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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.

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.

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.

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.

This chapter describes the micromorphic continuum and fractal properties of faults and earthquakes. Because the internal structures and discontinuities in the lithosphere seem to have an essential influence on the fracturing in the lithosphere, therefore, the notion of continuum with microstructure can be a suitable tool in describing earthquake phenomena. The generalized micromorphic continuum is especially suitable for introducing microstructure. The chapter introduces the rotational effects related to the internal microstructures at the epicenter zones and the equilibrium equations in terms of displacements (the Navier equation), which are derived from the concept of this micromorphic continuum. These equations are the generalization of Laplace equations in terms of displacements that generate several local diffusional conservation equations (Laplace equations) of strains within the lithospheric plate with microstructures. A generalization of Elsasser and Rice's model of stress diffusion and some scaling laws of fracturing in the lithosphere are also reviewed. The nonequilibrium macroscopic flux of strains evolves a local balance or stationary state of strains with the fractal fracturing within the lithospheric plate with microstructures.

Self-strains (thermo-nuclei, dislocations, disclinations, vacancies) generally are sources of internal stresses. This chapter presents an elastic continuum with some distribution of self-strain nuclei: thermal nuclei and defects (dislocations, disclinations and vacancies). Plastic deformations can be described using an elastic continuum with some distribution of dislocations and disclinations instead of using the plastic constitutive laws. In such an elastic continuum one cannot restrict only to strains, but also consider other deformation measures and the relations between them. The deformations expressed include total distortion β T related to displacement vector and total strain, rotation, and bend-twist. It is often convenient to divide the total deformations into elastic and self-deformation (plastic/defect-related) parts. Small elastic strains deform the structure and orientation of a crystal lattice, whereas plastic deformation (dislocations) can be introduced by the well-known procedure of cutting, shifting, and coalescing of material elements (with a possible addition or removal of some matter). The curvature of a body structure is expressed by gradients of small rotations associated with the bend-twist tensor.

This chapter describes evolution, propagation, and diffusion of dislocation fields. A time change of the dislocation density tensor follows from the conservation formula expressed by time change of the dislocation density integrated over a surface element perpendicular to the dislocation lines and the circuit integral enclosing this element representing an outflow of dislocations. The stress and dislocation fields depend on the production rate of dislocations and on the barrier content in an earthquake preparation zone that hampers the escape of dislocations in an outflow process. The integral relations between the dislocation density and stresses are commonly known and are in common use in computations; stresses are expressed by integrals in which the integrand is given by the dislocation density, as a source function, multiplied by the appropriate Green function. Self-strains cause a field of internal stresses; a thermal field is an example of a self-strain field. The dislocation density is related to self-distortion (plastic distortion) by the Kroner formula.

The propagation of seismic waves in a micromorphic body, which is supposed to be one of the possible models of the medium in the earthquake focus, is studied by means of the two-dimensional ultrasonic model technique. The results obtained indicate that such a medium is characterized by distinct elastic anisotropy and by increased attenuation. The propagation velocity of longitudinal waves decreases and their amplitudes change considerably. The prevailing frequency of the waves passing through the micromorphic medium tends smoothly to one value which depends on the space distribution, orientation and dimensions of the elements characterizing such a medium. The obtained results are confronted with the theory of a micromorphic medium.

Some features of the global distribution of earthquakes are discussed, and symmetry in the global distribution of tectonic fractures is noted. A new notion of seismic chimneys is defined and their role in global tectonics is considered. Deep horizontal displacements in the distribution of deep-focus earthquakes are also pointed out.

The tensor relations describing the shear deviatoric strains and rotation strains may be presented as vector relations in a special coordinate system, e.g., in the diagonal or off-diagonal one. However, these fields can be also presented in the 4D invariant forms by means of invariant Dirac tensors. We present 4D relativistic relations for the invariant shear deviatoric strain and rotation strain vectors closely related to a fracture process in solids and to the molecular strains (shear and rotational) in fluids. These shear and rotation strains may interact with the radial derivatives of pressure along the propagation directions.