PHYSICAL MODELS OF SOLID MASS AND RELATED PROCESSES IN INTERACTION W ITH FOUNDATIONS

In the article, interaction of foundations with foundation beds is described and represented on the basis of a general theory of mathematical modeling of large systems in performing three mathematical spaces: – qualitative characteristics; – discrete parameters (DP); – continuous parameters (CP). This representation was made for the first time in comparison with the known works devoted to the theory of interaction of foundations with foundation beds. We have developed the theory of mathematical modeling of large systems, including models of their interaction, of physical processes in the interaction and optimization mathematical models based on the vector criterion in performing three mathematical spaces. They are widely used in the field of mining sciences. We have proved that the issue of interaction of foundations and foundation beds is multidimensional not only in terms of physical processes, described and systematized, but also due to a variety of foundations in the multidimensional space of their qualitative characteristics. This allowed us to cover all foundation beds and their properties. This article provides all well-known models of solid deformable body behavior presented in a systematized form: elastic deformable body, viscoelastic deformable body, models of plastic and flowing medium, models of fluid motion and gas flow, models of thermal conductivity and heat and mass exchange models. We have presented mathematical models of physical processes occurring in a rock mass in the interaction of foundations with foundation beds.


INTRODUCTION
There are no comprehensive generalizing works on the interaction of foundation with foundation bed (rock formation) have been found that systematize and evaluate the study of this issue. In known sources, the necessity of considering this complex system from the position of mathematical modeling of large systems (LS) is not mentioned [1][2][3]. In terms of LS, parameters and characteristics are necessary to determine in performing three mathematical spaces [4,5] for studied systems: artificial system -foundation ( ); natural environment -foundation bed (rock formation) -. Namely: -qualitative characteristics that do not have a reference scale; -discrete parameters (DP); -Continuous parameters (CP).
Introduction of these three mathematical spaces requires the use of two concepts: a point and trajectory of their behavior under or in performing these spaces. Disorder and intuitive consideration of characteristics has led researchers to explicit and implicit contradictions [6][7][8]. For example, this applies to mathematical and physical models for determining the interaction parameters of and systems. This is especially typical for a random research on a huge variety of foundations, which characteristics have not been formalized in certain works [9][10][11]. Mutual interaction considered in a deterministic form is the second major drawback of the known works. Although, interaction of these two systems was proven to occur over the overlapping parameters, which are random variables and even functions that depend on time (t). Mathematical modeling of LS allows eliminating these two major drawbacks. Thus, the presented solution will contribute to more accurate description and, consequently, to the creation of a real situation. This article substantiates the way of overcoming the first drawback. In subsequent works, the possibility of eliminating the second drawback will also be considered.

MATERIALS AND METHOD
There are only three methods for correct and effective study of any system. The first method is a direct experiment on the system in real time. It is not suitable for such LS as the interaction of foundations with foundation bed, since such LS has only one trajectory of behavior. Consequently, we cannot compare the results of such a method, even if they are theoretically possible. There is no possibility to rebuild the structure with other parameters if it was built once. The second method is a large-scale modeling of a system based on the well-known theory of model and system similarity. This method is not suitable for our LS -foundations and foundation bed, because any reproduction of rock formation in another scale does not make sense. At the same time, we should recognize that the separate LS subsystems foundation + foundation bed" can be studied on large-scale models of separate foundations designed to study the nature of their loading in order to select the most appropriate mathematical models of physical processes. The study of soil behavior -foundation under load on rock samples -is an important part of this method.
The only drawback here is that the studied samples are subjected to an experimental study when they are without load in comparison to their real state during the operation of buildings and structures. In this regard, only the third method is useful in studying the presented two systems (foundations and -foundation bed) -mathematical modeling of LS. Obviously, we will be able to consider the entire variety of foundations interacting with any natural environment (rock formation) if we make an adequate description of physical processes and system operation (foundations and -foundation bed) by mathematical language. Besides, mathematical models of LS allow conducting any desired number of any experiments with a maximum approximation to their actual state. Experiments are available for any interval of LS "life" and for determining the sensitivity of all controlled parameters and characteristics of foundations and structures. Mathematical models of LS can be used in modeling the control and damping processes of ground vibrations in earthquakes and wind loads.

Parameters and characteristics of solid mass interacting with foundations
Any building technology Т1, …, Тn interacts with natural or artificial rock mass and with fluids -water, gas or solutions. In construction, there are various physical processes in the solid mass that require studying in order to adapt building technologies to the environment. The most important physical processes for any considered building technologies are: -state of the rock mass -soils, leading to a stressed state of rock formations; -diffusion dissolution of any components in rock formation; -filtration transfer of solutions or any fluids in porous medium.
Let us consider the well-known models of rock mass.

Models of solid deformable body
Х1 -elastic body; Х2 -viscoelastic deformable body; Х3 -elastic deformable body; Х4 -flowing deformable medium; Х5 -medium with a crack system; Х6 -block medium with various organization systems of blocks.

Models of thermodynamic processes in medium:
Х12 -thermal conductivity model; Х13 -heat and mass transfer models.
Let us consider the state of the rock mass on the quasiordered graph G = (X, Г).
If k = 4;  = 4, we will have the following possible number of graph paths from Х1 to Х6 on the quasiordered graph: Simple analysis shows that not all G paths are acceptable: For example, in terms of sub-graph G1  G, we will get: G1 = (X, Г); ; only 4 paths are acceptable: In terms of sub-graph G2 = (X, Г)  G, X = X5  X6 и Х5  Х6 = , the number of acceptable paths will be: ( ) Although, there might be mediums having any block system with its own system of cracks. Then the number of acceptable paths will be: ( ) The total number of solid deformable rock formations will be: Therefore, one has to find a model that corresponds to the real environment in each specific case of modeling foundation and foundation bed.
There are only 6 possible models to describe the processes of diffusion dissolution, filtration of fluids and sorption. They are represented on the sub-graph G3 = (X, Г);   (9) Therefore, the total number of possible models designed to describe physical processes in rock mass with any foundation will be: If k = 4;  = 4, the total number of rock mass models is: Determining 6 specific conditions is a challenge, as well as determining which model more effectively reflects the rock mass with its various properties. Only separate models are usually considered to simplify the process, but not their joint combinations. Thus, the most commonly used models Х1 ÷ Х4 are described in the book [12]. The models Х7, Х8, Х9, Х10 and Х11, thermal conductivity model -Х12, and heat transfer model -Х13 are considered in the same monograph [12]. A brief outline of these models is provided below. M.V. Kurleny and others have made a successful attempt to describe the model of technogenic geomechanical stress [6].
Here is a brief description of geomechanical models of rock deformation, according to an academy fellow M.V. Kurleny and to the monograph [6,12].
Continuum mechanics substantiates the following equilibrium equations at any point (x, y, z) for time t: Where: x, y, z -normal components of stress tensor; xy, xz, yz -shear components of stress tensor; u, ,  -projections of displacement vector on the axis x, y, z; X, Y, Z -projections of volume force vector on the axis x, y, z; If the right-hand sides of equations are not equal to zero, then (12) are dynamic equilibrium conditions; if the right-hand sides are equal to zero, we are dealing with a static equilibrium.
We have also found the following geometric relationships: Strain tensor components in (13) must satisfy the following conditions for medium continuity in the absence of discontinuities: Equations (12), (13) and (14) are common for any continuum -rock formations, water, air, etc.
Physical equations for a linear deformable homogeneous rock mass relate stresses and deformations at any point: Where: Е -Young's modulus (elasticity modulus); ν -Poison's ratio.
The listed equations (12)(15) conceptually suffices to solve any geomechanical problem about the stress-strain state (SSS) of some continuous rock mass volume. (12)(15) shows that the rock mass is characterized here only by the following parameters:
There are only three known force fields in a solid mass: force of gravity in terms of overlying rock mass and structures on the foundation, vertical force: Qz = -gM; -tectonic force in a folded rock mass, which has a very complex nature and a wide manifestation, especially in mountain regions [13,14], Qт(x, y, z); -seismic forces of two types -natural force during the earthquakes, and technogenic force during blasting operations, Qc(x, y, z).
In terms of viscoelastic bodies (rock formations), necessary medium parameters are expanded by two rheological characteristics from the equation [14]: Where:  and  under 0 <  < 1 are required to determine the SSS parameters of the solid mass.
Plastic and flowing medium are expanded by two parameters of rock formations: С -soil adhesion;  -angle of internal friction.
Two additional parameters are often used in SSS model of solid mass: K -bulk modulus of elasticity; G -shear modulus of elasticity.
These two parameters are related with already mentioned Е and ν parameters in the following simple dependencies: Therefore, in dealing with theoretical works in the field of stress-strain state of broken rock mass, one has to know seven parameters: and three vector fields of force distribution in soil mass: This paramount knowledge of medium will make it possible to solve any SSS problem of soil medium. However, this is possible only without taking into account the qualitative characteristics of the rock mass. We get an overall theoretical picture for describing geo-mechanical processes in the solid mass with due account for the possible qualitative characteristics displayed by acceptable sub-graph paths (G1 and G2), when the total number of models for studying SSS of the rock mass is: We should also emphasize that parameters (19) have their own values at every point of the space Т1 = (x1, y1, z1) -rock mass, which can be essentially different.
This circumstance and a lack of mathematical description of vector fields (19) lead us to a suggestion that complete and adequate description of geomechanical processes that develop in real interaction of foundations with foundation bed is challenging.
Apparently, research on SSS has been based on direct measurements and on experiments with the widest spectrum of recorded parameters (19) and characteristics (Х1Х 13) of the rock mass for a long time.

Models of solid deformable body
Most mathematical models of physical processes associated with the stress-strain state of a rock mass are based on the methods of continuum mechanics. We will briefly present the necessary information -its basics -based on the papers [7]. The position of each point of the body is determined by its radius -vector with the components x1 = x, x2 = y, x3 = z in a particular coordinate system. Any solid is deformed under external forces, namely -it changes its shape and volume.
The stress-strain state of a body at any of its points (x, y, z) is characterized by: Where: xx, yy, zz -normal stress; and xy, yz, xz -shear stress at the sites perpendicular to coordinate axes х, у, z; b) symmetric strain tensor Where: ( ) There are three mutually perpendicular sites with zero shear stress at each point of the medium. Directions of normal stress to these sites form the major axes of the tensor ik. Normal stresses on these sites are called the major ones. They are denoted by 1, 2, 3,. It is usually determined that shear stress is in the cross-sections, bisecting the angles between the principal planes: is called a mean stress. In the total number of components ik , only the diagonal components are nonzero by the major axes of the strain tensor ik: These components are called principal extensions. The sum of diagonal components is equal to the relative change in volume: The medium is subject to maximum (principal) shear on the sites located at the angle of 45 ° to the major axes of a strain tensor: Where: 1, 2, 3 -principal extensions.
Two more characteristics of the stress-strain state have to be considered.

Elastic deformable body
If the body subjected to deformation returns to the initial state after removing external loads, then deformations of this kind are called elastic. The Hooke's law is an equation for the state of an elastic body, under which stresses, and strains are related by linear dependence. In expanded form, these equations are like: Where: Ε -Young's modulus and v -Poison's ratio. Bulk modulus of elasticity K and shear modulus of elasticity G are often used in solving problems related with the following formulas: In the invariant form, Hooke's law can be represented like: Where: Т and Г are determined by formulas (29), (31). The specific value for accumulated energy of elastic deformation of a solid body in general form is found from the expression: In solving practical problems in terms of the theory of elasticity, the Hooke's law (32), (34) is used along with the equations of motion and equilibrium for an elastic body. Motion value of medium elements with density ρ and with applied force is determined by differential equations: In this case, summation is performed over the repeated index k. In mathematical models of mining processes, the plane problem of the theory of elasticity is often used. Let's assume that one of displacement vector components is equal to zero (uz), and all other variables depend only on х, у. In this case, we deal with a plane deformation. Thus, the strain tensor components zz, yz, xz and the stress tensor components yz, xz, vanish identically. The stress zz is non-zero and is determined by:

Viscoelastic deformable body
In general cases, viscous bodies are medium models, in which mechanical processes are considered in time. Such models can also be called rheological models. The change of SSS of deformable bodies based on the rheological models is studied by the creep theory of heredity, flow theory, hardening theory and the theory of aging. The creep theory of heredity has received the most theoretical justification and practical application in studies of rock deformation processes. It was developed by L. Boltzmann and V. Volterra in the form of a mathematical description of linear creep phenomena. Subsequently, Yu. N. Rabotnov has proved that the theory of linear heredity can be formally considered as a problem in the theory of elasticity, in which time operators with a creep kernel are necessary to use instead of elastic constants. This position was called the Volterra principle [13]. Zh. S. Yerzhanov (1964) has given a systematic construction of the theory of rock creep. He has showed that the dependence of stress and strain on time is a linear phenomenon: with Abel creep kernel Where ,  -rheological characteristics of rocks (0 <  < 1). Time operators  , , v E for the kernel in the form of a exponential function (40) can be represented as follows. In terms of the Young's modulus: In terms of Poison's ratio: In terms of shear modulus of elasticity: Typical creep curve for rock mass is shown in Figure 1. Section (3-4) -to stage III of progressive flow (destruction).

Models of plastic and flowing medium
In interaction of foundation with foundation bed, a significant part of deformed medium possesses dissipation properties. This includes plastic and flowing medium, which have to be considered separately.

Plastic deformable medium
If deformed body does not return to the initial state after removing the external load, but has permanent deformations, we are dealing with a plastic deformation. Plastic yield criterion is usually expressed mathematically by means of stress components.

Huber-Mises-Hencky plasticity condition:
The s is known not to differ much from max. φ -angle of internal friction.

Mohr-Coulomb yield criterion is usually used to describe the plastic deformation of rock mass
Mathematical theories of plasticity are based on the hypothetical understanding of rock formation properties. They include the so-called ideal rigid-plastic, elasto-plastic, hardening and softening solids with own functional relationship between stress and strain ( Figure 2).
that reflects the law of conservation of mass.
Viscoelastic body. In reaching the ultimate shearing resistance, further deformation can be represented as a viscous fluid motion. Viscous liquid deformation is described by Solutions for viscoplastic flow problems with Navier-Stokes equations are provided in [12][13][14]. It should be noted that there is a significant difficulty in determining viscosity factor of rock formations experimentally, especially since it is not a constant, but represents a particular function of stress and strain.

Flowing deformable medium
The study of soil behavior while foundation interacts with sand is associated with statics and dynamics of flowing medium.

Perfect flowing medium
Let's consider a bulk medium under a certain volume V. If the medium consists of particles, which size r satisfies the relation ( ) These sites correspond to two families of flow lines. Solution of practical problems of bulk mechanics is connected with the assumption of its compressibility. In terms of a compressed flowing medium, its density at each point is a functional relation of a stress state at this point = (). In general case, internal friction angle (σ) is also changing. It has been experimentally established that the angle of internal friction  = () increases in terms of a nonlinear relationship while compressive stress σ is also increasing.

Models of fluid motion and gas flow
A significant class of processes in solid mass is related to the mechanism of fluid motion and gas flow in a medium with variable porosity and permeability.

Filtration-diffusion models of medium
Currently, there are three groups of pores distinguished by physicochemical processes in porous medium: 1) filtration (macroscopic pores); 2) diffusion (Knudsen and Volmer); 3) sorption (microscopic pores). This fact calls for a consideration of a three-phase model of porous structure.
The model of such a medium can be represented by a system of three types of balls placed into each other. Non-deformable balls with radius r1 forming a sorption volume are placed in larger elastic deformable balls with radius r2 corresponding to a diffusion volume of pores. Elastic deformable balls are placed in plastic deformable balls with radius r3 corresponding to a filtration volume of pores. Parameters of the model with an average-static size of particles are determined according to differential porosity curve f(r). Thus, this model reflects the actual porous medium more adequately. The clearance determines the surface porosity, the area ratio of filter channels and the entire filter. In terms of a model medium, clearance is defined by formula (52) The properties of a deformable porous medium depend on the components of stress and strain tensors. In general cases, porosity is determined by  Where: moo, koo -porosity and gas permeability without mechanical stress ( = 0).
If strain components are known for each direction, then the permeability coefficient kjj with allowance for (53), (54) will be ( ).

Equations for fluid motion and gas flow
Fluid or gas internal processes are described by thermodynamic equations of state: Where: Ρ, ρ -medium pressure and density; Т -absolute temperature.
If the process is isothermal, then the temperature is constant at all points of the medium, and T is a parameter in equation (57) If we exclude dissipative processes from true fluid, in particular -heat exchange, we will have a so-called adiabatic process, which equation is Sorption processes are important in the mechanism of fluid motion and gas flow in the coal mass. They include surface and space absorption, as well as formation of a certain chemical compound sorbate/sorbent. Desorption is a process opposite to sorption -separation of previously absorbed substance. Sorption equilibrium is established as a result of opposite processes (sorption and desorption), which feature includes the equality of rates In assuming that the adsorption rate is proportional to the magnitude (1-) at any instant of time t and partial pressure P, and the desorption rate is proportional only to a degree of surface filling by adsorbed substance -, we will get a Langmuir equation Where: D -diffusion coefficient; С -Concentration of diffusible substance (gas or fluid).
Equation of continuity for non-equilibrium (with allowance for sorption processes) gas flow is Where: β -kinetic desorption coefficient.
The parameter β is often assumed to be β = 0. In this case, the system of equations (58), (61)  (64) reduces to one differential equation: If there is one-dimensional distribution of gas pressure near a moving exposed surface, then equation (64)

Thermodynamic models
Processes associated with heat and mass transfer are studied due to a necessity in maintaining prescribed temperature conditions, as well as due to a need to combat underground fires and other thermal phenomena. Thermodynamic models are a mathematical basis of these processes [13].

The law of conservation and transformation of energy in thermodynamics.
Let's consider a physical system interacting with the environment. In this case, there is an energy exchange U between the system and the environment in the form of heat and in the form of activity A dU = dQ-dA.
In applying the Ostrogradsky formula with due account for the randomness of V, we get the equation of thermal conductivity: ( ) In general cases, parameters λ, с, , f are functions of the temperature T, the radius vector r and the time t. For example, , the one-dimensional equation (76) is written in Cartesian coordinate system as follows: ( ). , The equation (76) with constant coefficient (λ, с, ) is:  -Laplace operator. Equation (78) characterizes the non-steady temperature field in threedimensional space without heat sources. Equation (76) shows that the steady temperature field will be described by expression.

(
) In this case, λ, f and Т depend on t. Moreover, if λ = const, we end with the Poisson's equation: , and equation (78) is normalized by the Laplace's equation: These expressions form a basis for one-dimensional and two-dimensional equations of thermal conductivity.

Heat and mass transfer models
Regular thermal conditions usually occur when the bodies are heated and cooled in a medium with constant temperature and with a constant heatexchange coefficient. Let's say that environment temperature is Ts, and the bulk body temperature -Τ v. In this case, regular thermal conditions are described by equation In integrating the system of equations, boundary and initial conditions that arise in connection with specific problems are required to be taken into account. The major problems of non-steady heat exchange are formed as co-stressed ones. This leads to the solution of equations with partial differential coefficients of various types. The general solution of the Fourier-Kirchhoff equation with due account for the convection in a moving medium is very difficult. Therefore, we are considering specific equations that accept solutions in the form of different thermal potentials. Heat transfer in a two-dimensional flow is an important case, because the Kirchhoff-Fourier equation can be simplified here by conformal transformations. Let's say that the medium is moving and the temperature T satisfies the equation Regular heat and mass transfer systems. Heat exchange under high temperatures is often accompanied by physical and chemical phenomena of medium destruction and mass transfer. In this connection, physical fields will be described by the corresponding system of differential equations. Let's consider a continuum with transfer process under the n generalized forces. The energy conservation law will be represented by k -particular constants (for example, is k -temperature, then k = ckk); fk = wk/k -internal energy sources.
Quadratic form allows transforming the system of differential equations (88) of heat and mass transfer into the system: T -intrinsic energy loss by conduction; () -losses related to internal energy exchange. In each particular case, the function us determined from physical considerations. In solving the system of differential equations (89), equations of mass continuity and multi-component mixture motion are also taken into account. If the latter is a viscous liquid, then the Navier-Stokes equation is used: Where: v  -kinematic viscosity factor; P -pressure; F -volume force.
The system (89) usually solved by the so-called eigen-functions or by numerical methods.