## The forecast of rock displacements to determine the underground working support load

The practice of designing the support of underground workings is based on the recommendations of regulations. The rock mass displacement towards the mined-out area are is the determining parameter. For its estimation the package of nomograms and correction factors is suggested, the choice of which is rather subjective, which reduces the reliability of the calculations. In the present work, on the grounds of theoretical and experimental investigations, the methods of estimating the coefficient of rock massif structural weakening is introduced and stress concentration at the outline of the underground working, which is based on the fractal investigations of the working geometry. Calculation formulae are adapted to various types of jointed structure of rock massifs: with parallel and chaotic systems of fissures, which form the blocky structure of the massif. The methods suggested don’t contradict the recommendations of regulations and significantly extend the calculation base of designing the loading on the underground workings support.

## Stability estimation of rock mass weakened by fissure

Stability of jointed rock mass outcroppings is determined, as a rule, by a process of rock shift along the surface of weakness (fissure). Shift parameters depend on the strength characteristics of enclosing rock and fissure geometry. The present article considers the procedure of analysis of rock mass stability by the example of the real conditions of an open pit at the deposit Udachnoye. The methods of division and quantitative estimation of roughness parameters and tortuosity of fissures are introduced on the basis of their trajectory fractal analysis. The procedure of building up the certificate of strength at the rock shift along the fissure is described. The real trajectory length is determined and the surface area of fissures with the account of their tortuosity and fractal dimension estimation. The analysis of stability of outcroppings is introduced under various value of the angle of fissure incidence. The results of the analysis make it possible to determine the coefficient of stability of the outcroppings, together with the allowable load on the benches of an open pit. The discussed methods of analysis can by applied for various conditions of opencast and underground development of jointed rock mass.

## The fractal coefficient of undergound workings shape

In the process of tunneling the mine workings with the help of drilling and blasting operations, the quality of delineation possesses great significance, i.e. obtaining the section which is maximum designated to the designed outline of the working. The quantitative measure of the degree of the outline curvedness of the working is its fractal size. Particularly, it determines the size of the working section perimeter in mining. In this connection in order to quantitatively estimate the outline deviation from the designed one, the outline blasting quality criterion in the shape of the fractal coefficient of the working shape is suggested. It represents the relation of the working conform representation section area to its perimeter: under the constant working section area, the worth the quality of the delineation is, i.e. the bigger the perimeter is, the smaller the fractal coefficient of the shape is. The surface heterogeneity of the working leads to the increase in the concentration of stresses in its outline. Consequently, fractal coefficient of the shape can serve as the estimation of the coefficient of the stresses concentration. In order to study the given problem, the measurements of the sections of 32 workings in Severouralsk bauxite mines have been fulfilled. The correlation analysis of the results has shown the reliable relation of the fractal coefficient of the shape of workings with the coefficient of the stresses concentration in its outline. The results acquired make it possible to estimate the quality of drilling and blasting operations and the stability of rocks in the working.

## Modeling and the forecast of rock strength at the shift of rocks along the fissure

Rock breaking in a mine working in the presence of developed fissure system happens, as a rule, due to the shift along the surface of weakness. Experimental investigation of the given process is complicated by a high labor intensity of full-scale experiments and impossibility of multiple shears along one and the same fissure. In this conditions imitational modeling of the process with the method of Monte-Carlo is an efficient means of investigation. For its realization the establishment of quantitative relations is required, which define the character of rock deformation before and after the achievement of ultimate shearing stress at the shear. On the basis of experimental investigations the present work determines the equations of ascending and descending branches of stress-strain curve. The equation of shear rigidity of fissures on the basis of the fractal dimension of its trajectory is obtained. Estimation procedure of remaining strength at rock shear along the fissure is substantiated. Model validity check by way of comparing the results of its realization with experimental data has shown their statistically reliable correspondence within the accuracy of observations. The application of the developed model allows investigating regularities of shear along the fissure process for various rocks and conditions. Averaging of multiple realizations of a model increases the reliability of received date.

## Statistic modeling of natural fissures

Condition and properties of rocks are significantly dependent on blocky structure and fissure structure of rock mass. Due to the complexity and low repeatability of real experiments, a more effective and informative means of investigation is statistic modeling of fissures on the basis of Monte Carlo method. The work argues the methods of modeling the trajectory of fissures on the basis f their characteristics as fractal objects. Mathematical frameworks for models are given on the basis of fractal brownian movement, approximation of fissures sinuosity by sectionally linear functions and their modeling with a method of middle shifts. Managing parameter of the model is fractal dimension of a fissure trajectory (Hölder condition). The means of generating open fissures edges coordinates are examined. Three model implementations are provided: mirror reflection tension joints edges; shift fissures edges modeling according to common algorithm and fissures midline determination under their edges shifts within established scale. Computer programs worked out allow modeling fissure structure of rocks, as well as the processes of fissures origination and development under various influences upon the massif in the course of its exploitation.

## Dilatancy examination at shift of rocks along the fissure

Dilatancy plays the most important role in the generation of strength and stress-strain state of rock massifs. It is determined by the tension crack of a fissure edges at shifting. The work examines the mechanism of dilatancy at tension crack of a fissure edges along the line of sinuosity and roughness. The results of experiments on the shift of rocks along the fissure are introduced. Linear and non-linear character of longitudinal and transversal deformations at the shift of rocks along the fissure is described. Dependence of dilatancy parameter on the coefficient of roughness and sinuosity is determined. The procedure of building up the certificate of rock strength at rock shift is revealed with the use of initially set value of dilatancy parameter with the help of fissures geometry characteristics. The possibility of direct use of fissure geometry characteristics is pointed out for the construction of two-stage shift of rocks at non-linear representation of the certificate of rock strength. The results of the investigations can be used for the forecast of strength and stability of fissured rock massifs.

## Examination of natural fissures surface as fractal object

Geometry of fissures surface determines the area of their edges contacts and in this way strength and stress-strain condition of rock massifs. The investigation aim is the substantiation of quantitative estimations of hard rock’s surface sinuosity and roughness. With this purpose fractal analysis of fissures geometry is fulfilled, comparative estimation of various ways of surfaces fractal dimension determination is given. Two groups of methods are examined – fractures plane section investigation and the whole surface coordinates analysis. More informative methods of fissure geometry estimation are ascertained. Spectral analysis of sections allows determining the topothesy of surface, the value of which is a criterion to divide fissures sinuosity and roughness. Triangulation method of fractal dimension determination allows estimating the real area of fissures surface and, consequently, their bearing capacity.

## The determination of fissure geometry to build up a certificate of rock strength

Stability of rock jointed rock mass is determined by the character of destruction mainly by means of rock shift along the fissures. In this case in order to build up a certificate of rock strength, quantitative estimation of fissure geometry is required; it is usually expressed in roughness coefficient and tortuosity coefficient. In order to divide these characteristics, the technique of spectrum analysis of natural fissures is offered, where the value of topothesy serves as a criterion. Tortuosity line is approximated by piece linear functions considering active direct stress at shift. The deviations of a fissure coordinates from a tortuosity line are evaluated by roughness coefficient, defined by fractal dimension of a fissure trajectory. The results of experimental investigations of rock shift along a fissure confirm the efficiency of the offered technique of fissure geometry determination to build up a certificate of rock strength at shift.

## Fractal dimension of a fissure as a measure of its roughness

The certificate of rock strength at the shift of rocks along the fissure is determined by its geometry, which is estimated by the indices of roughness and sinuosity. Their magnitude should be determined by comparing the trajectory of a real fissure and some samples. This kind of comparison is subjective to some extent. The article sets the goal to substantiate objective quantitative assessments of roughness and sinuosity of fissures. As the result of research performed it is determined that fissures are fractal objects and the coefficient of roughness is reasonably defined by their fractal dimension. The correlation function can be the probabilistic characteristic of fissure trajectory. Under its magnitude above 0,5, it is required to separate and exclude sinuosity line from the fractal analysis; the sinuosity line is approximated by sectionally linear functions. Average values of inclination angles of these functions diagrams are correspondent to the index of fissures sinuosity. The fulfillment of the said procedure allows determining the coefficients of roughness and sinuosity for typical (sample) fissures.