Ye. I. Rogov1, A. A. Kussainov2, V. V. Gumenyuk2*

1Institute of Mining named after D.A. Kunayev, Abai Ave., 191, 050046 Almaty, Kazakhstan.
2Kazakh Leading Academy of Architecture and Civil Engineering, Ryskulbekov Str., 28, 050043 Almaty, Kazakhstan.
*Corresponding author email:

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In the article, interaction of foundations with foundation beds is described and represented on the basis of a general theory of mathematical modeling of large systems in performing three mathematical spaces: – qualitative characteristics; – discrete parameters (DP); – continuous parameters (CP). This representation was made for the first time in comparison with the known works devoted to the theory of interaction of foundations with foundation beds. We have developed the theory of mathematical modeling of large systems, including models of their interaction, of physical processes in the interaction and optimization mathematical models based on the vector criterion in performing three mathematical spaces. They are widely used in the field of mining sciences. We have proved that the issue of interaction of foundations and foundation beds is multidimensional not only in terms of physical processes, described and systematized, but also due to a variety of foundations in the multidimensional spa ce of their qualitative characteristics. This allowed us to cover all foundation beds and their properties. This article provides all well-known models of solid deformable body behavior presented in a systematized form: elastic deformable body, viscoelastic deformable body, models of plastic and flowing medium, models of fluid motion and gas flow, models of thermal conductivity and heat and mass exchange models. We have presented mathematical models of physical processes occurring in a rock mass in the interaction of foundations with foundation beds.