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Basing on the Asymmetric Continuum Theory, we develop the thermodynamics including fragmentation spin fracture processes; applications for the earthquake source processes are considered. The fracture band model is used to describe a dislocation and disclination superlattice. The Gibbs free energy of defect formation is specified. A dynamic spin fracture criterion was formulated. Consequently, a dynamic model of rock fracture employing dislocations, disclinations, and cracks was constructed to describe slip and fragmentation fracture processes in the earthquake sources.

The Asymmetric Continuum Theory based on deformation fields includes the strain rotation as an equally important deformation part as the shear and confining strains; all these fields can be related to their origin in the fracture processes by some displacement motions in a source. Some of these motions may belong to an individual process, some to complex correlated events; in this latter case the displacements related to these strains could be shifted in phase. Moreover, we may expect an appearance of some molecular transport motions; the molecular transport may be helpful for understanding an interaction of the molecular processes and related molecular momentum flux. These correlated events should be mutually related in a source by the release-rebound mechanism. In particular, we consider the point fracture events as associated with a confining load or/and with the shear and rotation processes; we discuss the related effects and their meaning when discussing the fault plane mechanism and emitted waves.

Asymmetric continuum theory points to the equal roles of the rotation motions and those related directly to shear or confining strains. The strain motions could be quite independent or mutually related with the eventual phase shifts, while the displacements have only a mathematical sense; a real displacement may appear along the fracture slip only. Formally, any deformation could be presented as related to the displacements; however, its origin in a fracture source should be considered either as belonging to an individual process or to complex correlated events; in these cases, the confining, shear and rotation strains can be related mathematically to the different displacement fields. Some of these related deformations could be emitted from a source with a phase shift, while the observed displacements (deduced from records) result as a sum of these independent displacements. An important influence on a source process and on the premonitory micro-events has the material defects, their distribution, and mobility. The defect arrays lead to a concentration of stresses and their local reorganization. Thus, in this paper, we consider the induced stresses and strains related to defect content and to its modification and redistribution. Moreover, an important role in understanding the complex correlated events in a source plays the release–rebound mechanism. The release–rebound mechanism in an earthquake source processes leads to a possible direct or phase-shifted correlations between the emitted motions; in this aspect, a propagation of the coupled strain and rotation waves is discussed. In particular, we consider the point fracture events as related either to a confining load or/and to the shear and rotation processes; we discuss the related effects and their meaning when discussing the fault plane mechanism and emitted waves. In some important seismic regions, we have the recording system which permits to record the strains and rotations. However, we should point that the wordwide seismological network is not adequate to record the complex strain deformations released in the fracture processes and remains quite insufficient to understand the global stress changes and related strain waves of a very low period. Consideration on a recording mechanism of the long displacement waves indicates the insufficiencies of the present global recording system and points that recording of the global strain and rotation waves is an important and urgent task.

Induced strains play an important role in mining regions and water dam areas. We consider a mechanism of reorganization of the applied stress load by some changes in rock-body defect distribution caused by human activity. A defect content increases with increasing stress load and related deformations; hence, a relationship could appear between seismic risk and deformation level. Recent progress in the Asymmetric Continuum Theory permits to consider some internal reorganization of the applied load due to internal defect content and distribution; in this paper we consider an increase of internal defect densities due to mining works and the appearance of reorganized internal stress distribution.

We present a confrontation between the classic elasticity and asymmetric continuum theories in order to point out the abilities of the two approaches and their deficiencies. The role of independent strain and rotation waves is discussed and some related particular examples are given.

We follow our former considerations on rotational processes and emission of rotation and shear-twist waves as occurring in earthquake source zones. A concise presentation of the motion and constitutive relations is given for each type of the four fundamental point deformations: displacements, rotations, point stretch/squeeze, and string-string point deformations (shear nuclei).

We present a new development in fluid theory, incorporating into it the velocity and spin fields; special attention is given to the structure of transport.The theory includes asymmetric molecular stresses and independent rotation velocity, i.e., spin. Our approach is based on our former studies on the asymmetric continuum theory with the balance and constitutive laws for displacement velocity and independent rotation motion, and points out the role of a related characteristic length unit. It is assumed that the vorticity caused by velocities can induce a spin transport counterpart. Thus, under certain conditions, an additional transport term due to rotational velocity fields may be incorporated to the velocity transport, which may lead to the vortex fields included directly into the theory. The Coriolis effect, important for the vortex processes, is considered and it is demonstrated that the motion equations in our asymmetric theory include this effect automatically. When confinining to 2D case, some compatibilities are found between the relations derived for the rotation motions and the moment formed by the Coriolis forces and applied to such motions. This is an important argument supporting our approach. The obtained nonlinear vortex equations (solitons) are derived and discussed for a stationary case.

We present a new development in the asymmetric continuum theory with the shear oscillations (twist motions) and independent spin; these motions (displacement velocities and spin) can be shifted in phase to describe the independent rebound processes. Our approach provides an extension of the asymmetric continuum theory by including the microfragmentation processes with a double transport which may appear in an advanced fracture process under very high load. The related nonlinear equations, leading to soliton solutions, are derived.

The complexity of processes in a seismic source zone has inspired us to reconsider a class of basic motions and deformations in an asymmetric continuum, including simple motions (translation and rotation, named spin) and simple deformations. Simple deformations include the axial nuclei, that is, point extension/compression, and the shear nuclei related to string-string point deformations. The point deformation is further redefined as another kind of rotational motion, called twist, representing oscillations of the main shear axes and their amplitudes. Some remarks are added on the recording systems used to measure these motions and deformations.

We present a linear continuum theory incorporating asymmetric stress fields as well as symmetric strains and antisymmetric rotations. We discuss the related constitutive laws and balance equations. In this theory, the motion equation related to the balance of the antisymmetric part of stresses replaces that for the stress moments. Our theory proves that the rotation waves may exist even in a homogeneous elastic continuum. Different kinds of extreme deformations are considered. The wave solutions, including the coaction of the rotation and twist fields, are presented and discussed. The dislocation density–stress relations are derived with the help of the symmetric and antisymmetric parts of stresses. The synchronization solution, rotation, and twist, shifted in phase by π /2, are presented for a material in an advanced deformation state with granulation and microcracking. Some examples of the spin and twist motion records are reported that confirm this synchronization hypothesis.

We present a new approach to the continuum theories: for solids, our version includes the asymmetric stresses, symmetric strains and antisymmetric rotations, while for fluids we take the similar assumptions but related to the respective time rates of these fields. We consider the constitutive relations and the balance laws: those related to the antisymmetric stresses are equivalent and substitute the required relations for the angular moment; a similar approach with rates of these fields is applied for fluids. The stress-dislocation relations are derived and the role of rotation motions in the fracture processes in solids is studied. A new theory of the extreme motion phenomena in fluids is developed.

We shortly explain why the rotations strains should be included together with the shear strains in the theory of solid continuum; the related arguments are based both on the experimental and theoretical levels. In the presented Asymmetric Continuum Theory the wave propagation mechanism appears in a logical way with the respective shear and rotation strains. In this logical approach we include, beside the shear strains, also the rotation strains and their related wave propagation with shear and rotation strain interactions.

We consider the advantages of a formalism based on concept of the asymmetric continuum and we present some equivalence theorems relating it to the asymmetric elasticity and to micropolar and micromorphic theories as founded by Nowacki, Cosserats and Eringen.

We present a relation between stress moments and antisymmetric part of stresses and the related constitutive law joining those stresses with the particle/point rotations. Correspondence of the asymmetric continuum theory to the micromorphic theory is considered. An extension of the asymmetric continuum theory for the case of 2D anisotropy for antisymmetric stresses leads us to the problems of friction anisotropy and fracture pattern.

We present a consistent theory of continua with defect distribution including the density of rotation nuclei. The elastic and self-fields of stresses and strains become asymmetric; the tensor of incompatibility also becomes asymmetric. We derive the dislocation–stress relations and the equations of motion related to the momentum and moment of momentum. Some applications important for earthquake engineering are presented.

Contemporary studies of deep lithosphere structure in Europe focus on the so-called Teisseyre-Tornquist (or Tornquist-Teisseyre) line, named for its discoverers, Polish geologist Wawrzyniec Teisseyre (1860–1939) and Swedish magnetologist A. J. H. Tornquist (1868–1944). Deep seismic soundings, first performed there by A. Guterch from the Institute of Geophysics, Polish Academy of Sciences; and later in cooperation with M. Grad from the Institute of Geophysics, University of Warsaw, supported by the physical interpretation, established that this line relates to a 50- to 100-km-wide deep rift zone which appears to be part of the broad deformation zone now called the Trans-European Suture Zone (TESZ).

Solid Earth Physics, including Seismology, Physics of the Earth, Earth Magnetism to name a few more topical disciplines, strongly relies on mathematical and numerical possibilities of modeling very complex physical processes ongoing in the Earth interior. Tremendous progress in geophysical instrumentation and still increasing quality and quantity of observational data also prompts for advanced processing methods in order to get more reliable interpretations. The goal of this chapter is to review some contributions from the Institute of Geophysics, Polish Academy of Sciences (IGF PAS) to physical and mathematical concepts used in Solid Earth Physics. We have selected some topics which are general enough to be interesting for a wide range of readers, leaving many topical issues uncovered in this review.

A new joint approach to deformations and motions in solids and fluids is presented. For the theory of solids, in addition to the shear and rotation strains, we define the molecular transport, while for fluids we consider the transport motions and the shear and rotation molecular strains. In this way we arrive at a common asymmetric theory for solids and fluids. Thus, for solids we present the release-rebound relations and related wave equations for strains, while for fluids we present the Navier-Stokes transport relations; moreover, for solids the included molecular transport and for fluids the molecular rotation and shear strains are considered additionally. The molecular transport in solids helps us to understand the fracture preparation processes. Of course, to each of these continua we include an influence of pressure; thus, in fluids we have both the pressure and its time derivative, that is, the molecular pressure related to sound phenomena.

Asymmetric Continuum Theories - Fracture Processes in Seismology and Extreme Fluid Dynamics

We theoretically study spatio-temporal evolution of macroscopic stress in an elastic medium permeated by continuously distributed dislocations. We develop a system of equations governing stress evolution, taking account of growth and coalescence of dislocations. As dislocation distribution evolves, we observe the compatible evolution of stresses. Numerical calculations of these equations denote that stress evolution shows both an increase and a decrease inside the medium, which is due to dislocation interactions, for fixed-stress load on the boundary. This kind of approach will be useful to understand the preparatory process of shallow earthquakes from physical laws at the microscopic level.

Some examples of unusual telluric activity were found during the Italian-Polish magnetotelluric measurement campaign in the Friuli seismic zone, 1987–1988. The standard procedure for the elimination of the induction component from the telluric recordings and the calculation of the transfer functions were first applied. However, due to a high noise level in the observed telluric fields in the northern Alpine region, it appeared indispensable to develop new methods for the detection of any tectonically related component of the telluric field. The common function, defined as the average curve derived from two telluric stations and multiplied by the correlation coefficients (moving window procedure), allowed an increase in the telluric activity (time changes of variances) to be traced before and during seismic events in September, 1987, and February, 1988. Another approach was based on the time-frequency isoline calculations of the Poynting vector of electromagnetic field. Positive isoline values (opposite to the usual downward direction of this flux), were discovered and correlated with a course of seismic activity; some good results were found for the time intervals studied. Some considerations on a mechanism for the generation of an electric current in a zone of seismic preparation are discussed. An accumulation of stress leads to an oriented motion of the charged (charge-bearing) dislocations. The mechanism consists of the formation of a kind of polarization current induced by the motion of these dislocations.

We present a comparison between some experimental results on the interaction and synchronization of mechanical and electric fields; such a synchronization may appear with the phase delay by π/2, as shown in recent theoretical results. The solutions related to such a phase-shifted synchronization between some fields follow from the Asymmetric Continuum Theory. This theory concerns not only the mechanical fields, strains and rotations, but also other physical fields entering into interaction with stresses.

In the frame of the asymmetric continuum theory we present some aspects of the micro-fracture processes. An extension of this theory, accommodating for a significant increase of external load, can describe some features of a progressively granulated and fractured material. According to our theory, two kinds of motions – displacements and rotations – in some source processes, e.g., in an earthquake source, may be generated independently or with some phase shift due to the rebound release processes. The asymmetric theory includes a possible phase shift between the simultaneous solutions for the displacement and rotation motions. Equivalently, we present the solutions with the simultaneous strain and rotation motions or those with the π/2 phase shift between them. Such specific solutions can explain the synchronization action in fracture processes.

In Chapter 1 we have discussed the basic motions in an asymmetric continuum: the physical fields are represented by tensors. Any antisymmetric tensor preserves its general form; however, the symmetric tensors may have different forms in different systems: diagonal form, off-diagonal or mixed one. Therefore, it is useful to split a symmetric tensor into its axial, A ik E , and deviatoric, D ik E , parts