph Geol. 2018;1–14. mineral modal amounts, mineral chemistry, and/or microstructures, thought to reflect a state of equilibrium attained at some point during rock history (e.g., Spear, 1993). Key parameters such as thermodynamic pressure (P) and temperature (T) are often the outcome of complex wileyonlinelibrary.com/journal/jmg © 2018 John Wiley & Sons Ltd | 1 2 | calculations involving minimization of the Gibbs free energy of a system (e.g., Connolly, 2005; De Capitani & Brown, 1987; De Capitani & Petrakakis, 2010; Karpov, Chudnenko, Kulik, & Bychinskii, 2002; Kulik, 2002; White, Johnson, & Dantzig, 1958). In most of these calculations, it is implicitly assumed that the thermodynamic pressure of the reacting mixture is the same for all fluid/ mineral phases present. This is likely a good approximation in cases wherein the metamorphic assemblage is under low differential stress (<~1 kbar), the irreversible volumetric deformation is negligible and is, therefore, commonly employed in phase equilibria experiments. On the other hand, geodynamic models help us understand the thermomechanical evolution of rocks by using mass, energy, and momentum conservation equations as well as constitutive relations that describe material behaviour (e.g., Gerya, 2010; Turcotte & Schubert, 2014). Such models usually result in the calculation of stress and temperature evolution in deforming regions. A major challenge here is to establish a link between the stress state of a deforming rock and its thermodynamic pressure, i.e., the pressure that can be used in metamorphic phase equilibria. A common practice in geodynamic modelling is to use mean stress as a proxy for the thermodynamic pressure of the system considered. However, there are cases where this approximation may not be valid (e.g., Dahlen, 1992). Hobbs and Ord (2017) have recently suggested that thermodynamic pressure can be defined only for processes where entropy does not change with time (i.e., isentropic). Similarly, Powell, Evans, Green, and White (2017) have also stated that equilibrium thermodynamic calculations cannot be applied to situations where irreversible deformation, as an entropy‐producing process, takes place. If true, this would have significant consequences for earth sciences where natural processes are most probably never isentropic and entropy production is never zero due to irreversible processes, such as radiogenic heat production, chemical diffusion, viscous deformation, and the ever‐present heat conduction. Moreover, according to Hobbs and Ord (2017), the use of viscous incompressible formulation, a commonly used approximation in geodynamics to quantify the magnitude of mean stress in deforming rocks, is inappropriate to estimate thermodynamic pressure since volume deformation is not being taken into account. The aforementioned statements challenge the link between the common geodynamic models of rock deformation that involve dissipative processes, and calculations of metamorphic recrystallization. The purpose of this study is to demonstrate that thermodynamic pressure can be safely defined for nonisentropic processes, and that equilibrium thermodynamic MOULAS ET AL. calculations are certainly applicable when considering conventional systems for which differential stress is low and irreversible volumetric deformation is negligible. In addition, we show that the incompressible approximation of viscous flow yields accurate mean‐stress estimates for deforming viscous rocks that are elastically compressible. Finally, we establish that the thermodynamic pressure of conventional systems may be significantly different from the lithostatic pressure. An assessment of unconventional models that consider large differential stress or dramatic effects of low‐differential stress on mineral phase equilibria (e.g., Schmalholz & Podladchikov, 2014; Tajčmanová, Vrijmoed, & Moulas, 2015; Wheeler, 2014) is beyond the scope of this work. 2 | HYDROSTATIC STRESS AND LITHOSTATIC PRESSURE By definition, isotropic‐stress conditions are those where all normal stress components are equal to a mean‐stress value, and shear stresses are zero. This state of stress is known as hydrostatic. A fundamental result of hydrostatics, i.e., the study of fluids at rest, is that under the influence of gravity only, the stress state is isotropic and the mean stress at any point in a fluid is related to the weight of the overlying fluid column. The relation between hydrostatic pressure, which in this case is equal to mean stress, and depth is given by the formula (e.g., Batchelor, 1967, p. 19): dPh ¼ ρg dz (1) where Ph is the hydrostatic pressure (Pa), z the depth (m), ρ the density (kg/m3), and g the acceleration of gravity (m/s2). If one assumes constant density and constant gravity acceleration, the above relationship can be integrated to provide the expression for Ph (e.g., Batchelor, 1967, p. 20): Ph ðzÞ ¼ P0 þ ρgðz z0 Þ (2) where P0 is the reference hydrostatic pressure at depth z0. This expression has been extensively used in the geological literature for the calculation of depth of metamorphic recrystallization. Interestingly, it inherently contains the assumption that density remains constant during integration with respect to depth (z) and, therefore, the result is valid only for incompressible fluids at rest. When rocks are considered, Ph is commonly referred to as lithostatic pressure (PL), i.e., pressure that results only from the weight of static rocks (Bucher & Frey, 2002, pp. 72–73). In fact, there is no particular justification why stresses in the earth should be considered lithostatic except for simplicity reasons (e.g., Jaeger, 1969, p. 172). MOULAS | ET AL. 3 | THERMODYNAMIC RELATIONSHIPS du ¼ Tds By definition, state functions are independent of the path followed by the physical processes (i.e., isothermal or isentropic etc.). The experimental basis of thermodynamics is the recognition of the volume (m3), temperature (K), and thermodynamic pressure (Pa) as fundamental state variables. Thermodynamic pressure is defined as the force f (N) acting perpendicularly on a unit of area A (m2) (e.g., Kondepudi & Prigogine, 1998, p. 9): P¼ f A (3) This definition applies to gases, fluids, and solids under isotropic‐stress conditions and without irreversible volumetric deformation. Another experimental finding in thermodynamics is the establishment of path‐ and process‐ independent relationships between state functions, commonly known as equations of state (EOS). When expressed for density, EOS are functional relationships of the form: ρ ¼ ρðT; PÞ (4) A typical example for an EOS is the ideal‐gas law: ρðT; PÞ ¼ MP RT (5) where R is the universal gas constant and M is the molar mass of the gas. For more complex fluids, many other formulations can be used (e.g., Ferry & Baumgartner, 1987). By definition, EOS are independent of the path followed by the physical processes (i.e., isothermal or isentropic etc.). More complex EOS can be found in the literature regarding the density of minerals as a function of P and T (e.g., Anderson, 1995). Note that EOS apply regardless if the operating processes are infinitely slow or not. Having the EOS of a substance is sufficient to describe its density as a function of P and T but not its energy. In quasistatic fluids, for example, the work increment (dW) done by P is (e.g., Fermi, 1936): dW ¼ f dx ¼ PAdx ¼ PdV (6) where dx is the displacement in the direction of the force f and A is the area normal to f. The latter equation implies that P is the proportionality factor between mechanical work and volume change. For a finite volume change, the mechanical work can be obtained by integrating Equation 6. If chemical composition and temperature are taken into account, then, the variation of the specific internal energy (du) is given by (e.g., De Groot & Mazur, 1984, p. 458): Pdν þ ∑ck¼1 μk dwk 3 (7) where u (J/kg) is the specific internal energy of the fluid, T (K) is the temperature, s (J K−1 kg−1) is the specific entropy, P (Pa) is the thermodynamic pressure, ν (m3/kg) is the specific volume, μk (J/kg) is the chemical potential of the kth component, wk is the total mass of the kth component divided over the total mass of the system, c is the number of components and d represents the exact differential. Note that the energy (u), entropy (s), volume (ν), and chemical potential (μ) are given per unit mass (kg). Naturally, thermodynamic pressure can then be back‐calculated from internal energy by partial differentiation of Equation 7 (e.g., Callen, 1960, p. 31; Landau & Lifshitz, 1976, p. 42):   @u P¼ (8) @ν s;wk According to Equation 8, the thermodynamic pressure (P) is the proportionality factor (with a minus sign) of the internal‐energy increment to volume change when entropy (s) and composition (components wk) are kept constant during partial differentiation. The reason why we go at such length in clarifying the issue is the recent statement by Hobbs and Ord (2017) that: “the thermodynamic pressure, the chemical potentials, and the stress are only defined, inter alia, at constant entropy, so that although the entropy may change during the evolution of a metamorphic system as deformation and chemical reactions proceed we are only interested in states where the  entropy production is zero, expressed as s ¼ 0 where the over‐dot denotes differentiation with respect to time”. This reasoning led the previous authors to the conclusion that “the role of tectonic pressure is only relevant to phase equilibrium when an equilibrium thermodynamic pressure can  be defined” and thus for isentropic (i.e., s ¼ 0) processes only. Following the logic of Hobbs and Ord (2017), it is not entirely clear why entropy has to remain constant during an actual physical process in order for pressure (P) to be “usefully defined”. One way to interpret the statement of Hobbs and Ord (2017) is to relate it with the “definition” of P by Equation 8. This would imply that the previous authors interpreted the constraint of s being constant (during partial differentiation) in Equation 8 as a requirement of s to be constant in an actual physical process. If this is the case, then the aforementioned statement is incorrect on mathematical grounds for two reasons. First, entropy differ entiation with respect to time (expressed as s ¼ 0) does not necessarily mean that entropy production is zero (see Appendix 1 for details). Second, keeping s constant during partial differentiation (Equation 8) does not mean that P is defined only for processes where s is constant in time. 4 | MOULAS ET AL. From a mathematical perspective, partial differentiation of any function [e.g., u(s, ν, wk)] keeping certain variables constant (e.g., s), leads to a new function that contains the same variables as the original one. Therefore, Equation 8 that was derived from partial differentiation of Equation 7, keeping s constant, is a function of the same variables as those that define Equation 7, i.e., s, ν, wk, even though s was kept constant during partial differentiation. Keeping certain thermodynamic variables constant during partial differentiation does not place any constraints on the use of the derived functions with respect to the variables that were kept constant. By definition, state functions are functions of the state and are not sensitive to the process(es) that led to that particular state. A more familiar example in this case would be the usage of EOS for minerals where volume (V) is commonly given as a function of P–T. For illustration purposes, we will consider the EOS as it is used by Holland and Powell (1998), in order to calculate the bulk modulus KT:   1 1 @VðP; TÞ
¼ KT ðP; TÞ V ðP; T Þ @P T (9) 1 ¼ K298 ð1 1:5 10 4 ðT 298ÞÞ þ 4ðP 1Þ sound (c.f. Müller & Weiss, 2005, p. 3) and this is evident by the numerous industrial applications of thermodynamics, combustion engines and shock‐wave EOS for solids (c.f. Müller & Müller, 2009). In that respect, geodynamic processes take place at sufficiently slow rates for the application of equilibrium thermodynamics relationships and hence P is well‐defined for nonisentropic geodynamic processes. Last but not least, if Equation 8 was the only way to “usefully” define thermodynamic pressure in petrology, then the a priori knowledge of the specific‐energy function (u) is required in order to perform partial differentiation. In other words, one has to know u in order to be able to determine P. However, in petrology the free energy and other thermodynamic potentials are calculated by using EOS that assume knowledge of pressure (P). It is therefore the energy, which is usually calculated from the pressure and not vice versa. where K298 is the isothermal bulk modulus (bar) measured at 298 K and 1 bar. The value of K298 is tabulated by Holland and Powell (1998) for a large number of minerals. We emphasize here that although KT was calculated by partial differentiation by keeping temperature constant, it is a function of both P and T and can be used for any P–T path chosen including nonisothermal processes. Similar conclusions can be drawn if one uses different energy functions to calculate pressure (see Appendix 2 for details). Based on the arguments presented in the previous paragraph, it is not possible to justify the Hobbs and Ord (2017) statement regarding the definition of pressure on mathematical grounds. Alternatively, we can consider the statement that entropy has to remain “constant” for the definition of pressure stems from interpretations of equilibrium thermodynamics. It is common in classical equilibrium thermodynamics to consider sufficiently slow and quasistatic processes where it is assumed that time derivatives related to the physical processes vanish. But, does quasistatic mean that the physical processes are infinitely slow? If the thermodynamic pressure (among other variables) could be defined only for isentropic processes, as Hobbs and Ord (2017) suggest, then even the simplest examples from equilibrium thermodynamics would be invalidated. An example would be a thermodynamic cycle (e.g., Carnot, Otto) where s is generally changing during a cycle and yet P can be defined at every part of the cycle. However, sufficiently slow could be as fast as the speed of It is commonly held that equilibrium thermodynamic relationships (e.g., Equations 7 and 8) are justified only when a system is very close to thermodynamic equilibrium, or when all ongoing processes are reversible. In other words, equilibrium thermodynamics applies when metamorphic reactions have small overstepping with respect to the equilibrium‐reaction boundaries, chemical diffusion has homogenized minerals on a certain scale, thermal diffusion has equilibrated the temperature on the same scale and last, but not least, viscous relaxation has relaxed all differential stresses. In fact, this is a sufficient but not a necessary condition for the application of equilibrium thermodynamic relationships. In order to apply equilibrium thermodynamics in petrology, the concept of local equilibrium is used in one way or another, since metamorphic systems are never in global equilibrium and there are ever‐present dissipative processes like conductive cooling of the Earth. The framework of classical irreversible thermodynamics (CIT) is the established way to quantify far‐from‐equilibrium processes, such as thermal convection (e.g., De Groot & Mazur, 1984; Lebon, Jou, & Casas‐Vázquez, 2008; Müller & Müller, 2009). The foundation of CIT is the assumption of local thermodynamic equilibrium, according to which: “local and instantaneous relations between thermodynamic quantities in a system out of equilibrium are the same as for a uniform system in equilibrium” (Lebon et al., 2008, p. 39). In other words, CIT assumes that Equations 7 and 8 are 4 | THE ASSUMPTION OF LOCAL THERMODYNAMIC EQUILIBRIUM: WHAT DOES IT REALLY MEAN? MOULAS | ET AL. satisfied even in reacting and deforming systems far from equilibrium. The studies of CIT require constitutive relationships such as EOS (e.g., Equation 4). The applicability of such relationships applies also for dynamic systems as witnessed by the shock‐wave EOS for minerals (e.g., Ahrens, 1987, 1993), combustion thermodynamics (e.g., Powers, 2016) detonation waves (e.g., Müller & Müller, 2009; Powers, 2016), kinetics (e.g., Lasaga, 1998), and other dynamic processes. The simplest example of an irreversible geological process is the temperature evolution during simple heat conduction, where the caloric equation of state, an equilibrium thermodynamics relationship, must be used in order to relate changes in energy with changes in temperature. Similarly, simple chemical diffusion utilizes the equilibrium chemical potential (i.e., activity–composition) relationships when diffusion in minerals is quantified. Even if a body is capable of performing pure elastic deformation, thermal elasticity generally requires the development of temperature gradients under elastic strains. Consequently, temperature gradients will induce thermal diffusion and dissipation (e.g., Zener, 1937, 1938). Similar conclusions can be drawn for the case of chemical diffusion in minerals. Contrary to CIT, the applicability of equilibrium thermodynamics in dissipative systems has been recently challenged (e.g., Hobbs & Ord, 2017; Powell et al., 2017). If true, the ever‐present geothermal gradient, associated with dissipative heat conduction and radiogenic heat production, would preclude the usage of equilibrium thermodynamics during the whole history of the Earth (for a more formal definition of dissipative systems and the difference between entropy‐producing and nonisentropic processes see Appendix 1). 5 | THERMODYNAMIC AND GEODYNAMIC CALCULATIONS There are two types of calculations that are commonly employed in geology, namely thermodynamic and geodynamic. These calculations are based on different sets of assumptions, which can be justified by the accuracy and applicability of the computed results. Elaborate EOS quantifying volumetric changes are often involved in order to accurately predict phase boundaries at phase diagrams and physical properties according to experiments (e.g., Anderson, 1995). By contrast, geodynamic calculations often neglect the volumetric deformation of the materials (i.e., they follow the incompressible approximation) but still yield physically accurate and meaningful results that reproduce fluid‐dynamic experiments (e.g., thermal convection; Schubert, Turcotte, & Olson, 2001; Turcotte & Schubert, 2014) or other geologically relevant structures (e.g., Pollard 5 & Fletcher, 2005). Worth noting here is that the most‐commonly used expression for lithostatic pressure (Equation 2) in petrological interpretations is derived by integrating Equation 1 assuming constant density and this, strictly speaking, also follows the incompressible approximation. In the ensuing paragraphs, we present cases to demonstrate the applicability and accuracy of the incompressible approximation in geodynamic models. 5.1 | A static case In order to demonstrate that the incompressibility assumption is valid for first‐order approximations of geodynamic problems, we will modify Equation 1 to include the effects of elastic compressibility of solids. For this demonstration, the effects of increasing temperature with depth (namely thermal expansion) are neglected because the thermal expansion counteracts the effect of the P‐related increase in density with depth. Hence, the maximum effect of compressibility (assuming that composition is constant) is inherent in the following calculations. First, density is a function of P and is given by the relation: dρ ¼ βdP ρ (10) where β is the material compressibility (Pa−1) and an isotropic material has been considered. Compressibility is the inverse of bulk modulus and can also depend on P and other factors like temperature (e.g., Equation 9) or composition, but we will consider it constant for now. Integration of Equation 10, assuming a reference density (ρ0) measured at reference P0, yields the following expression for density as a function of P: ρ ¼ ρ0 eβðP P0 Þ (11) Inserting Equation 11 into Equation 1 and integrating (assuming g is constant) yields an expression that relates depth (m) to P (Pa): z¼ 1 eβðP0 βρ0 g PÞ (12) For comparison purposes, the corresponding expression assuming constant density yields the standard expression for the lithostatic pressure and its relation to depth: z¼ P0 P ρ0 g (13) Substitution of average values for feldspar (ρ0 = 2,620 kg/m3, β = 1.82·10−11 Pa−1) in Equations 12 and 13 shows that the P correction for elastic compressibility is negligible compared to the incompressible approximation (Figure 1). 6 | MOULAS (a) (b) F I G U R E 1 (a) Lithostatic pressure profiles (PL). The solid line (PcL ) is the lithostatic pressure predicted by a compressible formulation (Equation 12) and the black diamonds (PiL ) show the lithostatic pressure predicted by the incompressible approximation (Equations 2 and 13). The lithostatic pressure is given in GPa and the difference between PcL and PiL is almost indistinguishable. The acceleration of gravity (g) is taken as 9.81 m/s2 and the compressibility is approximated by a representative value for feldspars (55 GPa)−1 according to Angel (2004a, 2004b). (b) The pressure difference (MPa) between the two lithostatic pressures, PcL and PiL 5.2 | A dynamic case In the case of deforming rocks, such analytical integration as the one presented in Section 5.1 is not easily performed and more complex mechanical models are needed. Over recent years, a number of theoretical models based on continuum mechanics have been presented in an attempt to quantify mechanical and thermodynamic variables in evolving metamorphic systems where deformation and recrystallization occur simultaneously (Gerya, 2015; Mancktelow, 2008; Moulas, Podladchikov, Aranovich, & Kostopoulos, 2013; Tajčmanová et al., 2015, and references therein). In these studies, it has been suggested that field variables such as stress need not be homogeneously distributed and certainly mean stress cannot always be a simple function of depth. Several reasons can be responsible for large or small deviations from lithostatic pressure and they involve topography changes, density variations, tectonic loading, volumetric deformation, viscosity heterogeneities etc. Contrary to most engineering studies on solids, geological studies investigate processes where the shear strain during rock deformation can be orders of magnitude larger than volumetric strain. This difference in strain is the main reason why the incompressible approximation is widely used in geodynamic calculations. In addition, due to the large time‐scales involved in geological deformation, some ET AL. models consider only the viscous deformation of rocks by treating them as highly‐viscous fluids. The mean stress predicted from such models is usually equated with pressure derived from equilibrium phase assemblages (Burg & Gerya, 2005; Gerya, Perchuk, & Burg, 2008; Reuber, Kaus, Schmalholz, & White, 2016; Schmalholz, Duretz, Schenker, & Podladchikov, 2014; Sizova, Gerya, & Brown, 2012). Hobbs and Ord (2017) suggested that: “except for elastic deformations, mean stress is not useful in discussing mineral phase equilibrium”. Nevertheless, ignoring compressibility in models of viscous deformation does not mean that thermodynamic pressure cannot be defined or its corresponding value is inaccurate. As shown in Figure 1, considering a lithostatic pressure (PL) gradient assuming constant density (Equation 2), yields a PL estimate that is directly comparable to that calculated by additionally considering volumetric deformation (Equation 10). Similarly, the mean stress calculated from incompressible viscous deformation can be quite similar to the mean stress calculated from elastic volumetric deformation. To illustrate this point, let us ponder a simplified geological scenario of viscous deformation (for details on the model see Supporting Information Appendix S1). Here, a layer 10 times more viscous than its surroundings is compressed in pure shear under the influence of gravity (Figure 2). Both the matrix and the layer are taken to be isotropic and elastically compressible having no irreversible volumetric deformation. The materials are thus elastic with respect to volumetric deformation and purely viscous with respect to shear deformation. This approximation is commonly employed when long geodynamic time‐scales are considered (>~1,000 year). In this example, pressure is the mean stress that is used in the elastic EOS and therefore, for this example, mean stress (σm) and thermodynamic pressure (P) are synonymous. We will examine three cases of varying bulk modulus along with an incompressible case for reference. In the absence of tectonic loading, the calculated mean stress is equal to the lithostatic pressure (Figure 3a). Note that, in this simplified example, there is no topography and there are no horizontal gradients of density that would require differential stresses to be supported. Once a tectonic compressive stress is imposed at the sides, the strong viscous layer exhibits a mean stress different from the lithostatic (Figure 3b). In this example, we consider temperature and composition to be constant, hence mean stress and P can be calculated using Equation 10. Since it is commonplace in mechanics to incorporate the bulk modulus (i.e., the inverse of compressibility) in the calculation of P, Equation 10 can be modified as: dP ¼ Kb dV V (14) where Kb is the isothermal bulk modulus (Pa) and the materials are assumed to be isotropic. The second term in MOULAS | ET AL. F I G U R E 2 Two‐dimensional model configuration. Pure shear of a viscous layer embedded in a viscous matrix. The bulk rate of shortening, ɛ_ b , is 10−14 s−1. Both the layer and the matrix are elastically compressible and they have the same bulk moduli. Free slip is imposed at all boundaries. The shear viscosity of the layer is 10 times larger than the one of the matrix. Gravity acts downwards in a direction parallel to y. The reference density (ρ0) of the two materials is considered to be the same and equal to 2,700 kg/m3. Each side of the box is 1 km long. The description of the numerical solution can be found in the Supporting Information Appendix S1 the right‐hand side of Equation 14 is the volumetric strain due to elastic deformation. Equation 14 dictates that materials with larger bulk moduli will develop larger P for a given elastic volumetric strain. The P evolution in the modelled layer has been investigated for three different bulk moduli, namely 30, 60, and 100 GPa (Figure 4). Initially, the model domain is subjected only to the gravitational force (as in Figure 3a); P is close to the lithostatic 7 approximation from Equation 2, but not exactly equal to it because density changes as a function of P. Once the domain is compressed at the sides (as in Figure 3b), the mean stress (σm; also equal to P in this case) in the highly viscous layer deviates from the lithostatic pressure (PL) and increases with time (Figure 4). Mean stress builds continually up to the value predicted for the incompressible case for all three compressible materials and henceforth remains constant provided that the layer keeps being compressed (Figure 4). At a given time of 20 kyr, we remove the exerted tectonic load and mean stress retreats back to its lithostatic value. The mean‐stress deviation that develops in the viscous layer is proportional to the layer's far‐field stress, which depends on the viscosity difference between the layer and the matrix and its strain rate (e.g., Schmalholz & Podladchikov, 1999; Smith, 1975). The time needed for the mean‐stress buildup/relaxation depends on the different characteristic Maxwell times of the system (Figure 5). For the studied case, the characteristic time is given by: tM ¼ μl Kb (15) where μl is the shear viscosity of the layer. This formulation looks similar to the Maxwell characteristic time for shear deformation but we note that our model is purely viscous with respect to shear deformation and purely elastic with respect to volumetric deformation. The applicability of the lithostatic pressure formula (Equation 2) is actually also restricted to time‐scales larger than this Maxwell relaxation time, because the lithostatic formula assumes that differential stresses have been relaxed. 5.3 | Applications to systems with large mean‐stress variations The direct relation between thermodynamic pressure (P) and mean stress (σm) may not apply to systems under high F I G U R E 3 Mean stress (σm) calculated for the case of viscous incompressible deformation. (a) In the absence of imposed tectonic stress, σm is given by the lithostatic formula. (b) In the case of pure shear of a viscous layer, the mean stress (σm) in the layer is different to the lithostatic. In this example, the mean stress is equal to the thermodynamic pressure (P) that is used in the EOS if the material is considered to be compressible. The vectors show the local velocity directions 8 | MOULAS F I G U R E 4 Mean‐stress evolution (σm) in the centre of the viscous layer (μl = 1021 Pa·s) for the case of compressible and incompressible materials for the model configuration displayed in Figure 2. Three different curves correspond to three different bulk moduli and the solid black line indicates mean‐stress evolution predicted in the viscous layer by the incompressible approximation. The dashed horizontal line indicates the lithostatic pressure in the absence of tectonic loading. Tectonic compression by pure shear stops at a given time (here 20 kyr) and mean stress relaxes back to the lithostatic pressure differential stress (e.g., Sekerka & Cahn, 2004) and currently there are different views on the importance of the differential stress on metamorphic phase equilibria (c.f. Dahlen, 1992; Fletcher, 2015; Hobbs & Ord, 2015; Wheeler, 2014). This situation becomes more complex for fluid–rock interactions in porous rocks where interface conditions must be considered (c.f. Dahlen, 1992). Nevertheless, there are particular cases where rocks can experience locally large mean‐stress variations from the lithostatic (a) ET AL. pressure while themselves being under low differential stress. Geologically relevant examples are shear zones and inclusions that support low differential stress. As shown already by Schmalholz and Podladchikov (2013), a shear zone that develops during continental shortening will have in general low differential stress but its mean stress may be different from the lithostatic (Figure 6a,b). Although this may be counterintuitive, it has been demonstrated that weak zones (i.e., zones that cannot support high differential stress) may develop significant mean‐stress deviations from the far‐field mean stress, which in turn, will be different from the lithostatic (Moulas, Burg, & Podladchikov, 2014; Schmid & Podladchikov, 2003). This is also the case when compressible materials are considered (Figure 7). The reason why this is possible is the ability of the surrounding strong rocks to support high differential stresses (Figures 6c,d and 7). Factors that control the magnitude of the mean‐stress variation in viscously deforming rocks are: (a) the shape and orientation of the weak zones, (b) the viscosity ratio between the weak rocks and their surroundings, and (c) the magnitude of the far‐field stress. Therefore, a weak viscous inclusion or a weak shear zone within a strong rock unit will experience a mean stress, which may be considerably different from the lithostatic pressure and, in extreme cases, can reach up to two to three times the lithostatic pressure value (c.f. Moulas et al., 2014). If the volumetric irreversible deformation of the weak zone is considered to be negligible (see also Supporting Information Appendix S2 for details), the mean stress will be approximately equal to the thermodynamic pressure since the stress components within the weak zone are almost equal (i.e., small differential stress; Figures 6d and 7). We have therefore demonstrated, in marked contrast with Hobbs and Ord (2017), that thermodynamic pressure, associated with low differential stress, can control metamorphic (b) F I G U R E 5 Mean‐stress difference (δσm) evolution in the layer during compression (a) and relaxation after compression has ceased (b) for the model configuration displayed in Figure 2. The parameter δσm is the mean stress at the centre of the layer minus the mean stress of the matrix at the top of the model. The lithostatic pressure (PL) within the layer has been subtracted and the result has been normalized to the layer far‐field stress (2(μl − μm) ɛ_ b ). Time has been normalized using the characteristic Maxwell time of each material (Equation 15). A simple scaling relationship is also shown with a solid line for reference. Time in (b) is shifted, so that t = 0 coincides with the termination of compression MOULAS | ET AL. (a) (b) (c) (d) 9 F I G U R E 6 (a) Tectonic overpressure (P0) calculated from the data of Schmalholz and Podladchikov (2013) for the case of continental shortening. The lithostatic pressure is PL and PMoho is the initial (lithostatic) pressure at the Moho before compression starts. Variations of P0 indicate variations of mean stress (σm) from the lithostatic (PL). (b) Differential stress calculated from the data of Schmalholz and Podladchikov (2013). The differential stress has been normalized by PMoho. (c) Pressure of a weak viscous elliptical inclusion subjected to far‐field stress (τff). Pressure has been normalized to the far‐field stress of the strong surrounding rocks (for details see Moulas et al., 2014). (d) Mohr diagram for the weak elliptical inclusion shown in (c). For the calculation of the Mohr diagram, stress is taken positive in compression. Note that the weak elliptical inclusion has a homogenous state of stress (constant mean stress) that is represented by the small red circle. The dashed circle corresponds to the state of stress of the matrix at the location indicated by a red dot in (c), which is at the contact with the inclusion. The far‐ field stress is indicated by a solid black circle. The mean stress of the far‐field has been subtracted from the solution, hence the circle is centred at 0. For details on the analytical solution, refer to Schmid and Podladchikov (2003) reactions during dissipative deformation. Such a situation is not restricted to the large scale and to viscous rheology as demonstrated by models of plastic extrusion and elastoplastic pressure vessels (e.g., Mancktelow, 2008; Tajčmanová et al., 2014). 6 | DISCUSSION We have presented theoretical arguments based on calculus and thermodynamics in an attempt to clarify misconceptions regarding the definition of thermodynamic pressure from internal energy in dissipative systems. The thermodynamic pressure is a state function and can therefore be independent of the type of process affecting a system, in contrast to the assertions of Hobbs and Ord (2017) for the opposite. Additionally, we have presented arguments regarding the use of equilibrium thermodynamic relations (such as EOS) in dynamic processes according to theories coming from nonequilibrium thermodynamics. Having a dissipative system far from equilibrium does not mean that equilibrium thermodynamic relations do not 10 | MOULAS ET AL. F I G U R E 7 Mean stress (σm) in a viscous layer containing three weak viscous inclusions. The model configuration is as in Figure 2, the main difference being the addition of these weak inclusions that have different aspect ratios and orientations. Two of the inclusions (left and right) have an aspect ratio of 2 while the one in the centre is circular. The left inclusion has a long axis parallel to the layer. The inclusion in the right has a long axis perpendicular to the layer. The viscosity of the inclusions and the matrix is 1019 Pa·s and the viscosity of the layer is 1021 Pa·s. The far‐field bulk‐shortening rate is 10−13 s−1. The bulk modulus of the materials is 50 GPa. The governing equations are given in Supporting Information Appendix S1. (a) Mean‐stress (σm) distribution at the end of the simulation. (b) Mean‐stress (σm) evolution in the layer and within the weak inclusions. (c) Differential stress (σd) evolution in the layer and the inclusions. The weak circular inclusion (labelled with black square) yields a similar mean stress as the strong layer (labelled with magenta circle, b), but the differential stress in the inclusion is negligible compared to the differential stress in the layer (c) hold. Simple examples to demonstrate this point is the modelling of chemical/thermal diffusion processes where entropy production and dissipation are nonzero even if the studied thermodynamic variables do not evolve with time. If the definition of thermodynamic pressure was restricted to isentropic processes only, as Hobbs and Ord (2017) have proposed, then EOS would not be applicable to natural processes and therefore any relation of thermodynamic pressure to depth would be invalid. Geodynamic models often assume that geomaterials are incompressible. This simplification is commonly used in models where mean stress is calculated using the lithostatic approximation (Equation 2) or in models where the viscous deformation of rocks is considered. We have demonstrated that for conventional systems (Figures 1 and 4) thermodynamic pressure, calculated from compressible models, is essentially identical to the mean stress that is calculated from corresponding incompressible models; apart from an initial transient period which is controlled by the Maxwell relaxation time, tM. This was found to be the case for either static or viscously deforming rocks. In deforming materials with variable density and topography however, the mean stress will deviate from the lithostatic pressure. Here, the compressibility of the deforming materials will have an effect on the time needed for these mean‐stress deviations to develop (Figure 4). In the example shown, the time for the mean‐stress buildup/relaxation is proportional to the characteristic Maxwell time defined in Equation 15 and its duration is about five times the Maxwell time (Figure 5). For typical lithospheric viscosities and bulk moduli (1021 Pa·s and 40 GPa, respectively), this stress buildup lasts almost 4 kyr. To resolve this, stress buildup in typical MOULAS numerical geodynamic models would require time steps in the order of a few hundred years. Such small time steps are, however, infeasible for most current models, since it would require more than 10,000 time steps to simulate a deformation process lasting longer than 10 Ma. Our results suggest that the incompressible approximation provides reasonable estimates for the mean stress in viscous geological materials with elastic volume changes, which deform for times longer than a few thousands of years, and can be far more accurate than simple lithostatic approximations. Although the link between mean stress and thermodynamic pressure is not clear when differential stresses are high, the mean stress that develops in systems with small differential stress will be almost equal to the thermodynamic pressure, if irreversible bulk deformation is negligible. For specific geodynamic situations, rocks with small differential stress can have a mean stress significantly different from the lithostatic. Examples are mechanically weak shear zones on all geological scales, which form in stressed regimes, or weak inclusions inside stressed rocks with higher differential stress. For such weak rocks, the phase equilibria will record the actual mean stress and not necessarily the lithostatic pressure. 7 | ET AL. | CONCLUSIONS The recent challenging of the relation between mean stress, that evolves in space and time as it is calculated by geodynamic models, and thermodynamic pressure, used in conventional metamorphic phase equilibria, would have, if true, severe implications for the foundations of metamorphic petrology. Indeed, if thermodynamic pressure could not be “usefully defined” in deforming rocks, then the majority of thermodynamic calculations could not be performed for any geodynamic setting, ranging from collisional orogenic belts to divergent plate margins. Similarly, thermodynamic calculations could not be applied to heat conducting, hence dissipating rock either in the lithosphere or in the convecting mantle/core until the complete cooling of the earth. Here, we have clarified that the definition of thermodynamic pressure does not depend on any constraint concerning variations in entropy since it is a state function. Therefore, the thermodynamic pressure is well defined for processes exhibiting entropy variations with time and for dissipative processes. Hence, lines of argumentation supporting the view of thermodynamic pressure being only defined for isentropic processes are incorrect. The mean stress in a deforming incompressible viscous material is essentially identical to the mean stress in a deforming elastically compressible viscous material, if the irreversible volumetric deformation is negligible, and if the deformation time is approximately five times longer than 11 the Maxwell time. For typical lithospheric rocks, this deformation time is smaller than ten thousand years. Therefore, the mean stress calculated during numerical simulations of long‐term (>10 kyr) geodynamic processes, which assume incompressible deformation, provides accurate mean stress estimates. Stress relaxation towards lithostatic pressure does not only require the consideration of time‐scales that are five times longer than the Maxwell time, but it also requires the absence of all causes for the nonhydrostatic stress state. Causes for such nonhydrostatic stress state can be lateral variations of topography and of crustal thickness that may take billions of years to disappear. Moreover, the nonhydrostatic stress state is maintained as long as far‐field tectonic stresses are present, which can be significantly longer than a few ten thousand years. Consequently, conventional metamorphic systems under low differential stress may experience thermodynamic pressure significantly different from the lithostatic pressure over geological time‐scales. Such thermodynamic pressure is well approximated by the mean stress, which can be calculated with common geodynamic models. ACKNOWLEDGEMENTS We acknowledge the participants of the workshop “Interplay between mineral reactions and deformation” (September 2017 – ETH Zurich) for the stimulating discussions that motivated us to write this manuscript. E.M. S.M.S., Y.P., and L.B. acknowledge the University of Lausanne for financial support. L.T. acknowledges the ERC starting grant (335577). Constructive reviews by Ross Angel & John Wheeler, comments of an anonymous reviewer, and editorial handling by D. Robinson are greatly appreciated. ORCID Evangelos Moulas http://orcid.org/0000-0002-2783-4633 REFERENCES Ahrens, T. J. (1987). Shock-wave techniques for geophysics and planetary physics. In C. G. Sammis & T. L. Heyney (Eds.), Methods of experimental physics (vol. 24(A), pp. 185–235). Academic Press. https://doi.org/10.1016/S0076-695X(08)60587-6 Ahrens, T. J. (1993). Equations of state. In J. R. Asay & M. Shahinpoor (Eds.), High-pressure shock compression of solids (pp. 75– 114). New York, NY: Springer-Verlag. https://doi.org/10.1007/ 978-1-4612-0911-9 Anderson, O. L. (1995). Equations of state of solids for geophysics and ceramic science (p. 405). New York, NY and Oxford, UK: Oxford University Press. Angel, R. J. (2004a). Equations of state of plagioclase feldspars. Contributions to Mineralogy and Petrology, 146, 506–512. https://doi. org/10.1007/s00410-003-0515-5 12 | Angel, R. J. (2004b). Equations of state of Plagioclase feldspars – ERRATUM. Contributions to Mineralogy and Petrology, 147, 241. https://doi.org/10.1007/s00410-004-0564-4 Batchelor, G. K. (1967). An introduction to fluid dynamics. Cambridge, UK: Cambridge University Press. Bucher, K., & Frey, M. (2002). Petrogenesis of metamorphic rocks (7th edn). Berlin Heidelberg: Springer, 428 pp. https://doi.org/10. 1007/978-3-662-04914-3 Burg, J.-P., & Gerya, T. V. (2005). The role of viscous heating in Barrovian metamorphism of collisional orogens: Thermomechanical models and application to the Lepontine Dome in the Central Alps. Journal of Metamorphic Geology, 23, 75–95. https://doi.org/ 10.1111/j.1525-1314.2005.00563.x Callen, H. B. (1960). Thermodynamics: an introduction to the physical theories of equilibrium thermostatics and irreversible thermodynamics. New York: John Wiley & Sons, Inc., 376 pp. Connolly, J. A. D. (2005). Computation of phase equilibria by linear programming: A tool for geodynamic modeling and its application to subduction zone decarbonation. Earth and Planetary Science Letters, 236, 524–541. https://doi.org/10.1016/j.epsl. 2005.04.033 Dahlen, F. A. (1992). Metamorphism of nonhydrostatically stressed rocks. American Journal of Science, 292(3), 184–198. https://doi. org/10.2475/ajs.292.3.184 De Capitani, C., & Brown, T. H. (1987). The computation of chemical equilibrium in complex systems containing non‐ideal solutions. Geochimica et Cosmochimica Acta, 51(10), 2639–2652. De Capitani, C., & Petrakakis, K. (2010). The computation of equilibrium assemblage diagrams with Theriak/Domino software. American Mineralogist, 95, 1006–1016. https://doi.org/10.2138/am.2010.3354 De Groot, S.R., & Mazur, P. (1984). Non-equilibrium thermodynamics. New York: Dover, 511 pp. Fermi, E. (1936). Thermodynamics. New York: Dover, 160 pp. Ferry, J. F., & Baumgartner, L. (1987). Thermodynamic models of molecular fluids at the elevated pressures and temperatures of crustal metamorphism. Reviews in Mineralogy, 17(1), 323–365. Fletcher, R. C. (2015). Dramatic effects of stress on metamorphic reactions: COMMENT. Geology, 43(1), e354. https://doi.org/10. 1130/G36302C.1 Gerya, T. (2010). Introduction to numerical geodynamic modelling. Cambridge, UK: Cambridge University Press, 345 pp. Gerya, T. (2015). Tectonic overpressure and underpressure in lithospheric tectonics and metamorphism. Journal of Metamorphic Geology, 33, 785–800. https://doi.org/10.1111/jmg.12144 Gerya, T. V., Perchuk, L. L., & Burg, J.-P. (2008). Transient hot channels: Perpetrating and regurgitating ultrahigh‐pressure, high‐ temperature crust–mantle associations in collision belts. Lithos, 103, 236–256. https://doi.org/10.1016/j.lithos.2007.09.017 Hobbs, B. E., & Ord, A. (2015). Dramatic effects of stress on metamorphic reactions: COMMENT. Geology, 43(11), e372. https:// doi.org/10.1130/G37070C.1 Hobbs, B. E., & Ord, A. (2017). Pressure and equilibrium in deforming rocks. Journal of Metamorphic Geology, 35, 967–982. https://doi.org/10.1111/jmg.12263 Holland, T. J. B., & Powell, R. (1998). An internally consistent thermodynamic data set for phases of petrological interest. Journal of Metamorphic Geology, 16, 309–343. Jaeger, J. C. (1969). Elasticity, fracture and flow, 3rd ed. London: Butler & Tanner Ltd, 268 pp. MOULAS ET AL. Karpov, I. K., Chudnenko, K. V., Kulik, D. A., & Bychinskii, V. A. (2002). The convex programming minimization of five thermodynamic potentials other than Gibbs energy in geochemical modeling. American Journal of Science, 302, 281–311. https://doi.org/ 10.2475/ajs.302.4.281 Kondepudi, D., & Prigogine, I. (1998). Modern thermodynamics. United Kingdom: John Wiley & Sons, Inc., 552 pp. Kulik, D. A. (2002). Gibbs energy minimization approach to modeling sorption equilibria at the mineral‐water interface: Thermodynamic relations for multi‐site‐surface complexation. American Journal of Science, 302, 227–279. https://doi.org/10.2475/ajs.302. 3.227 Landau, L. D., & Lifshitz, E. M. (1976). Course of theoretical physics – Statistical Physics p.1, 3rd edn. Oxford: Pergamon, 544 pp. Landau, L. D., & Lifshitz, E. M. (1986). Course of theoretical physics – theory of elasticity, 3rd edn. Oxford: Butterworth-Heinemann, 187 pp. Lasaga, A. C. (1998). Kinetic theory in the earth sciences. Princeton, New Jersey: Princeton University Press, 811 pp. https://doi.org/10. 1515/9781400864874 Lebon, G., Jou, D., & Casas-Vázquez, J. (2008). Understanding non-equilibrium thermodynamics (p. 325). Berlin Heidelberg, Germany: Springer-Verlag. https://doi.org/10.1007/978-3-54074252-4 Mancktelow, N. S. (2008). Tectonic pressure: Theoretical concepts and modelled examples. Lithos, 103, 149–177. https://doi.org/10. 1016/j.lithos.2007.09.013 Moulas, E., Burg, J. P., & Podladchikov, Y. Y. (2014). Stress field associated with elliptical inclusions in a deforming matrix: Mathematical model and implications for tectonic overpressure in the lithosphere. Tectonophysics, 631, 37–49. https://doi.org/10.1016/ j.tecto.2014.05.004 Moulas, E., Podladchikov, Y. Y., Aranovich, L. Y., & Kostopoulos, D. K. (2013). The problem of depth in geology: When pressure does not translate into depth. Petrology, 21, 527–538. https://doi. org/10.1134/S0869591113060052 Müller, I., & Müller, W. H. (2009). Fundamentals of thermodynamics and applications. Berlin Heidelberg, Germany: Springer-Verlag, 404 pp. Müller, I., & Weiss, W. (2005). Entropy and energy – A universal competition (p. 273). Berlin Heidelberg, Germany: Springer-Verlag. Pollard, D. D., & Fletcher, R. C. (2005). Fundamentals of structural geology. Cambridge, UK: Cambridge University Press, 514 pp. Powell, R., Evans, K. A., Green, E. R. C., & White, R. W. (2017). On equilibrium in non‐hydrostatic metamorphic systems. Journal of Metamorphic Geology, 36, 419–438. https://doi.org/10.1111/ jmg.12298 Powers, J. M. (2016). Combustion thermodynamics and dynamics. Cambridge, UK: Cambridge University Press, 464 pp. Reuber, G., Kaus, B. J. P., Schmalholz, S. M., & White, R. W. (2016). Nonlithostatic pressure during subduction and collision and the formation of (ultra)high‐pressure rocks. Geology, 44(5), 343–346. https://doi.org/10.1130/G37595.1 Schmalholz, S. M., Duretz, T., Schenker, F. L., & Podladchikov, Y. Y. (2014). Kinematics and dynamics of tectonic nappes: 2‐D numerical modelling and implications for high and ultra‐high MOULAS | ET AL. pressure tectonism in the Western Alps. Tectonophysics, 631, 160–175. https://doi.org/10.1016/j.tecto.2014.05.018 Schmalholz, S. M., & Podladchikov, Y. (1999). Buckling versus folding: Importance of viscoelasticity. Geophysical Research Letters, 26, 2641–2644. https://doi.org/10.1029/1999GL900412 Schmalholz, S. M., & Podladchikov, Y. (2013). Tectonic overpressure in weak crustal‐scale shear zones and implications for the exhumation of high‐pressure rocks. Geophysical Research Letters, 40(10), 1984–1988. https://doi.org/10.1002/grl.50417 Schmalholz, S. M., & Podladchikov, Y. (2014). Metamorphism under stress: The problem of relating minerals to depth. Geology, 42(8), 733–734. https://doi.org/10.1130/focus0822014.1 Schmid, D. W., & Podladchikov, Y. Y. (2003). Analytical solutions for deformable elliptical inclusions in general shear. Geophysical Journal International, 155(1), 269–288. https://doi.org/10.1046/j. 1365-246X.2003.02042.x Schubert, G., Turcotte, D. L., & Olson, O. (2001). Mantle convection in the earth and planets. Cambridge, UK: Cambridge University Press, 940 pp. https://doi.org/10.1017/CBO9780511612879 Sekerka, R. F., & Cahn, J. W. (2004). Solid‐liquid equilibrium for non‐hydrostatic stress. Acta Materialia, 52, 1663–1668. https://d oi.org/10.1016/j.actamat.2003.12.010 Sizova, E., Gerya, T., & Brown, M. (2012). Exhumation mechanisms of melt‐bearing ultrahigh pressure crustal rocks during collision of spontaneously moving plates. Journal of Metamorphic Geology, 30, 927–955. https://doi.org/10.1111/j.1525-1314. 2012.01004.x Smith, R. B. (1975). Unified theory of the onset of folding, boudinage and mullion structure. Geological Society of America, Bulletin, 86, 1601–1609. https://doi.org/10.1130/0016-7606(1975)86&lt;1601: UTOTOO&gt;2.0.CO;2 Spear, F. S. (1993). Metamorphic phase equilibria and pressuretemperature-time paths. USA: Mineralogical Society of America, 799 pp. Tajčmanová, L., Podladchikov, Y. Y., Powell, R., Moulas, E., Vrijmoed, J., & Connolly, J. A. D. (2014). Grain‐scale pressure variations and chemical equilibrium in high‐grade metamorphic rocks. Journal of Metamorphic Geology, 32, 195–207. https://doi. org/10.1111/jmg.12066 Tajčmanová, L., Vrijmoed, J., & Moulas, E. (2015). Grain‐scale pressure variations in metamorphic rocks: implications for the interpretation of petrographic observations. Lithos, 216–217, 338–351. https://doi.org/10.1016/j.lithos.2015.01.006 Thompson, W. (1852). On the universal tendency in nature to the dissipation of mechanical energy. Philosophical Magazine, Series, 4, 511–514. Turcotte, D. L., & Schubert, G. (2014). Geodynamics, 3rd edn. Cambridge, UK: Cambridge University Press, 623 pp. https://doi.org/ 10.1017/CBO9780511843877 Wheeler, J. (2014). Dramatic effects of stress on metamorphic reactions. Geology, 42(8), 647–650. https://doi.org/10.1130/ G35718.1 White, W. B., Johnson, S. M., & Dantzig, G. B. (1958). Chemical equilibrium in complex mixtures. The Journal of Chemical Physics, 28, 751–755. https://doi.org/10.1063/1.1744264 Zener, C. (1937). Internal friction in solids. I. Theory of internal friction in reeds. Physical Review, 52(3), 230–235. https://doi.org/10. 1103/PhysRev.52.230 13 Zener, C. (1938). Internal friction in Solids. II. General theory of thermoelastic internal friction. Physical Review, 53(1), 90–99. https://d oi.org/10.1103/PhysRev.53.90 SUPPORTING INFORMATION Additional supporting information may be found online in the Supporting Information section at the end of the article. Appendix S1. Model formulation for isothermal compressible viscous deformation. Appendix S2. Mean stress and thermodynamic pressure for materials with dissipative volumetric deformation. How to cite this article: Moulas E, Schmalholz SM, Podladchikov Y, Tajčmanová L, Kostopoulos D, Baumgartner L. Relation between mean stress, thermodynamic, and lithostatic pressure. J Metamorph Geol. 2018;00:1–14. https://doi.org/ 10.1111/jmg.12446 APPENDIX 1 CLASSICAL IRREVERSIBLE THERMODYNAMICS AND ENTROPY EVOLUTION The purpose of Appendix 1 is to show the difference between entropy production and entropy change in time. This difference becomes important when isentropic processes are considered. We would first like to refer to some basic principles from what is known in the literature as classical irreversible thermodynamics (CIT, e.g., De Groot & Mazur, 1984; Lebon et al., 2008; Müller & Müller, 2009). CIT is based on principles that allow the use of local thermodynamic equilibrium in dissipative systems. Dissipative systems are those where mechanical or other forms of energy (e.g., chemical) dissipate due to irreversible processes (e.g., Thompson, 1852). Irreversible processes produce entropy and therefore, dissipation is associated with entropy production. The framework of CIT combines fundamental physical laws (conservation of mass, momentum, energy, etc.) and respects that the entropy production for every irreversible process must be greater than zero. The entropy production is not to be confused with entropy change in space or time. The evolution of entropy is given by a partial differential equation of the form (e.g., De Groot & Mazur, 1984, their eq. 10): 14 | MOULAS @ρs ¼ @t @Jis;tot þ Qs @xi APPENDIX 2 (1.1) where ρ is density, s is the mass‐specific entropy (in J K−1 kg−1), Jis;tot is the total entropy flux in the ith direction and Qs is the entropy production (repeated indices imply summation). This equation can be envisaged as the definition of the entropy production, Qs. According to CIT (c.f. De Groot & Mazur, 1984, their eq. 11): Qs ≥ 0 ET AL. (1.2) where the equality holds only if all the processes that operate in the system are reversible. Note that entropy production is positive for every irreversible process, so that the entropy production of one irreversible process cannot reduce the entropy production of another irreversible process. This CIT principle “is valid for all systems, regardless of the boundary conditions” (e.g., Kondepudi & Prigogine, 1998, p. 336). However, entropy in nature varies with time. For example, entropy decreases in a glass of water that is cooling and freezing in the fridge. Inspection of Equation 1.1 in that case reveals that in order to decrease entropy in a given volume, the divergence of the total flux of entropy must be larger than the production term. A common misconception in the literature is that the entropy production term (Qs) is confused with entropy variation with time (see for example Hobbs & Ord, 2017; their eq. 7). This misconception can be demonstrated if we consider a stationary state (where the time derivative of entropy is zero) for heat production in the Earth. Setting the time derivative of Equation 1.1 to zero reveals that the divergence of the entropic flux must compensate for the entropy production. Therefore, even in the case where locally, entropy does not change with time (time derivative is zero), local entropy production may still exist and thermodynamic relationships are still valid in that locality according to CIT. Consequently, statements like the one given by Hobbs and Ord (2017) that “two or more dissipative processes balance with respect to dissipation” or “all dissipation stops when T s_ ¼ 0” disagree with the definition of the entropy production (Qs). THERMODYNAMIC RELATIONS INVOLVING HELMHOLTZ FREE ENERGY The purpose of Appendix 2 is to show how pressure can be related to the changes of Helmholtz free energy at constant temperature (T). Given the internal energy (U) and the Helmholtz free energy (F) of a homogeneous elastic body (both in J), the total stresses can be calculated from:     1 @F 1 @U σ ij ¼ ¼ (2.1) V @ɛij T;M k V @ɛij S;M k where V (m3) is volume, Mk (kg) is the total mass of component k, σij (Pa) are the total stress tensor components and ɛij (dimensionless) are the strain components for small deformations. In that case, the internal energy variation can be written as (e.g., Landau & Lifshitz, 1986, pp. 8–9): dU ¼ TdS þ Vσ ij dɛij þ ∑ck¼1 μk dMk (2.2) and the corresponding Helmholtz free energy variation can be expressed as: dF ¼ SdT þ Vσ ij dɛij þ ∑ck¼1 μk dMk (2.3) For a purely volumetric deformation, the volumetric strain components are related to volume changes by dV=V ¼ dεii and the stress components are related to the thermodynamic pressure by σij = − Pδij, so that finally Vσij dɛij = −PdV (Einstein's summation rule on repeated indices apply and δij is Kronecker's delta, i.e., it is equal to 1 when i = j and zero otherwise). In that case, thermodynamic pressure can be obtained by partial differentiation of Helmholtz energy (at constant temperature):   @F P¼ (2.4) @V T;Mk Although temperature is kept constant during partial differentiation, pressure is still defined if temperature varies with time. The corresponding Helmholtz free energy variation is: dF ¼ SdT PdV þ ∑ck¼1 μk dMk (2.5) APPENDIX S1 MODEL FORMULATION FOR ISOTHERMAL COMPRESSIBLE VISCOUS DEFORMATION Here we present the set of equations that were used in order to obtain the two-dimensional stress field the results of which are shown in Figures 3-5 and 7. In order to calculate stresses and velocity, we solved the equations of momentum and mass balance assuming plane strain. The equation of mass balance for one phase is: � + � �� = � (S1.1) where � is the ith coordinate, � is time, � is density and � is the velocity component in the ith direction (repeated indices imply summation). In the absence of inertial forces, the conservation of linear momentum becomes: � − �� = � (S1.2) Where � are the total stress tensor components and � is the acceleration of gravity in the jth direction. The stress tensor � is symmetric as required by the conservation of angular momentum. Total stress (� ) can be decomposed as: � = −�� � + � ˙′ (S1.3) where � is the shear viscosity of the material and �� is the mean stress taken as positive in compression. The mean stress is obtained by: � �� = − The strain rate deviator ( ˙′ ) is given by: � � + � � ˙′ = − � � � (S1.4) (S1.5) Mean stress and volumetric deformation are related via an EOS of the form: ��� = �� �� � (S1.6) where �� is the isothermal bulk modulus and assuming that the material is isotropic. Differential stress is given by the formula � = √ � −� 2 1 + �2 (S1.7) This closed system of equations was solved using an iterative finite difference approach (c.f. Chorin, 1997; Poliakov, Cundall, Podladchikov, & Lyakhovskii, 1993) and employing a regular, staggered-grid discretization (Virieux, 1986). Since we are considering only small deformations, we did not consider advection. APPENDIX S2 MEAN STRESS AND THERMODYNAMIC PRESSURE FOR MATERIALS WITH DISSIPATIVE VOLUMETRIC DEFORMATION In this section we show the influence of bulk viscosity, on mean stress and pressure calculations. We consider isotropic materials for simplicity. We would like to emphasise that different mechanical models (Maxwell, Kelvin etc) have different results with respect to the calculated values for mean stress and therefore we give the two most basic models below. When materials experience irreversible volumetric deformation (e.g. viscous compaction) mean stress (�� ) is not generally equal to the thermodynamic pressure (�). A common relation that is given for fluids is (e.g. Batchelor, 1967, p. 154): �� = � + �� = −�� − � ̇ (S2.1) where � is the bulk viscosity (in Pa·s) and the material is assumed to be isotropic. In this formulation mean stress is decomposed into two parts, one is the thermodynamic pressure � (also termed equilibrium pressure; Batchelor, 1967, p. 154) and the other is a pressure that results from viscous volumetric deformation �� (e.g. Müller & Müller, 2009, p. 344). When we consider an elastic isotropic material with negligible bulk viscosity then �� reduces to � (as shown also in Appendix 2). This formulation (Equation S2.1) is a Kelvin-type model for viscoelastic volumetric deformation, which means that both bulk deformations are arranged in parallel. For such parallel arrangement the volumetric strain and strain rate is equal for viscous and elastic bulk deformation, however, the corresponding pressures are different. Similar formulations can be obtained if the plastic volumetric deformation is considered, however, these terms are usually assumed to be negligible. An alternative formulation for viscoelastic irreversible deformation is the Maxwell model in which the volumetric strains are arranged in series. For this arrangement, both pressures are identical, �� = �, but the corresponding volumetric strain and strain rates are different. In such a model the volumetric deformation rate is the sum of viscous and elastic deformation as shown below �� �̇ (S2.2) ̇ = ̇� + ̇ = − − � �� However, we note that under constant load this model would constantly creep and, for example, the volume of any material under the influence of gravity would eventually reduce to zero. Evidently, this is not the case and this is the reason why a simple Maxwell model is not used to describe large volumetric deformations. Following a similar logic, plastic dilatational deformation must also become negligible when rocks are under high confining pressure and when they experience a few percent strain (e.g. Vermeer & de Borst, 1984, pp. 14–15). REFERENCES Batchelor, G. K. (1967). An introduction to fluid dynamics. Cambridge, UK: Cambridge University Press. 615 pp. Chorin, A. J. (1997). A numerical method for solving incompressible viscous flow problems. Journal of Computational Physics, 135, 118–125. https://doi.org/10.1006/jcph.1997.5716 De Groot, S.R., & Mazur, P. (1984). Non-equilibrium thermodynamics. New York: Dover, 511 pp. 2 Landau, L. D., & Lifshitz, E. M. (1987). Course of theoretical physics – fluid mechanics, 2nd edn. Oxford: Pergamon, 539 pp. Müller, I., & Müller, W. H. (2009). Fundamentals of thermodynamics and applications. Berlin Heidelberg, Germany: Springer-Verlag, 404 pp. Poliakov, A. N. B., Cundall, P. A., Podladchikov, Y. Y., & Lyakhovskii, V. A. (1993). An explicit inertial method for the simulation of visco-elastic flow: An evaluation of elastic effects on diapiric flow in two- and three- layers models. In D. B. Stone, & S. K. Runcorn (Eds.), Flow and creep in the solar system (pp. 175–195). Dordrecht, The Netherlands: Kluwer. Vermeer, P. A., & de Borst, R. (1984). Non‐associated plasticity for soils, concrete and rocks. Heron, 29, 1–64. Virieux, J. (1986). P‐SV wave propagation in heterogeneous media: Velocity‐stress finite‐ difference method. Geophysics, 51, 889–901. https://doi.org/10.1190/1.1442147 3