Defining the out-of-limit state of the rocks in the vicinity of the working contour

The problem of defining the stress-strain state of a rock mass near the working contour is solved by the Cauchy stress vector and the displacement vector set on it. To do this, you also need to know the elastic properties and the passport curve of the material “tangent stress – shift” with a section of extreme deformation. The information obtained in solving the problem allows us to judge the condition and remaining reserve of strength of the material both on the contour of the rock mass itself and in its vicinity.


Introduction
In the development of minerals by open or mine methods, questions about the state of the rock mass near the outlines of mine workings are relevant. This information is necessary to prevent collapses, rock bursts, sudden emissions of gas and coal. Traditionally, it is obtained from solving boundary value problems of rock mechanics either in the Dirichlet formulation, when a displacement vector is specified on the entire working contour (the 2nd boundary value problem), or in the Neumann formulation, when the Cauchy stress vector (1st boundary value problem) is set, or in the formulation of Robin, when a vector of Cauchy stresses is specified on a part of the boundary, and on the other one -a displacement vector (3rd or mixed problem) [1][2][3]. When solving these boundary-value classical problems, it is required to know the entire geometry of the computational domain; moreover, it is also required to know the loading conditions of the rock mass at "infinity". Both are not always achievable. At the same time, obtaining information on the state of the rock mass near the mine working is a priority. In this situation, the essential solution to the problem can be obtained by setting the Cauchy stress vector and simultaneously the displacement vector on the circuit. Let us give a solution to this problem in the case when the rock mass near the working contour is in a state of destruction -at the out-of-limit stage of deformation. If the stress expresses the resistance of the material to deformation, then at this stage, the resistance of the material to deformation decreases with increasing strain, i.e., here comes what is traditionally called destruction. The proposed statement of the problem may seem impossible. But it's enough to recall the simplest situation: there is a surface of the Earth free of stresses (the Cauchy vector is equal to zero vector); on the other hand, there are 3D measurements of displacements of the Earth's surface made from satellites (a given vector of displacements of the Earth's points). A similar picture can be seen in mine conditions -there is a working contour free of stress, there are displacements. So, this statement of the problem is feasible.  Figure 1 shows the diagrams of sulfide ore deformation in the form of two dependencies ,

Experimental data of A. N. Stavrogin and his followers on hard loading of rocks [4]
where the axes 1, 2, 3 correspond to the axes , , z r  of the cylindrical coordinate system.
The diagrams are obtained with axial and lateral compression of cylindrical samples. The samples were initially subjected to hydrostatic compression, then, at a constant level of lateral pressure, they were brought to destruction by axial compression with a controlled change in the axial displacement (hard loading). The stress and strain tensors in these experiments were as follows: then the dependencies between the coordinates of the tensors T  , T  will have the form like in Figure  1, i.e., they are not passport, they depend on the level of lateral pressure. If we take as a basis  It is seen that the basis (2) is again not proper, since the curves in Figure 2 depend on the level of lateral pressure. The next step is to turn the basis (2) If 40    , then the initial dependences of Figure 1 in the new basis (3) will take the form of Figure   3. We can see that the behavior of the curves does not depend on the level of lateral pressure. One curve is a straight line (on the left), the other is a "single curve". Linear sections on these curves determine the theory of elasticity of different -modulus materials, sections up to the limits of strength -the theory of plasticity, beyond the limit of strength we have the equations of out-of-limit deformation.

Figure3.
Diagrams of stress and strain changes for sulfide ore in the new basis (3).

Plane strain of out-of-limit deformation
As a simplified version let's consider plane strain. For it, instead of (2), we have a tensor basis of the form Here, 1, 2 are the main axes of the stress and strain tensors (the axes coincide 3 0   ). The defining deformation relations for the rotated basis (4) have the form: in the case of elasticity in the case of out-of-limit deformation leads the problem of out-of-limit deformation to a characteristic equation of the form Then we assume that if a system of differential equations has characteristics, they are arranged symmetrically relative to the first main direction for the stress tensor. Substitution ( ) tg      turns (7) to the biquadrate equation 4 2 where 4 2 0 Solving (8) and ratios on the characteristics: where 2 2 cos sin , Thus, we obtain four characteristics of the out-of-limit deformation and four relations on them that connect the maximum tangential stress T , the angle  , the average stress  and the rotation angle .
It is shown that the system of equations (9) at each point of the destruction area has the unique solution (its determinant is not zero) provided that both the Cauchy stress vector and the displacement vector are set at the same time as the body boundary.

Defining of the stress-strain state in the vicinity of the working contour
We consider a three-dimensional body with a boundary S. Let on this boundary at some point 0 M the Cauchy stress vector with coordinates is set simultaneously with the vector of displacement n u  with coordinates