The value epsilon means dimensionless product of structure size into wavenumber of usual S waves in continuous medium. Value x means the real value of product of structural wavenumber into structure size. Value y is the imaginary part of it.
This chapter is devoted to study the properties of structured continuum, with specific surface and characteristic size of structure. This linear dimension means the absence of automatic transforming difference relations into differential equations. It is impossible to apply conservation laws at any point of the real structural body, because any closed points in vicinity of inner surface can represent both solid and liquid (gas) phases. We need use some representative minimal volume, which characterized the complicate body at hole. This approach leads to differential equations of motion of the infinite order. Solutions of them, along usual P and S waves, contain many waves with abnormally low velocities, which are not bounded below. It is shown that in such media, weak perturbations can increase or decrease without limit. The reason of the infinite order of differential equations is many degrees of freedom in such media. Catastrophes correspond to unstable solutions equations of motion. Plasticity begins in elastic state like continuous phenomenon, and there is a finite distance between the sliding lines on the contrary with classic plasticity, where distances between sliding lines are infinitely small.
- structure of pore space
- porous and cracked media
The main idea of continuous mechanics is that any volume is the representative one. It means that the integral of loadings, which concentrates on the surface and bounds mentioned volume, is equal to zero in statics or to inertial forces in dynamics. The evident disagreement that the surface forces and inertial ones apply to different points (inertial forces apply to center of gravity of volume) overcomes due to an assumption about infinite small sizes of the mentioned volume. This assumption gives us a possibility to equal the volume forces (divergence of the stress tensor), which was created by the internal stresses, and the inertial forces, according to the second Newton law. Mathematical technique is based on the Gauss theorem about relation between the field flux across surface and divergence of this field in the volume, which is bounded by closed surface. However, in the structured bodies, there is a fundamentally different situation. The representative volume must contain some set of elementary structures. Otherwise, a small volume will contain only one of the phases, for example, liquid in the pores or the solid skeleton without liquid, and will not characterize the properties of the structured body. The characteristic size of the structure leads to fact that the average distance is between one of the cracks to another and one pore to another given by the specific surface of the sample. It is necessary to connect the integral geometric properties of a medium with physical processes of such bodies deforming. On the contrary with a classic continuum of Cauchy and Poisson, the new continuum for structured or blocked media must contain many degrees of freedom. It is evident because elementary blocks may translate the motion by contact interactions, by rotations, and by group of particle’s motion. It means that the energy contents not in first derivatives (strains) only. The potential energy contents in the second derivatives (curvatures) and other orders of ones. It means that the equation of motion of a blocked medium should contain many derivatives; in other words, the equation of motion may have been very high, probably, the infinite order. The static and dynamic processes in the classic continuum are divided by the Great Wall of China from each other. The equation of equilibrium never will pass in the equation of motion. However, it is evident that the dynamic processes often arise very slow and are quasi-static motions. It would be nice to destroy this mentioned wall by a newly structured continuum. It would be a good idea to destroy the abovementioned wall by means of justification of the newly structured continuum. The seismic emission, which causes due to static loading, maybe not a bad example of such phenomena, which are existed between statics and dynamics.
2. Equations of motion for structured media
In Figure 1, an element of the volume of structured body is shown, in which
The distinction between classic and structured continuums is clear, see Figure 1. In the volume, which is inside into surface
The idea of creation of the new model of space is as follows: consider some finite volume of the body (a sphere on a figure with radius
We need to translate the surface forces to the center of the structure by a special operator, and after this, it is possible to apply the law of conservation for some structural image continuum and to act as in a typical classical model of space. The main feature of this approach is to fill all the space, including the pores and cracks by field force. Because of it, we have a continuous image of a very complex media and a possibility to apply the physical laws into an image of the media.
The one-dimensional operator of field translation from point
The operator is . The difference operator is a difference between two translation operators
This is a first difference for finite distance between two points. The second difference may be represented as quadrate of the first difference,
The formally expansion in Taylor’s series gives a finite increment of field. This expansion contains the infinite number of derivatives with different powers of
The analogous operator of translation for some spheres is given by expression
Because there is a Poisson formula 
In the formula (7), parameters are some quantities. However, in Eqs. (6) and (7), parameters play the role of differential operators. The relation between quantities and operators is established by Maslov . Hence,
In the classic continuum, we apply the impulse conservation law to any element of the medium. In this situation, we need to fill all pores over space by a force field. Instead of real stresses, which are changing very fast from one point to another, we can construct the continual image of real stresses. Namely, we use a continuous field, which is constructed by the application of the operator
In a more detailed form Eq. (9) can be rewritten as follows
No wonder that Eq. (9) contains derivatives of the infinite order. This circumstance is due to many degrees of freedom for structured bodies. At , we have the usual equations of motion for classic continuous model of space.
3. Fundamental solutions
We can pass to the image space, following Hooke’s law and applying the Fourier transform along three coordinates, as 
where . The operator
This allows us to calculate the Fourier transform for the fundamental solution of the system Eq. (9):
At very small values,
We can rewrite Eq. (15) in a different form with
Equation (15) obviously has many real roots corresponding to
The growing of ratio
At the same time, Eqs. (14) and (15) likewise have complex roots. The first Eq. (15) shows that complex roots arise only at some values of
Complex roots can mean either damping or unlimited growth of wave amplitude, of course, in the presence of an energy-unbounded source. The minimum damping (growth) corresponds to (2.0288)−1 or about a half of the normal velocity. The same process can be expected to cause both excitation and damping in porous and cracked media, depending on the phase of stationary oscillations.
4. One-dimensional case: plane wave and instabilities
In one-dimensional case, the Eq. (10) takes more simple expression
This equation by substitution gives us the dispersion equation for an unknown wave number
It is evident that by
This effect is more for
It is evident that at ,
Hence, if there is a source of sufficient energy, even some small oscillations can produce catastrophes. It is interesting that nonlinear deforming of samples decreases this effect, because a wave velocity for rocks is decreasing, by growing amplitude of wave. It means that the wave number is growing by the same frequency in the pure elastic process. In Figure 6, the real roots of dispersion, Eq. (18) are shown. The vertical axis shows a dimensionless frequency, namely
5. Pointing vector and equation of equilibrium for blocked media
The equation of equilibrium for micro-structured media can be written from Eq. (9) as
The inverse operator contains zero, but not zero only. It contains some periodic functions and the average value equal to zero. For example, such construction satisfies to Eq. (19)
In Eq. (21) is a quadrat of specific surface; is the pointing vector of usual continuous body, is a dimensionless constant, which must be obtained. These values we can put as constants in small structure volume. The integration with respect to spherical angles gives us a result that the imaginary part of exponent is zero in average sense, namely
Taking into account that the sizes of area much more, than sizes of structure, the area of integration is the infinite large one. In this case, integral Eq. (13) practically is the Fourier transform of fundamental solution of usual elastic equilibrium equations
In Eq. (24) the imaginary part of the exponent is used. Hence, the additional value in average sense is equal to zero. Using relation Eq. (1) , we get a partial solution, which depends on porosity only
If these indexes coincide, we get
Take into account that the average value of a quadrat of cosine is . There is a summation with respect to and is a Pointing vector for usual continuous model of the medium. This value is a small one of the second order compared to usual displacement, because a Pointing vector, divided on the shear module is order to strain, multiplied to size of structure .
Strains. By differentiating of an integral Eq. (23) take into account that the main part of the field contains in fast changing exponent, not in Green tensor itself, i.е.,
According to Eq. (9) the additional dilatation is
Let us integrate the normal component of the Pointing vector on the small sphere with radius
The additional dilatation due to randomly oriented volume forces (an average value of these forces is zero) may be written as
In Eq. (32) the symbol means an integral sine of argument . The left hand in Eq. (32) is an additional expansion or compression, so called as dilatancy. It depends on the potential energy of the continuous body
More strong effect is related with product of high-changing volume force (equal to zero in average) into displacement. This product in not equal to zero in average, because it contains a quadrat of high-changing sine, which is equal to number one third in three dimension space.
If indexes coincide, , we get the additional potential energy, due to fluctuations
The summation with respect to index from unit up to infinity gives
In spite of a fact that the Pointing vector is the small value of more high order, than stresses, the high value (quadrat of specific surface) in Eq. (35) can produce not small common effect. The indefinite factor depends on the real structure of pore space and macro-stress-strain state. However, in some simple situations, it can calculate elementary. For example, at rigid pressing of globe by spherical force (radial displacements are constants), the stress-strain state is a hydrostatic state in average, but not such state at any point. The compressional energy is proportional to compress module of skeleton and its volume plus the incompressibility of fluid and its volume, namely [8, 9, 10, 11]
Indexes unit and zero in Eq. (36) mean solid and liquid parameters. The dilatation of two-phase body gives by the formula
If we have uniform random distribution of phases, the average energy is
In Eq. (38) is the porosity and are the energies of solid and liquid. The dispersion of random value relates with random volume forces, i.e.,
Equation (39) gives the additional energy for very simple macro-hydrostatic state in average. This is the additional of interphase acting. It is equal to additional energy, which is given by Eq. (15). It is reasonable that at unit or zero porosity, an additional energy is equal to zero. The second result is, if the phase energy is equal, the mentioned additional one is equal to zero too. Hence, the indefinite factor given by the simple equation is
6. The arriving of plasticity
In spite of that, the additional average strains is small, does not means, that these strains are small in the any point of the volume. Equations (28) and (29) show that on the planes (
In plane situation, the role of these planes plays orthogonal lines .
The series Eqs. (28) and (29) with respect to
The model of the structured continuum with specific surface of the blocked medium or average size of structure, gives us the differential equations of motion of the infinite order. This model includes collective properties of pore space like the porosity and specific surface and predicts besides usual elastic waves many unusual waves with very small velocities.
This model predicts the decreasing of the Poisson ratio (up to negative values) due to finite size of microstructure. The reason for this is the decreasing of wave velocity with finite specific surface of the rock.
The localization of stresses and strains in structured media begins in elastic state of deforming.
The small areas of a stress-strain concentration looks like usual orthogonal sliding lines in classic plasticity. However, they have a finite effective thickness, which depends on the average size of the structure and the elastic strain limit. Besides, there is a finite distance between analogs of sliding lines, which is equal to the average distance from one pore to another one, or between cracks.