Research Article | Open Access

Sergey V. Simonenko, "Fundamentals of the Thermohydrogravidynamic Theory of the Global Seismotectonic Activity of the Earth", *International Journal of Geophysics*, vol. 2013, Article ID 519829, 39 pages, 2013. https://doi.org/10.1155/2013/519829

# Fundamentals of the Thermohydrogravidynamic Theory of the Global Seismotectonic Activity of the Earth

**Academic Editor:**Umberta Tinivella

#### Abstract

The article presents the fundamentals of the cosmic geophysics (representing the deterministic thermohydrogravidynamic theory intended for earthquakes prediction) based on the author's generalized differential formulation of the first law of thermodynamics extending the classical Gibbs' formulation by taking into account (along with the classical infinitesimal change of heat and the classical infinitesimal change of the internal energy ) the infinitesimal increment of the macroscopic kinetic energy , the infinitesimal increment of the gravitational potential energy , the generalized expression for the infinitesimal work done by the nonpotential terrestrial stress forces (determined by the symmetric stress tensor ) acting on the boundary of the continuum region , and the infinitesimal increment of energy due to the cosmic and terrestrial nonstationary energy gravitational influence on the continuum region during the infinitesimal time . Based on the established generalized differential formulation of the first law of thermodynamics, the author explains the founded cosmic energy gravitational genesis of the strong Chinese 2008 and the strong Japanese 2011 earthquakes.

#### 1. Introduction

The problem of the long-term predictions of the strong earthquakes [1, 2] is the significant problem of the modern geophysics. The analysis of the period 1977–1985 revealed [3] the strongly nonrandom tendencies in the earthquake-induced geodetic changes (owing to the mass redistribution of material inside the Earth) related to the change of the Earth’s rotation and the Earth’s gravitational field. The analysis of the period 1977–1993 (characterized by 11015 major earthquakes) revealed [4] the strong earthquakes’ tendency to increase the Earth’s spin (rotational) energy. The analysis of the same period 1977–1993 revealed [5] “an extremely strong tendency for the earthquakes to decrease the global gravitational energy” confirming the inherent relation of the earthquakes with the transformation of the Earth’s gravitational energy into the seismic wave energy and frictional heat. The previous analysis of the principal geological features of the past years revealed [6] the geological evidence for a pulsating gravitation related to periodic variation of the Earth’s radius during the geological evolution of the Earth. The combination of satellite and gravimetric data revealed [7] the free-air anomalies of the Earth’s gravitational field.

It is well known that “the deterministic prediction of the time of origin, hypocentral (or epicentral) location, and magnitude of an impending earthquake is an open scientific problem” [8]. It was conjectured [8] that the possible earthquake prediction and warning must be carried out on a deterministic basis. However, it was pointed out [8] with some regret that the modern “study of the physical conditions that give rise to an earthquake and the processes that precede a seismic rupture of an ordinary event are at a very preliminary stage and, consequently, the techniques of prediction of time of origin, epicentre, and magnitude of an impending earthquake now available are below standard”. The authors [8] argued that “a new strong theoretical scientific effort is necessary to try to understand the physics of the earthquake”. It was conjectured [8] that the present level of knowledge of the geophysical processes “is unable to achieve the objective of a deterministic prediction of an ordinary seismic event, but it certainly will in a more or less distant future tackle the problem with seriousness and avoiding scientifically incorrect, wasteful, and inconclusive shortcuts, as sometimes has been done”. Sgrigna and Conti conjectured [8] that “it will take long time (may be years, tens of years, or centuries) because this approach requires a great cultural, financial, and organizational effort on an international basis”. It was conjectured [8] that a possible contribution to a deterministic earthquake prediction approach is related to observations and physical modelling of earthquake precursors to formulate, in perspective, “a unified theory able to explain the causes of its genesis, and the dynamics, rheology, and microphysics of its preparation, occurrence, postseismic relaxation, and interseismic phases”. It was conjectured [8] that “the study of the physical conditions that give rise to an earthquake and of the processes that precede a seismic rupture is at a very preliminary stage and, consequently, the techniques of prediction available at the moment are below standard”. However, Sgrigna and Conti believe [8] that “it should be better to pursue the deterministic prediction approach even if a reliable deterministic method of earthquake prediction will presumably be available only in the more distant future”.

It was pointed out [9] that the gravity changes (derived from regional gravity monitoring data in China from 1998 to 2005) exhibited noticeable variations before the occurrence of two large earthquakes in 2008 in the areas surrounding Yutian (Xinjiang) and Wenchuan (Sichuan). A recent research [10] by Zhan and his colleagues demonstrated that significant gravity changes were observed before all nine large earthquakes that ruptured within or near mainland China from 2001 to 2008. It was pointed out [9] that the past experience and empirical data showed that “earthquakes typically occur within one to two years after a period of significant gravity changes in the region in question”. It was concluded [9] that the “additional research is needed to remove the subjective nature in the determination of the timeframe of a forecasted earthquake”.

The need of the thermohydrogravidynamic approach [11, 12] is confirmed by previous studies [3–5, 13] and by noticeable variations of gravitational field identified [9, 10] before strong earthquakes in China from 2001 to 2008. The necessity to consider the gravitational field (during the strong earthquakes) is also related to the observations of the slow gravitational [14, 15] ground waves resulting from strong earthquakes and spreading out from the focal regions [16, 17] of earthquakes. Lomnitz pointed out [16] that the gravitational ground waves (related to great earthquakes) “have been regularly reported for many years and remain a controversial subject in earthquake seismology”. Richter presented [18] the detailed analysis of these observations and made the conclusion that “there is almost certainly a real phenomenon of progressing or standing waves seen on soft ground in the meizoseismal areas of great earthquakes”. Lomnitz presented [17] the real evidence of the existence of the slow gravitational waves in sedimentary layers during strong earthquakes. The fundamental connections of the geodynamics, seismicity, and volcanism with gravitation (and the slow gravitational ground waves resulting from strong earthquakes) are presented in the works [19–22].

It was conjectured [23] that the recent destructive earthquakes occurred in Sichuan (China, 2008), Italy (2009), Haiti (2010), Chile (2010), New Zealand (2010), and Japan (2011) “have shown that, in present state, scientific researchers have achieved little or almost nothing in the implementation of short- and medium-term earthquake prediction, which would be useful for disaster mitigation measures”. It was conjectured [23] that “this regrettable situation could be ascribed to the present poor level of achievements in earthquake forecast”. It was pointed out [23] that “although many methods have been claimed to be capable of predicting earthquakes (as numerous presentations on earthquake precursors regularly show at every international meeting), the problem of formulating such predictions in a quantitative, rigorous, and repeatable way is still open”. It was formulated [23] that “another problem of practical implementation of earthquake forecasting could be due to the lack of common understanding and exchange of information between the scientific community and the governmental authorities that are responsible for earthquake damage mitigation in each country: they operate in two different environments, they aim at different tasks, and they generally speak two different languages”. It was pointed out [23] that “the way how seismologists should formulate their forecasts and how they should transfer them to decision-makers and to the public is still a tricky issue”. It was clearly formulated [23] that “the formulation of probabilistic earthquake forecasts with large uncertainties in space and time and very low probability levels is still difficult to be used by decision-making people”. It was conjectured [23] that “in real circumstances the authorities deal with critical problems related to the high cost of evacuating the population from an area where the scientific methods estimate an expected rate of destructive earthquake as one in many thousand days, while they require much more deterministic statements”. In the special issue [23] of the International Journal of Geophysics, Console et al. assessed the status of the art of earthquake forecasts and their applicability. They invited authors “to report methods and case studies that could concretely contribute or, at least seemed promising, to improve the present frustrating situation, regarding the practical use of earthquake forecasts” [23].

In this article, by accepting with gratitude the personal invitation from Dr. Reem Ali and Dr. Radwa Ibrahim (representing the Editorial Office of the International Journal of Geophysics) to submit an article to the special issue on “Geophysical Methods for Environmental Studies”, the author presents the fundamentals of the thermohydrogravidynamic theory intended for deterministic prediction of earthquakes. The thermohydrogravidynamic theory is based on the established generalized differential formulation [11, 12, 24–26] of the first law of thermodynamics (for moving rotating deforming compressible heat-conducting stratified macroscopic continuum region subjected to the nonstationary Newtonian gravity):

extending the classical formulation [27] by taking into account (along with the classical infinitesimal change of heat and the classical infinitesimal change of the internal energy ) the infinitesimal increment of the macroscopic kinetic energy , the infinitesimal increment of the gravitational potential energy , the generalized expression for the infinitesimal work done by the nonpotential terrestrial stress forces (determined by the symmetric stress tensor ) acting on the boundary of the continuum region , and the infinitesimal increment of energy due to the cosmic and terrestrial nonstationary energy gravitational influence on the continuum region during the infinitesimal time .

In Section 2 we begin by considering the inherent physical incompleteness of the classical expression [28, 29] for the macroscopic kinetic energy per unit mass defined (in classical nonequilibrium thermodynamics) as the sum of the macroscopic translational kinetic energy per unit mass of the mass center of a continuum region and the macroscopic internal rotational kinetic energy per unit mass , where is the speed of the mass center of a small continuum region, is an angular velocity of internal rotation [29, 30], and is an inertia moment per unit mass of a small continuum region [28]. The classical de Groot and Mazur expression has inherent physical incompleteness [24, 31] related to the questionable assumption about the rigid-like rotation of a small continuum region. The classical de Groot and Mazur expression [28] does not consider the nonequilibrium component of the macroscopic velocity field related to the velocity shear defined by the rate of strain tensor . In Section 2 the macroscopic kinetic energy per unit mass is presented [31] as a sum of the macroscopic translational kinetic energy per unit mass of the mass center of a continuum region, the classical macroscopic internal rotational kinetic energy per unit mass [28], the new macroscopic internal shear kinetic energy per unit mass [31], and the new macroscopic internal kinetic energy of shear-rotational coupling per unit mass [31] with a small correction. The presented expression for and its particular form for homogeneous continuum regions of spherical and cubical shapes generalized [31] the classical de Groot and Mazur expression in classical nonequilibrium thermodynamics [28, 29] by taking into account the new macroscopic internal shear kinetic energy per unit mass , which expresses the kinetic energy of irreversible dissipative shear motion, and also the new macroscopic internal kinetic energy of shear-rotational coupling per unit mass , which expresses the kinetic energy of local coupling between irreversible dissipative shear and reversible rigid-like rotational macroscopic continuum motions.

Following the “Statistical thermohydrodynamics of irreversible strike-slip-rotational processes” [11] and the “Thermohydrogravidynamics of the Solar System” [12], in Section 2.2 we present the generalized differential formulation of the first law of thermodynamics (in the Galilean frame of reference) for nonequilibrium shear-rotational states of the deformed finite one-component individual continuum (characterized by the symmetric stress tensor ) region moving in the nonstationary gravitational field. In Section 2.3 we present the generalized differential formulation [11, 12] of the first law of thermodynamics (in the Galilean frame of reference) for nonequilibrium shear-rotational states of the deformed finite individual region of the compressible viscous Newtonian one-component continuum moving in the nonstationary gravitational field. We present the generalization [11, 12] of the classical [27] expression by taking into account (for Newtonian continuum) the infinitesimal works and , respectively, of acoustic and viscous Newtonian forces acting during the infinitesimal time interval on the boundary surface of the individual continuum region bounded by the continuum boundary surface . Based on the generalized differential formulation of the first law of thermodynamics, in Section 2.4 we present the analysis [11, 12] of the gravitational energy mechanism of the gravitational energy supply into the continuum region owing to the local time increase of the potential of the gravitational field inside the continuum region subjected to the nonstationary Newtonian gravitational field.

Following the “Statistical thermohydrodynamics of irreversible strike-slip-rotational processes” [11] and the “Thermohydrogravidynamics of the Solar System” [12], in Section 3 we present the fundamentals of the cosmic energy gravitational genesis of earthquakes. Using the evolution equation of the total mechanical energy of the macroscopic continuum region (of the compressible viscous Newtonian continuum), we demonstrate the physical adequacy [11, 12] of the rotational model [2] of the earthquake focal region for the seismic zone of the Pacific Ring. We present the thermodynamic foundation [11, 12] of the classical deformational (shear) model [1] of the earthquake focal region for the quasi-uniform medium of the Earth’s crust characterized by practically constant viscosity. We present the generalized thermohydrogravidynamic shear-rotational model [11, 12] of the earthquake focal region by taking into account the classical macroscopic rotational kinetic energy [28, 29], the macroscopic nonequilibrium kinetic energies [24, 31], and the external cosmic energy gravitational influences [12, 25, 26] on the focal region of earthquakes.

In Section 4 we present the fundamentals of the cosmic geophysics [12] applicable for the planets of the Solar System. In Section 4.1 we consider the energy gravitational influences on the Earth of the inner planets and the outer planets of the Solar System. In Section 4.1.1 we present the relation for the energy gravitational influences (on the Earth) of the inner and the outer planets in the second approximation of the elliptical orbits of the planets of the Solar System. In Section 4.1.2 we present the evaluation of the relative maximal planetary instantaneous energy gravitational influences on the Earth in the approximation of the circular orbits of the planets of the Solar System. In Section 4.1.3 we present the evaluation of the relative maximal planetary integral energy gravitational influences on the Earth in the approximation of the circular orbits of the planets of the Solar System. In Section 4.2 we consider the energy gravitational influences on the Earth of the Moon. In Section 4.2.1 we present the evaluation of the relative maximal instantaneous energy gravitational influence of the Moon on the Earth in the second approximation of the elliptical orbits of the Earth and the Moon around the combined mass center of the Earth and the Moon. In Section 4.2.2. we present the evaluation of the maximal integral energy gravitational influence of the Moon on the Earth in the approximation of the elliptical orbits of the Earth and the Moon around the combined mass center of the Earth and the Moon. In Section 4.3 we demonstrate the reality of the cosmic energy gravitational genesis of preparation and triggering of earthquakes owing to the energy gravitational influence on the Earth of the Moon and the planets of the Solar System. In Section 4.3.1 we demonstrate the real cosmic energy gravitational genesis of preparation of earthquakes by considering the energy gravitational influence on the Earth of Venus. In Section 4.3.2 we demonstrate the real cosmic energy gravitational genesis of triggering of the preparing earthquakes. In Section 4.4 we demonstrate the cosmic energy gravitational genesis of the seismotectonic activity induced by the nonstationary cosmic energy gravitational influences on the Earth of the Sun, the Moon, Venus, Jupiter, and Mars. In Section 4.4.1 we present the evaluations of the time periodicities of the maximal (instantaneous and integral) energy gravitational influences on the Earth of the Sun-Moon system, Venus, Jupiter, and Mars. In Section 4.4.2 we present the empirical time periodicities of the seismotectonic activity for various regions of the Earth. In Section 4.4.3 we present the set of the time periodicities of the periodic global seismotectonic (and volcanic) activity and the global climate variability of the Earth induced by the different combinations of the cosmic energy gravitational influences on the Earth of the Sun and the Moon, Venus, Jupiter, and Mars. In Section 4.5 we present the evidence of the cosmic energy gravitational genesis of the strong Chinese 2008 earthquakes. In Section 4.6 we present the evidence of the cosmic energy gravitational genesis of the strongest Japanese earthquakes near the Tokyo region. In Section 5 we present the summary of main results and conclusion.

#### 2. The Generalized Formulation of the First Law of Thermodynamics for Moving Rotating Deforming Compressible Heat-Conducting Macroscopic Individual Continuum Region Subjected to the Nonstationary Newtonian Gravitational Field

##### 2.1. The Generalized Expression for the Macroscopic Kinetic Energy of a Small Continuum Region in Nonequilibrium Thermodynamics

De Groot and Mazur defined the macroscopic kinetic energy per unit mass as [28] the sum of the macroscopic translational kinetic energy per unit mass of a continuum region (particle) mass center and the macroscopic internal rotational kinetic energy per unit mass :

where is the speed of the mass center of a small continuum region, is an angular velocity of internal rotation [29], and is an inertia moment per unit mass of a small continuum region [28]. Gyarmati’s definition [29] of the macroscopic kinetic energy per unit mass is analogous to de Groot and Mazur’s one. The classical de Groot and Mazur’s and Gyarmati’s definition (2) of the macroscopic kinetic energy per unit mass for a shear flows has some inherent physical incompleteness associated with the assumption about the rigid-like rotation of the continuum region with the angular velocity vector . This definition is based on the assumption of local thermodynamic equilibrium since it does not consider the nonequilibrium shear component of the macroscopic continuum motion related to the rate of strain tensor . However, the assumption of local thermodynamic equilibrium, as noted by de Groot and Mazur [28], may be justified only by reasonable agreement of the experimental results with the theoretical deductions based on this assumption.

Landau and Lifshitz defined [32] the macroscopic internal energy of a small macroscopic continuum region as the difference between the total kinetic energy of the continuum region and kinetic energy of the translational macroscopic motion of the continuum region. According to Landau and Lifshitz’s definition [32] of the macroscopic internal energy, the term in the expression (2) is the internal energy of the macroscopic (hydrodynamic) continuum motion. The classical definition [28, 29] of the macroscopic internal rotational kinetic energy per unit mass is consistent with the Landau and Lifshitz’s definition of the macroscopic internal energy. We shall use further the Landau and Lifshitz’s definition [32] of the macroscopic internal energy.

Following the works [11, 12, 24–26], we shall present the foundation of the generalized expression for the macroscopic kinetic energy in nonequilibrium thermodynamics. We shall assume that is a small individual continuum region (domain) bounded by the closed continual boundary surface considered in the three-dimensional Euclidean space with respect to a Cartesian coordinate system . We shall consider the small continuum region in a Galilean frame of reference with respect to a Cartesian coordinate system centred at the origin and determined by the axes , , and (see Figure 1).

The unit normal -basis coordinate vectors triad , , and is taken in the directions of the axes , , and , respectively. The -basis vector triad is taken to be right-handed in the order , , and ; see Figure 1. is the local gravity acceleration.

An arbitrary point in three-dimensional physical space will be uniquely defined by the position vector originating at the point and terminating at the point . The continuum region-affixed Lagrangian coordinate system (with the axes , , and ) is centered to the mass center of the continuum region . The axes , , and are taken parallel to the axes , , and , respectively: the axis parallel to the axis , where . The unit normal -basis coordinate vector triad , , and is taken in the directions of the axes , , and , respectively. The -basis vector triad is taken to be right-handed in the order , , and . The mathematical differential of the position vector , , expressed in terms of the coordinates () in the -coordinate system, originates at the mass centre of the continuum region and terminates at the arbitrary point of the continuum region.

The position vector of the mass center of the continuum region in the -coordinate system is given by the following expression: where is the mass of the continuum region , is the mathematical differential of physical volume of the continuum region, is the local macroscopic density of mass distribution, is the position vector of the continuum volume , and is the time. The speed of the mass centre of the continuum region is defined by the following expression: where is the hydrodynamic velocity vector and the operator denotes the total derivative following the continuum substance [33]. The relevant three-dimensional fields such as the velocity and the local mass density (and also the first and the second derivatives of the relevant fields) are assumed to vary continuously throughout the entire continuum bulk of the continuum region . The instantaneous macroscopic kinetic energy of the continuum region (bounded by the continuum boundary surface ) is the sum of the kinetic energies of small parts constituting the continuum region when the number of the parts, , tends to infinity and the maximum from their volumes tends to zero [33]:

where is the local hydrodynamic velocity vector, is the local mass density, and is the mathematical differential of physical volume of the continuum region. We use the common Riemann’s integral here and everywhere.

For the analysis of the relative continuum motion in the physical space in the vicinity of the position vector of the mass centre we have the Taylor series expansion (consistent with the Helmholtz’s theorem [30, 34]) of the hydrodynamic velocity vector for each time moment : where , , is the hydrodynamic velocity vector at the position vector , is the differential of the position vector , is the angular velocity of internal rotation (a half of the vorticity vector) in the -coordinate system at the position vector , is the local vorticity in the -coordinate system at the position vector , is the rate of strain tensor in the -coordinate system at the position vector , (), is the gradient operator, and

is the small residual part of the Taylor series expansion (7), where , (), is the diameter of the continuum region and the vector originates at point and terminates at point of the surface . The linear of terms of the Taylor series expansion (7) is presented in the classical form [33].

Substituting formula (7) into the formula (6) and integrating by parts, then we obtain the following expression [24, 31]: where is the mass of the continuum region and is the -component of the classical inertia tensor depending on the mass distribution in the continuum region under consideration: where , are the , -components of the vector , respectively, in the -coordinate system, is the Kronecker delta tensor, is the third-order permutation symbol, and is the , -component classical centrifugal tensor depending on the mass distribution in the continuum region under consideration:

is a small residual part of the macroscopic kinetic energy after substituting the Taylor series expansion (7) into formula (6).

Formula (13) states that the macroscopic kinetic energy of the small continuum region is the sum of the macroscopic translational kinetic energy of the continuum region the macroscopic internal rotational kinetic energy of the continuum region the macroscopic internal shear kinetic energy of the continuum region and the macroscopic kinetic energy of shear-rotational coupling of the continuum region

The macroscopic internal rotational kinetic energy is the classical [28, 29] kinetic energy of reversible (equilibrium) rigid-like macroscopic rotational continuum motion. The macroscopic internal shear kinetic energy expresses the kinetic energy of irreversible (nonequilibrium) shear continuum motion related to the rate of strain tensor . The macroscopic internal kinetic energy of the shear-rotational coupling expresses the kinetic energy of the local coupling between irreversible deformation and reversible rigid-like rotation.

The deduced expression (13) for confirms the postulate [35] that the velocity shear () represents an additional energy source taking into account the Evans, Hanley, and Hess’s extended formulation [35] of the first law of thermodynamics for nonequilibrium deformed states of continuum motion. The energies , , , and are the Galilean invariants with respect to different inertial -coordinate systems.

We obtained [31] from (13) the following expression for the macroscopic kinetic energy per unit mass : where is the -component of the classical inertia tensor per unit mass of the continuum region , is the -component of the classical centrifugal tensor per unit mass of the continuum region , is the macroscopic translational kinetic energy per unit mass of the continuum region (moving as a whole at speed of the mass center of the continuum region ), is the macroscopic internal rotational kinetic energy per unit mass of the continuum region , is the macroscopic internal shear kinetic energy per unit mass of the continuum region ,

is the macroscopic internal kinetic energy of the shear-rotational coupling per unit mass (of the continuum region ), and is the residual correction. The energies , , , and are the Galilean invariants with respect to different inertial -coordinate systems. We have , , , and , when , where is the defined diameter of the continuum region .

For a homogeneous continuum region of simple form (sphere or cube) we have Formula (17) for the macroscopic internal rotational kinetic energy is reduced to the classical expression [29] where . Formula (18) for the macroscopic internal shear kinetic energy is reduced to the expression [24, 31] which is proportional to the local kinetic energy dissipation rate per unit mass in an incompressible viscous Newtonian continuum, where is the molecular viscosity. The macroscopic internal kinetic energy of shear-rotational coupling vanishes for the homogeneous continuum region of the form of the sphere or cube. The macroscopic kinetic energy for the homogeneous continuum region of the shape of sphere or cube is given by following expression [24, 31] Hence, the macroscopic kinetic energy per unit mass for the homogeneous continuum sphere or cube is expressed as the sum of explicit terms [24, 31]

where is the macroscopic translational kinetic energy per unit mass of the continuum region ; ; ; is the classical [28, 29] macroscopic internal rotational kinetic energy per unit mass of the continuum region ; is the macroscopic internal shear kinetic energy per unit mass of the homogeneous continuum sphere or cube [24, 31].

We have the following expression for the macroscopic internal kinetic energy of the homogeneous continuum region of the shape of sphere or cube [24, 31] The macroscopic internal kinetic energy per unit mass for the homogeneous continuum region of the shape of sphere or cube is given by the sum of explicit terms [24, 31]:

Compare formula (31) with the de Groot and Mazur’s definition (2). Expression (31) is reduced to de Groot and Mazur’s definition (2) under condition

of local thermodynamic equilibrium. Therefore, we can conclude that the definition (2) of the macroscopic kinetic energy per unit mass in classical nonequilibrium thermodynamics [28, 29] is based on the assumption of local thermodynamic equilibrium [24, 31, 35].

The obtained formula (20) for and its particular form (31) (obtained for homogeneous continuum regions of spherical and cubical shapes) generalized [24, 31] the classical de Groot and Mazur expression (2) in classical nonequilibrium thermodynamics [28, 29] by taking into account the irreversible dissipative shear component of the macroscopic continuum motion related to the rate of strain tensor . The expression (20) for contains the new macroscopic internal shear kinetic energy per unit mass , which expresses the kinetic energy of irreversible dissipative shear motion, and also the new macroscopic internal kinetic energy of the shear-rotational coupling per unit mass , which expresses the kinetic energy of local coupling between irreversible dissipative shear and reversible rigid-like rotational macroscopic continuum motions.

The macroscopic internal shear kinetic energy per unit mass (for homogeneous continuum regions of spherical and cubical shapes) is proportional to the kinetic energy viscous dissipation rate per unit mass in an incompressible viscous Newtonian continuum characterized by the kinematic viscosity . We have shown [24] that the proportionality is the basis of the established association [36, 37] between a structure and an order (and, hence, the associated macroscopic kinetic energy), on the one hand, and irreversible dissipation, on the other hand, for the dissipative structures of turbulence in viscous Newtonian fluids.

##### 2.2. The Generalized Differential Formulation of the First Law of Thermodynamics (in the Galilean Frame of Reference) for Nonequilibrium Shear-Rotational States of the Deformed One-Component Individual Finite Continuum Region (Characterized by the Symmetric Stress Tensor T) Moving in the Nonstationary Newtonian Gravitational Field

Following the works [11, 12, 25, 26], we shall present the foundation of the generalized differential formulation of the first law of thermodynamics (in the Galilean frame of reference) for nonequilibrium shear-rotational states of the deformed finite one-component individual continuum region (characterized by the symmetric stress tensor ) moving in the nonstationary Newtonian gravitational field. We shall consider the deformed finite one-component individual continuum region in nonequilibrium shear-rotational states characterized by the following condition: Considering the graphical methods in the thermodynamics of fluids, Gibbs [27] formulated the first law of thermodynamics for the fluid body (fluid region) as follows (in Gibbs’ designations):

where is the differential of the internal thermal energy of the fluid body, is the differential change of heat across the boundary of the fluid body related to the thermal molecular conductivity (associated with the corresponding external or internal heat fluxes), and is the differential work produced by the considered fluid body on its surroundings (surrounding fluid) under the differential change of the fluid region (of volume ) characterized by the thermodynamic pressure .

The formulation [32] of the first law of thermodynamics for the general thermodynamic system (material region) is given by the equivalent form (in Landau’s and Lifshitz’s designations [32])

where is the differential work produced by the surroundings (surroundings of the thermodynamic system) on the thermodynamic system under the differential change of volume of the thermodynamic system characterized by the thermodynamic pressure ; is the differential heat transfer (across the boundary of the thermodynamic system) related to the thermal interaction of the thermodynamic system and the surroundings (surrounding environment); is the energy of the thermodynamic system, which should contain (as supposed [32]) the kinetic energy of the macroscopic continuum motion.

We shall use the differential formulation of the first law of thermodynamics [28] for the specific volume of the compressible viscous one-component deformed continuum with no chemical reactions: where is the specific (per unit mass) internal thermal energy, is the thermodynamic pressure, is the viscous stress tensor, is the hydrodynamic velocity of the continuum macrodifferential element [28], and is the differential change of heat across the boundary of the continuum region (of unit mass) related to the thermal molecular conductivity described by the heat equation [28]: where is the heat flux [28]. The viscous stress tensor is taken from the decomposition of the pressure tensor [28]:

where is the Kronecker delta tensor.

Considering the Newtonian viscous stress tensor of the compressible viscous Newtonian continuum with the components [29] the differential formulation (41) of the first law of thermodynamics (for the continuum region (of unit mass) of the compressible viscous Newtonian one-component deformed continuum with no chemical reactions) can be rewritten as follows: where is the coefficient of the molecular kinematic (first) viscosity and is the coefficient of the molecular volume (second) viscosity [38]. The first and the second terms in the right-hand side of relation (45) are analogous to the corresponding respective first and second terms in the right-hand side of the classical formulations (39) and (40). The third term in the right-hand side of relation (45) is related to the “internal” heat induced during the time interval by viscous-compressible irreversibility [24]. The fourth term in the right-hand side of relation (45)

is related to the “internal” heat induced during the time interval by viscous-shear irreversibility [24]. The differential formulation (45) of the first law of thermodynamics (for the continuum element of the compressible viscous Newtonian one-component deformed continuum with no chemical reactions) takes into account (in addition to the classical terms) the viscous-compressible irreversibility and viscous-shear irreversibility inside the continuum element of the compressible viscous Newtonian one-component deformed continuum with no chemical reactions.

Using the differential formulation (41) of the first law of thermodynamics [28] for the total derivative (following the liquid substance) of the specific (per unit mass) internal thermal energy of a compressible viscous one-component deformed continuum with no chemical reactions, the heat equation (42) [28], the general equation of continuum movement [29] for the deformed continuum characterized by the symmetric stress tensor of general form (in particular, with the components [29] for the compressible viscous Newtonian one-component continuum) and taking into account the time variations of the potential of the nonstationary gravitational field (characterized by the local gravity acceleration vector ) inside an arbitrary finite macroscopic individual continuum region , we derived [11, 12] the generalized differential formulation (for the Galilean frame of reference) of the first law of thermodynamics (for moving rotating deforming compressible heat-conducting stratified macroscopic continuum region subjected to the nonstationary Newtonian gravitational field): where is the differential work done during the infinitesimal time interval by nonpotential stress forces (pressure, compressible, and viscous forces for Newtonian continuum) acting on the boundary surface of the continuum region ; is the differential element (of the boundary surface of the continuum region ) characterized by the external normal unit vector ; is the stress vector [29], [29], where is the pressure tensor characterized (in particular, for the model of the compressible viscous Newtonian continuum characterized by the coefficients of kinematic viscosity and the volume viscosity ) by components (obtained from(49)) is the differential change of heat of the macroscopic individual continuum region related to the thermal molecular conductivity of heat across the boundary of the continuum region , where is the heat flux [28] (across the element of the continuum boundary surface ); is the macroscopic potential energy (of the macroscopic individual continuum region ) related to the nonstationary potential of the gravitational field; is the classical microscopic internal thermal energy of the macroscopic individual continuum region ;

is the instantaneous macroscopic kinetic energy of the macroscopic individual continuum region . The instantaneous macroscopic kinetic energy is given by the relation (13) [24, 31] for the small macroscopic individual continuum region .

The generalized differential formulation (50) of the first law of thermodynamics can be rewritten as follows [11, 12]: extending the classical [12] formulations (39) and (40) by taking into account (along with the classical infinitesimal change of heat and the classical infinitesimal change of the internal energy ) the infinitesimal increment of the macroscopic kinetic energy , the infinitesimal increment of the gravitational potential energy , the generalized infinitesimal work done on the continuum region by the surroundings of , and the infinitesimal amount of energy [11, 12]

added (or lost) as the result of the Newtonian nonstationary gravitational energy influence on the continuum region during the infinitesimal time interval .

The generalized differential formulation (50) of the first law of thermodynamics can be rewritten as follows [11, 12]:

The equivalent generalized differential formulations (50), (56), and (59) of the first law of thermodynamics take into account the following factors:(1)the classical heat thermal molecular conductivity (across the boundary of the macroscopic continuum region ) related to the classical infinitesimal change of heat : (2)the classical infinitesimal change of the internal energy of the macroscopic continuum region : (3)the established [11, 12] infinitesimal increment of the macroscopic kinetic energy of the macroscopic continuum region : (4)the established [11, 12] infinitesimal increment of the gravitational potential energy of the macroscopic continuum region : (5)the established [11, 12] generalized infinitesimal work done on the macroscopic continuum region by the surroundings of : (6)the established [11, 12] infinitesimal amount of energy added (or lost) as the result of the Newtonian nonstationary gravitational energy influence on the macroscopic continuum region during the infinitesimal time interval :

The generalized differential formulations (50), (56), and (59) of the first law of thermodynamics (given for the Galilean frame of reference) are valid for nonequilibrium shear-rotational states of the deformed finite individual continuum region (characterized by the symmetric stress tensor in the general equation (48) of continuum movement [29]) moving in the nonstationary gravitational field. The generalized differential formulations (50) and (56) of the first law of thermodynamics [11, 12] are the generalizations of the classical formulations (39) and (40) of the first law of thermodynamics taking into account: the generalized expression (51) for the differential work done during the infinitesimal time interval by nonpotential stress forces acting on the boundary surface of the individual continuum region and the time variations of the potential of the nonstationary gravitational field inside the individual continuum region due to the deformation of the individual continuum region and due to the external (terrestrial and cosmic) gravitational influence on the individual continuum region moving in the total (internal + external) nonstationary gravitational field.

##### 2.3. The Generalized Differential Formulation of the First Law of Thermodynamics (in the Galilean Frame of Reference) for Nonequilibrium Shear-Rotational States of the Deformed Finite Individual Region of the Compressible Viscous Newtonian One-Component Continuum Moving in the Nonstationary Gravitational Field

There is evidence [39] that the rocks of the Earth’s crust at protracted loadings may be considered as fluids characterized by the very high viscosity. According to the classical viewpoint [39], the local mechanism of creation of the earthquakes is related to the release of the accumulated potential energy of the elastic deformation during the sudden local break (i.e., the discontinuous shear) of the Earth’s crust (or the sudden increase of fluidity in the local region of the Earth’s crust) accompanied by viscous relaxation and generation of seismic waves. It was conjectured [40] that “more punctual and refined methods of the mathematical analysis are obligatory” for “the practical assessment of the seismic hazard”.

Taking into account the established [31] conception of the macroscopic internal shear kinetic energy (per unit mass) related to the rate of medium deformation (i.e., with the rate of strain tensor , where is the deformation tensor [30]), we have elucidated [41] from the viewpoint of nonequilibrium thermodynamics the mechanism of generation of seismic waves from the separate deformed finite zone of the Earth’s crust. The proportionality (37) takes place also for deformed compressible finite region of the Earth’s crust for sudden rise of fluidity (in a local region of the Earth’s crust) related to the local sudden medium deformation in the separate seismic zones of the seismic activity. Taking into account the established [31] proportionality (37), we have assumed [31] that the accumulated potential energy of the elastic deformation (related to the deformation tensor ) converts to the macroscopic internal shear kinetic energy (related to the rate of strain tensor ) in the seismic zone simultaneously with the damping of by viscous dissipation and radiation of seismic waves during several oscillations. In Section 3 we shall evaluate this mechanism on the basis of the generalized differential formulation (50) of the first law of thermodynamics for nonequilibrium shear-rotational states of the deformed finite individual continuum region (characterized by the symmetric stress tensor ) moving in the nonstationary gravitational field.

Following the works [11, 12, 25, 26], we shall present the foundation of the generalized differential formulation of the first law of thermodynamics for nonequilibrium shear-rotational states of the deformed finite individual region of the compressible viscous Newtonian one-component continuum moving in the nonstationary gravitational field. The generalized differential formulation (50) of the first law of thermodynamics (formulated for the Galilean frame of reference) is valid for arbitrary symmetric stress tensor , in particular for nonequilibrium shear-rotational states of the deformed finite individual region of the compressible viscous Newtonian one-component continuum moving in the nonstationary gravitational field. The coefficient of molecular kinematic (first, shear) viscosity and the coefficient of molecular volume (second) viscosity are assumed to vary for each time moment as an arbitrary continuous functions of the Cartesian space (three-dimensional) coordinates.

The differential work for the Newtonian symmetric stress tensor (characterized by the components (49)) is presented by three explicit terms [11, 12]: where is the differential work of the hydrodynamic pressure forces acting on the boundary surface of the individual continuum region (bounded by the continuum boundary surface ) during the infinitesimal time interval ; is the differential work (related to the combined effects of the acoustic compressibility, molecular kinematic viscosity, and molecular volume viscosity) of the acoustic (compressible) pressure forces acting on the boundary surface of the individual continuum region during the infinitesimal time interval ;

is the differential work of the viscous Newtonian forces (related to the combined effect of the velocity shear, that is, the deformation of the continuum region , and the molecular kinematic viscosity) acting on the boundary surface of the individual continuum region during the infinitesimal time interval .

Along with (45) the differential formulation of the first law of thermodynamics [28] for the total derivative (following the continuum substance) of the internal thermal energy per unit mass of the one-component deformed continuum with no chemical reactions, the thermohydrodynamic theory [28] contains additionally the equations of the mass and momentum balances:

The generalized differential formulation (50) of the first law of thermodynamics (together with the generalized differential work given by the expression (66)) is valid for nonequilibrium shear-rotational states of the deformed finite individual region of the compressible viscous Newtonian one-component continuum moving in the nonstationary gravity field. The coefficient of molecular kinematic (first, shear) viscosity and the coefficient of molecular volume (second) viscosity are assumed to vary for each time moment as an arbitrary continuous functions of Cartesian space (three-dimensional) coordinates.

The generalized differential formulation (50) takes into account the dependences of the hydrodynamic pressure on the hydrodynamic vorticity and on the rate of strain tensor by means of the component (in the expression (66) for ) given by the expression (67). The presence of the third term (given by the expression (69) and related to the combined effect of the molecular kinematic viscosity and the deformation of the continuum region defined by the rate of strain tensor ) in the expression (66) for generalizes essentially the classical formulations (39) and (40) of the first law of thermodynamics by taking into account the differential work of the viscous Newtonian forces acting on the boundary continuum surface of the individual continuum region .

The general equation (48) of continuum movement [29] for the compressible viscous Newtonian one-component continuum is reduced to the following equation:

where is the internal multiplication of the vector and the rate of strain tensor in accordance with the corresponding definition [29]. Equation (72) generalizes the Navier-Stokes equation (71) (given for ) by taking into account the dependences of the coefficient of molecular kinematic viscosity and the coefficient of molecular volume viscosity on the space (three-dimensional) Cartesian coordinates.

The relevant example for illustration of the significance of the term (in the expression (66) for the differential work ) is related to the thermodynamic consideration [12] of the processes of the energy exchange [42] between the oceans and the lithosphere of the Earth. According to the expression (69) for the term , the energy exchange between the oceans (and the atmosphere) and the lithosphere of the Earth is possible only under the presence of the medium acoustic compressibility (i.e., ) and the medium deformations (i.e., ) in the boundary regions of fluid (in the oceans), air (in the atmosphere), and the compressible deformed lithosphere of the Earth. According to the generalized expression (66) for the differential work , the energy exchange between the oceans (and the atmosphere) and the lithosphere of the Earth is impossible for nondeformed () and noncompressible () lithosphere.

We have the evolution equation for the total mechanical energy () of the deformed finite individual macroscopic continuum region [11, 12]: obtained from the generalized differential formulation (50) for the compressible viscous Newtonian one-component continuum moving in the nonstationary gravitational field. In Section 3 we shall use the evolution equation (73) of the total mechanical energy to found the rotational, shear, and the shear-rotational models of the earthquake focal region.

##### 2.4. Cosmic and Terrestrial Energy Gravitational Genesis of the Seismotectonic (and Volcanic) Activity of the Earth Induced by the Combined Cosmic and Terrestrial Nonstationary Energy Gravitational Influences on an Arbitrary Individual Continuum Region (of the Earth) and by the Nonpotential Terrestrial Stress Forces Acting on the Boundary Surface of the Individual Continuum Region

Following the works [11, 12], we shall present the physical mechanisms of the energy fluxes to the continuum region related to preparation of earthquakes. The equivalent generalized differential formulations (50) and (59) of the first law of thermodynamics show that the nonstationary gravitational field (related to the nonstationary gravitational potential ) gives the following gravitational energy power:

associated with the gravitational energy power of the total combined (external cosmic and terrestrial and internal related to the macroscopic continuum region ) gravitational field. If the macroscopic continuum region is not very large, consequently, it cannot induce the significant time variations to the potential of the gravity field inside the continuum region . According to the equivalent generalized differential formulations (50) and (59) of the first law of thermodynamics and to the evolution equation (73) for the total mechanical energy () of the deformed finite individual macroscopic continuum region , the energy power of the gravitational field may produce the fractures in the continuum region . We shall consider this aspect in Section 3.

The generalized differential formulation (59) of the first law of thermodynamics and the expression (74) for the gravitational energy power show that the local time increase of the potential of the gravitational field is the gravitational energy mechanism of the gravitational energy supply into the continuum region . Really, the local time increase of the potential of the gravitational field inside the continuum region () supplies the gravitational energy into the continuum region . Consequently, according to the generalized differential formulation (59) and to the evolution equation (73), the total energy () of the continuum region and the total mechanical energy () of the continuum region are increased if .

According to the generalized differential formulation (59) of the first law of thermodynamics and to the evolution equation (73), the gravitational energy supply into the continuum region may induce the formation of fractures in the continuum region related to the production of earthquake. This conclusion corresponds to the observations [1, 3–5, 9, 10, 13] of the identified anomalous variations of the gravitational field before strong earthquakes.

According to the generalized differential formulation (59) of the first law of thermodynamics and to the evolution equation (73), the supply of energy into the continuum region may occur also by means of the work done by nonpotential stress forces (pressure, compressible, and viscous forces for Newtonian continuum) acting on the boundary surface of the continuum region during the time interval ().

The considered mechanisms of the energy supply to the Earth’s macroscopic continuum region should result in the irreversible process of the splits formation in the rocks related to the generation of the high-frequency acoustic waves from the focal continuum region before the earthquake. Taking this into account, the sum in the expression (45) is interpreted [11, 12] as the energy flux (related to the compressible and viscous forces acting on the boundary surface of the continuum region ) [38]

directed across the boundary (see Figures 2 and 3) of the continuum region .

The considered mechanisms of the energy supply to the Earth’s macroscopic continuum region should result in the significant increase of the energy flux of the geo-acoustic energy from the focal region before the earthquake. The deduced conclusion is in a good agreement with the results of the detailed experimental studies [43].

#### 3. Generalized Thermohydrogravidynamic Shear-Rotational and Classical Shear and Rotational Models of the Earthquake Focal Region

##### 3.1. The Generalized Thermohydrogravidynamic Shear-Rotational and the Classical Shear (Deformational) Models of the Earthquake Focal Region Based on the Generalized Differential Formulation of the First Law of Thermodynamics

Following the works [11, 12], we shall present the foundation of the generalized thermohydrogravidynamic shear-rotational model of the earthquake focal region based on the generalized differential formulation (59) of the first law of thermodynamics. Using the evolution equation (73) of the total mechanical energy of the subsystem (the macroscopic continuum region ) of the Earth, we shall show now that the formation of fractures (modeling by the jumps of the continuum velocity on some surfaces) is related to irreversible dissipation of the macroscopic kinetic energy and the corresponding increase of entropy. We consider at the beginning the analysis of formation of the main line flat fracture (associated with the surface of the continuum velocity jump) inside the macroscopic continuum region (bounded by the closed surface ). The macroscopic continuum region may be divided into two subsystems and by continuing mentally the surface by means of surface crossing the surface of the macroscopic region . The surface of the subsystem consists of the surface (which is the part of the surface ) and the surfaces and . The surface of the subsystem consists of the surface (which is the part of the surface ) and the surfaces and .

Using the formulation (73), we have the evolution equations for the total mechanical energy of the macroscopic subsystems and : where is the external unit normal vector of the surface (of the subsystem ) presented by surfaces and and is the external unit normal vector of the surface (of the subsystem ) presented also by surfaces and . Adding (77) (by using the equality of the elements of area of surfaces and ) we get the evolution equation for the total mechanical energy of the macroscopic region consisting of subsystems and interacting on the surface of the tangential jump of the continuum velocity:

where is the vector of the continuum velocity on the surface in the subsystem , and is the vector of the continuum velocity on the surface in the subsystem .

The evolution equation (78) takes into account the total mechanical energy of the macroscopic region consisting of subsystems and interacting on the surface of the tangential jump of the continuum velocity. The first term in the right-hand side (of (78)) describes the evolution of the total mechanical energy of the macroscopic continuum region due to the continuum reversible compressibility; the second and third terms express the dissipation of the macroscopic kinetic energy by means of the irreversible continuum compressibility and the velocity shear. The forms of three primary terms in the right-hand side (of (78)) are related to the considered model of the compressible viscous Newtonian continuum. The fourth, fifth, and sixth terms in the right-hand side (of (78)) are the universal terms for arbitrary model of continuum characterized by symmetrical stress tensor . The fourth term expresses the power

of external (for the continuum region ) nonpotential stress forces acting on the boundary surface of the macroscopic continuum region . The fifth term expresses the power of external (for the continuum region ) forces on different sides of the surface characterized by the velocity jump during the fracture formation. The sixth term in (78) presents the power of the total mechanical energy added (or lost) as the result of the Newtonian nonstationary gravitational energy influence on the macroscopic continuum region related to variations of the potential of the combined gravitational field in the continuum region .

Consider (78) for one continuum velocity jump on the nonstationary surface during the time interval (). Taking into account the form of the fifth term on the right-hand side of the evolution equation (78), we obtained [11, 12] the expression for the work (done during the time interval () by the external (for the continuum region ) nonpotential stress forces acting on different sides of the velocity jump on the surface ): which is reduced to the following expression:

To test the formula (81), let us calculate the energy , which dissipates during formation of the surface dislocation on the small surface during the time interval (). Using the theorem about the average value and integrating the internal integral on time, we obtained from relation (81) for the following relation [11, 12]: where is the average value of the stress vector for the element of area of the two-side surface , and and are the vectors of the continuum displacement on different sides of the element of area of the two-side surface in the points characterized by normal unit vectors and . Using the obvious expression for “linear” time average as the arithmetical average of the values of the stress vectors on the different sides from the surface of the jump of the continuum velocity, we obtained [11, 12] the expression for the elementary work of the external nonpotential stress forces on the two-side surface of dislocation: This expression was obtained [44] in the frame of the classical linear approach to formation of surface dislocations in rigid compressible continuum on the small area of surface . It is clear that the assumption (83) is valid only for weak tangential jumps of the continuum displacement. Consequently, we can consider the expression (80) as the natural nonlinear generalization of expression (84) for arbitrary surface of dislocation and for strong tangential jumps of the continuum displacement on the surface of dislocation. The work (80) of the external (for the continuum region ) nonpotential stress forces should be negative. The sufficient energy needed for formation of the surface of dislocation is equal to the work of the internal forces in the macroscopic continuum region . The energy should be positive and equal to the expression (80) with the sign “−”:

The formulae (80), (84), and (85) are obtained (taking into account the generalized differential formulation (50) of the first law of thermodynamics) for the model of continuum characterized by an arbitrary symmetrical stress tensor .

The macroscopic internal shear kinetic energy (of the subsystem ), the macroscopic internal rotational kinetic energy (of the subsystem ), and the macroscopic kinetic energy of shear-rotational coupling (of the subsystem ) are the significant components of the macroscopic internal shear-rotational kinetic energy [11, 12, 24, 31]:

taken into account (along with the classical internal thermal energy of the macroscopic continuum region , the macroscopic potential energy of the macroscopic continuum region , and the macroscopic translational kinetic energy of the continuum region (of a mass ) moving as a whole at speed equal to the speed of the center of mass of the continuum region ) in the generalized differential formulation (50) of the first law of thermodynamics for the macroscopic continuum region .

The macroscopic internal shear kinetic energy (of the subsystem ), the macroscopic internal rotational kinetic energy , and the macroscopic kinetic energy of shear-rotational coupling are the significant components of the macroscopic internal shear-rotational kinetic energy [11, 12, 24, 31]:

taken into account (along with the classical internal thermal energy , the macroscopic potential energy , and the macroscopic translational kinetic energy of the continuum region (of a mass ) moving as a whole at the speed of the center of mass of the continuum region ) in the generalized differential formulation (50) of the first law of thermodynamics for the macroscopic continuum region .

The macroscopic internal shear kinetic energy (of the subsystem ), the macroscopic internal rotational kinetic energy , the macroscopic kinetic energy of shear-rotational coupling , the macroscopic translational kinetic energy , the macroscopic potential energy , the macroscopic internal shear kinetic energy (of the subsystem ), the macroscopic internal rotational kinetic energy , the macroscopic kinetic energy of shear-rotational coupling , the macroscopic translational kinetic energy , and the macroscopic potential energy are the significant energy components taken into account in the presented thermohydrogravidynamic shear-rotational model described by the evolution equation (78) for the total mechanical energy () of the macroscopic region consisting of interacting subsystems and .

##### 3.2. The Rotational Model of the Earthquake Focal Region Based on the Generalized Differential Formulation of the First Law of Thermodynamics

Following the works [11, 12], we shall present the foundation of the rotational model [2] of the earthquake focal region for the seismic zone of the Pacific Ring. It was noted [2] that the studies of the dislocation models of the focal regions of strong earthquakes showed the bad correspondence with the model of flat endless dislocation in the uniform continuum [45–47]. The analysis [2] showed that the conditions exist to realize the rotational mechanism related to the rotation of the geoblocks by means of the stress forces related to the Earth rotation in the vicinity of the seismic zone of the Pacific Ring. It was noted [2] that the rotational mechanism can be more real compared to the conventional mechanism related to the formation of the main line flat fracture inside the focal region.

Let us consider the energy thermodynamic analysis of the rotational mechanism [2] of the earthquake focal region, related to formation of the circular continuum velocity jump revealed in the form of circular dislocation after relaxation of the seismic process in the earthquake focal region. The developed and tested (in this section) mathematical formalism of description of the main line flat fracture may be generalized on the closed surfaces of the continuum velocity jumps. Following the rotational model [2] of the earthquake focal region, we consider the separate geoblock of the seismic zone. If the external influences of the nonstationary gravitational forces (on the geoblock ) and the nonpotential stress forces (on the boundary of the geoblock ) exceed the certain critical value then the geoblock may rotate and slip relative to the surrounding fine plastic layer (subsystem) with the tangential continuum velocity jump on the boundary surface of the geoblock . We assume that fine plastic layer (subsystem) is limited by external surface of the considered thermodynamic system consisting of the macroscopic subsystems and .

Using the evolution equation (73) of the total mechanical energy of the subsystem , we obtained [11, 12] the evolution equations for the total mechanical energy of the macroscopic subsystems and :