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Published on 25 April 2020

# SYSTEM APPROACH TO THE DEVELOPMENT OF MATHEMATICAL SUPPORT FOR AVALANCHE ZONING GIS BASED ON THE STRESS- STRAIN STATE OF SNOW ON THE SLOPES OF MOUNTAIN TERRITORIES

## K. V. Khalkechev

Analysis of existing theoretical and experimental studies has shown that the mathematical basis of existing geographic information systems is not able to solve the problem of identifying avalanche hazard zones for mountainous areas.
To solve this problem, in presented research paper it has been developed the mathematical basis for the geographic information system of avalanche zoning in mountainous areas. This basis is formed on a mathematical model of the freshly fallen snow stress-strain state on the slopes of mountainous territories. During the development of the model, the follow-ing assumptions were made: freshly fallen snow is heterogeneous; structural components of snow (ice crystals, snow pellets and hoarfrost) are anisotropic; deformations are elastic; structural components interact with each other through an elastic field; the elastic properties of the structural components and the external stress field are considered known from the experiment; unknown is the stress field in the structural components. Texture components are snowflakes. In some cases, they are adjacent to each other, in others – are at a distance from each other. To develop this model, a systematic approach is used. Freshly fallen snow is divided into two nested subsystems of structural and texture levels. The mathematical model responsible for the structural level of freshly fallen snow is reduced to a system of equations in tensor values. This system includes: the equilibrium equation; Hooke's law in general tensor form, that written relative to local and average values; dependence between deformations and displacements. A complex self-consistent field method is used to solve this system of equations.
As a result, final expressions for calculating stress tensors and effective elastic modules are obtained. The main difference of the proposed solution method is the simultaneous consideration of the own field and the field induced by other in homogeneities. The mathematical model responsible for the texture level also reduces to a system of equations in tensor form. The solution of this system of equations allows one to obtain final expressions for calculating the effective elastic compliance tensor and the effective elastic modulus tensor. All obtained solutions allow carrying out avalanche zoning in mountain territories on the basis of stressed-strain condition.

 City Moscow Country Russia UDC 551.578.48 Issue 2020, № 1 (Т. 12) Key words systematic approach, geographic information system, avalanche zoning, stress-strain state, freshly fallen snow.
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Tags: системный подход, геоинформационная система, лавинное районирование, напряженно-деформированное состояние, свежевыпавший снег.

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ISSN 1998-4502 (Print)                                                            ISSN 2499-975Х (Online)