Explosive technologies in transport construction

Performing different works (construction of road embankments, canals, tunnels, moving some soil mass in a given direction), as well as construction of various towers, power transmission supports, foundations for bridge supports, special pits for anchors are carried out by explosive technologies. The paper analyzes in detail a new method of determining the boundary of the explosion excavation. Its distinguishing features are the use of an impulse-hydrodynamic model for setting explosion problems and use of boundary problems of the theory of analytical functions. The new method is used in solving the problem of determining the boundary of the ejection excavation for the case of a buried charge. The results of calculations for the main geometric parameters of the explosion are given. The prospects for the development and application of the method are discussed.


Introduction
The energy of an explosion is a special type of energy that can be easily metered, its transfer does not require stationary communications, and its ability to perform work eliminates the need for complex working machines. Its high concentration and enormous power characteristics have also determined the special field for the use of the energy of explosion where other energy sources are ineffective.
Explosive technologies have found wide application at development of mineral resources [1,2], at consolidation of grounds [3,4], they are used at reception of underground tanks [5,6], in tests of building designs on impulse influence [7]. At manufacture of some works (building of road embankments, channels, foundation pits) there is a task of moving some weight of a ground in the set direction. Construction of various towers, power transmission towers, and foundations for bridge supports, special foundation pits for anchors and their concreting is carried out using explosive technologies. During excavation works related to the construction of road embankments, trenches and wells, so-called cord charges are used. As experiments show, the movement of the medium accompanying the explosion can be divided into two stages. The first, short-term, is characterized by the propagation of the stress wave and a relatively small increase in displacements and velocities of the particles. At this stage, reflections may occur and destruction may occur.
The first stage is associated with numerical modeling of detonation [8], in particular, with the grid modeling of detonation waves [9], with the study of shock waves [10,11], and also with the structure of shock waves [12,13]. A number of works are devoted to the analysis of borehole charge initiation [14], the influence of shock wave shapes on explosion parameters [15], and the construction of shock wave computational schemes in special cases [16][17][18].
The second stage is ballistic. There is either non-camouflage explosion or individual blocks are thrown away from the point of explosion. The study of the second stage is of interest not only in the construction, but also in the study of the formation of craters of celestial bodies [19][20][21]. At the end of the first stage, a velocity field "initial" for the ballistic stage is produced. At the first stage, the fields of stress and velocity are determined mainly by the inertial resistance of the medium, so the compressibility can be neglected. Since the pressure at the initial stage of the explosion is very high, a second assumption is made to consider the strength effects as secondary and to describe the state of the ball pressure tensor. All this makes it possible to assume that the medium is incompressible, that the medium is ideal (that is, there are no tangential stresses), and that deformations and displacements remain small. These assumptions make it possible to use the model of an ideal incompressible fluid to calculate the size, embankment, tunnel or trench when calculating the explosion for an ejection. M. A. Lavrentyev introduced an additional strength characteristic of the soil -critical velocity.
Built by Lavrentyev M. A., the model is called impulse-hydrodynamic and applied by V. M. Kuznetsov [22]. Due to the large magnitude of the explosive loads and the short duration of their action at the initial stage of the explosion medium can be considered an ideal incompressible liquid, then its movement is described by the equation.
If we apply the impulse model of problems of hydromechanics, then we can obtain the field velocity through potential.
There are different varieties of impulse-hydrodynamic model. The paper uses a solidliquid model, where the ground is described by the equations of ideal incompressible liquid, only in some area near the charge. Outside of this area, the soil behaves like an absolutely rigid body; the boundary separating the liquid is a solid wall, which is located from the condition that the velocity module on it is equal to a critical value. The critical velocity is a strength characteristic of the medium and is defined as follows. Fracture is considered to occur when the specific kinetic energy of the medium particles � � �� � � exceeds the limit specific energy � * � � * � /�2�� required for the destruction of the medium (here E-Young's module, � * -yield strength limit). Then from the condition� � � * critical velocity can be found by formula � * � � * /���. The aim of the work is to develop a new method for calculating canals, trenches and sinkholes obtained by blasting.

Methods
Explosive technologies are successfully used to create pits and trenches in transport construction. To solve these problems, methods of the theory of functions of a complex variable may be successfully used. Let us solve the Hilbert problem with discontinuous coefficients for a bipartite domain and apply the solution to the problem of explosion. Without reducing generality, we may assume that We will solve the problem by the Muskhelishvili method, reducing it to the Riemann problem for a double-periodic function. Denote by rectangle, symmetric about the axis, and introduce the function (2) This function is analytic in the regions and . Let us continue the function up and down periodically by assuming, where is an integer.
Thus we obtain a bi-periodic function with periods , which satisfies the symmetry condition, i.e. . The boundary condition (1) can be given the form  (6) are, respectively, the Weirstrass zeta and sigma functions for periods and . The function will be biperiodic if  (8) is a solution of the Hilbert problem under study and will be bi-periodic if [24]: (9) Here C is an arbitrary complex constant. Function (8) will be analytic in the domain if the expression in curly brackets of the right-hand side of formula (8) is zero at points and at a point has zero of order . Consequently, the following conditions must be satisfied: Here we assume In addition to these conditions, we must also satisfy (9), which is, as is easily seen, a valid condition. It is also easy to check that each of the conditions (10) and (11) are real.
It is clear from the above that the problem under consideration is solvable if conditions (9) and (10)-(13) are satisfied. By singling out the real and imaginary parts in (12) and (13), we arrive at the real relations. One of the conditions (10), (11) can be satisfied by selecting . Thus, the problem under study is solvable if the real solvability conditions are satisfied. In this case it has a single solution defined by formula (8). The other cases are treated similarly. As an example of the application of the above results, let us consider the problem of the explosion of an infinitely long buried charge within the framework of the solid-liquid model. The motion is assumed to be plane-parallel, so it is sufficient to study it in the plane perpendicular to the axis of the cord charge, and take this plane as the plane of the complex variable . Let's direct the real axis along the plane coinciding with the ground surface, the charge will be considered to be located in the half-plane of . The fluid flow is described by the complex potential . Let the charge in the lower half-plane occupy the position , which makes an angle with the axis .   Using this formula it is possible to obtain all mechanical and geometric parameters of the explosion. Table 1 shows the results of calculations to determine the geometric parameters of trenches.

Results and discussion
The use of force is widespread in building technologies [25]. In this work, the methods of the theory of analytical functions in the framework of the solid-liquid model were used to solve the problem of determining the excavation of the ejection during the explosion of a buried charge. Methods of the theory of analytical functions are widely used in practice [22,[26][27][28][29]. Due to the importance of boundary value problems, research is conducted in the 0 1 y C x direction of solving Hilbert, Riemann boundary value problems with an infinite index [30,31], and also solutions in the case of special behavior of the boundary value coefficients [32]. In these works the behavior of singular integrals at special points [33] and peculiarities of application of the Christoffel-Schwarz formula [34] are used.
In the present work the boundary Hilbert problem with discontinuous coefficients for a bipartite domain was solved, which allowed to carry out a complete analysis of the buried charge explosion problem, to determine the boundary of the ejection notch, as well as to find the coordinates of important geometric characteristics of the notch.

Conclusion
In the process of construction of road embankments, canals and foundation pits there is a problem of moving some soil mass in a given direction and calculating the geometry of channels and foundation pits. To calculate the ejection of a buried cord charge we propose a method that uses the solution of the boundary Hilbert problem with discontinuous coefficients for a bipartite domain. The solution is obtained in this study.
In the study, calculated schemes are established and simulation of the process of formation of the ditch (trench, channel) is carried out, geometric and physical parameters affecting the shape and size of the ditch are determined, formulas for finding in the physical plane of all the interesting features of the ditch are obtained, numerical calculations have been performed.