Mathematical modeling of the impact of blast waves on the underground pipeline was composed from general equations of continuum mechanics, shell theory and hydraulics equations. The problem is formulated in the plane formulation for direct integration of the native system of equations chosen method of finite differences. At the contact of the array and the pipeline, boundary conditions of slippage and rigid clamping are considered.

The interaction of the flexible traction element with bottom mining device by the solution of the set of equations, which allows to determine the nature of the movement and the force between them, and also described the situation flexible traction element as it moves in the water.

The mathematic model of explosion, spreading in a rock mass with excavations and solid inclusions was visualized. The program displays the input data with the ability to control the process of visualization.

The mass velocity diagrams transformation in the layered border zone of the mining system in the time of tension explosion waves influence is considered. The explosion wave front is suggested to be parallel to the proximate handing roof environs discontinuity. Under the admitted assumption the mass velocity epures transformation is investigated in the layered roof stone of the mining system and in the dividing pillar.

In this paper, the joint effect of static and dynamic stress fields on the excavations is considered, and tensile stress regions in the contour zones of the excavations are modeled.

The joint impact of static and dynamic stress fields on the mine workings is considered, tensile stress zones are determined, and the process of brittle failure of rock in the contour areas of the mine workings is modeled. The study of such influences on the mine workings is based on the methodological principles of mathematical modeling of physical processes in complex systems formulated in the article [1].

Finite-difference and numerically analytical methods for solving dynamic and elastic-plastic problems of geomechanics are proposed. Using the finite difference method and on the basis of the principles of the mathematical modeling of physical processes the tensor components, which characterize the dynamic stress field in rocks around the vaulted excavation of the type under the oblique action of the stress wave on it, were determined. An iterative process, which allows calculating the components of the stress tensor and the velocity of the plastic flow of rocks around mine workings at great depths, is designed with the help of numerical analytical method. Reliability and validity of the method is confirmed by comparing the obtained calculations with the results of solutions found by other authors and experimental data.

When a body moves in a gas at high supersonic speed, the gas located in the area between the surface of the body and the shock wave heats up to a temperature of several thousand degrees. At such a temperature, the flow parameters can be significantly affected by the heat exchange of radiation between the gas particles, as well as between the gas and the surface of the streamlined body. The heat emission of the gas in the shock layer leads to a decrease in its temperature and an increase in density.

Let us consider elastic transverse (bending) free vibrations of a plate having the shape of a circular ring, pinched along the inner contour of radius B, with a free outer contour of radius C...

In the study of some problems of streamline bodies with supersonic gas flow, it is more convenient to write down the system of hydromechanics equations in natural coordinates rather than in Cartesian coordinates. Similar coordinates were used in solving problems of the flow of flat and axisymmetric bodies with hypersonic gas flow by the “boundary layer” method.

When bodies move with large supersonic velocities, the gas between the body surface and the shock wave is heated to several thousand degrees Celsius. At such temperatures, the heat exchange by radiation between gas particles, as well as between the gas and the body surface, has a significant effect on the gas parameters in the shock layer. In this case, in order to solve the problem of streamline of a blunt-nosed body by a hypersonic gas flow, it is necessary to involve the full system of gas dynamics equations taking into account gas viscosity and thermal conductivity and radiation field.