123

ISSN 0536-1028 (Print)
ISSN 2686-9853 (Online)

УДК 622.83+539.3 DOI: 10.21440/0536-1028-2019-3-22-29

Introduction. The class of ill-posed problems in geomechanics is very wide because of the lack of efficient numerical methods and a desire to obtain analytical solutions.
Research aim. The theory of problems in mathematical physics intended to consider infinite bodies using the class of Cauchy problems with the Cauchy conditions or Cauchy data that vanish at infinity. The violation of this conditions leads to ill-posed problems. The research aims to propose a system of integral equations which will make it possible to reduce the level of assumptions when solving some problems of geomechanics.
Methodology. The correctness of common solutions to a number of geomechanical problems has been analyzed. This class of problems includes rock mechanics problems considering a half-plane, a plane with holes, and a space with a plane. In these cases, it is required to solve an auxiliary problem by Mikhlin and Kristianovich.
Results. With the numerical methods, for example, finite element method, and the commercial programs at hand, another extreme appears. In calculation of stress-strain state in surrounding rock around underground excavations researches used to pose boundary conditions without concern about their compatibility. In the meanwhile, the requirements of the theory of problems in mathematical physics must be satisfied. The analysis shows that the new extreme has even increased the number of ill-posed problems.
Conclusions. This study puts forward the integral equations that allow withdrawing ill-posed problems before solving and discusses the class of such problems to solve the problems of rock mechanics as well.


Key words: underground excavation, displacements, Cauchy problem, boundary conditions, the correctness of the solution.

 

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