Limit load in the problem of penetration of a wedge-shaped tool into the rock mass

The problem of penetration of an absolutely rigid round cone shaped indenter into a weighted rock (an axisymmetric plasticity problem) is being solved. Limit load, depending on the angle of internal friction of the rock, adhesion, angular opening at the top of a cone, rock weight, is determined. The dependence of loading on these parameters is given.


Introduction
Problems of pressing in a rigid punch into a deformable medium were considered in [1][2][3][4][5][6], problems of a rigid wedge intrusion in [7][8][9][10], and axisymmetric problems of pressing in punches were studied in [11][12][13]. The main features of the solutions [1][2][3][4][5][6][7][8][9][10][11][12][13] are: the application of mathematical models of plasticity according to the scheme of a rigid-plastic body ignoring elastic deformations and carrying out calculations with the use of characteristics method. In geoproblems, these mainly relates to calculations of bases and foundations [12,13]. Meanwhile, calculations of rock-breaking tools, energy supply necessary for the destruction of a particular rock are of great importance for mining. At the same time, there is a need to study the influence of rock weight on the values of limit loads.

Mathematical model and solution
Let a rigid cone-shaped tool with the apex angle 2g intrudes into the rock mass. To simplify calculations, we neglect friction at the contact "tool -rock". It is required to determine such a load, applied to the tool along the boundary EOA , at which a plasticity domain is formed around the cone, as shown in the Figure 1. The problem is solved in an axisymmetric setting. In this case, the stress and strain tensor has the form 0 where r u r where k , ψ are constants of rock; n σ , n τ are normal and tangential stresses on the platform with a normal n  . Further, since in state (2) as follows from the Mohr diagram for stresses [2], instead of (2) we obtain the equation (4) is not affected by the second main stress 2 σ . On the other hand, in the theory of plasticity there is such a thing as a state of complete plasticity, for which the two main stresses ( 2 σ and 3 σ ) coincide. For such a state the system of equilibrium equations for stresses becomes hyperbolic [2,14,15]. This hypothesis is also accepted in the work. Since the stress ϕ σ due to (1) is the main one, the equality of the two main stresses means that To describe the states (4), (5), we introduce the following notations where θ is the angle between the first direction for the tensor T σ and the axis r in the Figure 1.
where в g is a specific wheight of the medium. As a result, the characteristics of the system of differential equations of equilibrium are found in the form: where angles µ and ψ are connected by the relation Besides, on each of the characteristics (7) for characteristic 2 λ : where cos a k ψ = . The further task is to solve the problem applying the relations (7) - (10). In order to do this we move from the points on the boundary AD to the points on the boundary OA (along the line PQRS ). At the boundary AD stresses z σ , zr τ are equal to zero, the circuit is stress-free. On the other hand, when the wedge EOA is inserted into the rock mass, the triangle ACD will be compressed from the sides, i.e. the tangential stress r σ at the boundary AD will be compressive. This means that the first principal stress on AD will be the stress z σ and the angle between the first principal direction for tensors T σ and T ε becomes equal / 2 π , i.e.
In this case, the characteristics forming an acute angle with an axis Or will be the characteristics of the 2 λ kind; they form an angle / 2 with the axis Or . At the boundary AD , the plasticity condition (4), where 1 0 Let us consider the triangle OAB . Here, the directions of the main axes 1, 3 are indicated in the Figure 1 (along the segment OA the tangential stress t σ will be negative, but its absolute value is less than the absolute value of the compressive stress 3 σ that coincides with the normal stress n σ on OA ). The value of the angleθ on OA , due to the specified location of the main axes, is equal to / 2 where 2g -an angular opening at the top O of the cone shaped tool inserted into the rock mass.
The angle, formed by the characteristic of the 2 λ kind with the axis Or on the basis of (7), (13) Further, in triangles OAB and ACD we assume that the angle θ is constant.
Let us consider the centered field BAC . To describe it, we use polar coordinates ρ , c : Substituting (15) into (7), we find the dependences ( ) ( )  Figure 1. Then, to solve the problem we should, with the help of (10), connect all these three states distributed in triangles ACD , OAB and sector BAC into one whole. We start the calculation by moving from the boundary AD , assuming that the angle / 2 θ π = in the entire triangle ACD . Substituting this value into (10), we obtain the equation after integrating it we find expression forσ in the form where C is the integration constant. At the point P shown in the Figure 1 values As p Q r r ≤ , than Q p σ σ ≥ .
Let us consider a segment SR of a polyline PQRS . Here, just as with the triangle ACD , it is assumed that in the triangle OAB the angle θ also is a constant value (13), therefore 0 dθ = and the equation (10) The integration constant C in (22) is obtained from (14): ( ) sin cos 2 cos sin 2 sin 2 ctg 2 sin( ) cos cos 2 sin cos 2 cos sin( ) 2sin where p is an unknown pressure at the point S shown in the Figure 1, S r -value of the polar radius at the point S . Applying (22), (23) we find the value at the point R .
The movement along the arc RQ shown in the Figure 1 is examined. For this case The function v is restored from the first equation (26) and u from the second one [16]. Thus , the function · t u v = on the segment QR is determined. Eventually, an analytical solution of the problem was obtained in all strain areas: in the triangle ACD shown in the Figure 1 in the form (20), in the centered field BAC in the form (26), in the triangle OAB in the form (22), (26) Solutions in each domain depend on their "own" constants, which are found from the conditions of continuity of solutions at the boundaries AC , AB . The constant in the triangle ACD is determined by the boundary conditions at the boundary AD . The load on AO corresponds to the solution in the triangle OAB . There is a remark: from the solutions (20), (22), (26) given above, it is clear that the pressure at the point S shown in the Figure 1 depends on the coordinate r of the point P . Due to this fact the pressure at each point on the edge of the wedge will be "its own". To find the full load (or force) on OA , it will be necessary to integrate all normal stresses 3 σ along OA . The paper presents the results of calculating the ultimate load depending on the input parameters.

Сonclusions
For a weighty medium, the relations on the characteristics of a hyperbolic system of inelastic deformation of the rock mass around a cone-shaped tool are integrated. It is shown that these relations are linear differential equations of the first order. The stress field is investigated in the case of simple stress states with a single loading, as well as in the case of a centered field.