One of the secondary damaging factors of explosions at the explosion and fire hazardous production facilities is the fragmentation field. However, the calculations currently being performed to assess the damaging effect of window glass fragments are limited, as a rule, to two-dimensional simplified models.

The purpose of the article is to describe the mathematical model of fragments scattering as secondary damaging factors, to perform the numerical calculation, and to determine the area of fragments scattering in 3D space at the given parameters.

To achieve this goal, the mathematical model of window glass fragments scattering under the influence of the air shock wave is proposed. The model is based on the regularities of the distribution and dissipation of energy, which the fragments initially receive when a shock wave is exposed to the window opening due to explosion. Further calculations are carried out by solving the system of differential equations describing the fragment movement in projections in three directions of its motion.

Numerical solution of the system of differential equations with the given initial conditions is obtained by integrating the system of ordinary differential equations using the Runge — Kutta formula of the 4th and 5th order. When modeling, the system of equations is represented in the form of the vector function. In this case, the following is set: the vector of initial conditions, the time interval, on which it is required to find a solution, and, also the additional options. Numerical calculation ends at the moment when the calculated fragment reaches the ground.

Thus, the calculations performed according to the proposed mathematical model allow to substantiate the range, altitude and flight time of glass fragments in case of explosions. In future, the software implementation will allow in an expeditious manner to make assessment and forecast calculations for analyzing possible technogenic factors and their consequences, which in turn will help the timely development of relevant regulations on assessment of the facilities safety, performance of civil engineering defense measures, and also accelerate the development of an industrial safety declaration.

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