PSI - Issue 65

Anvar Chanyshev et al. / Procedia Structural Integrity 65 (2024) 56–65 Anvar Chanyshev / Structural Integrity Procedia 00 (2024) 000–000

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2

position of the mine workings and inclusions, such as mineral deposits, is known a priori. Another range of problems refers, for example, to geological surveys, where there is a need to determine unknown loads acting on the rock mass, its structure, and to indicate oil- or gas-containing cavities, including their geometry, internal pressure, and also identifying solid inclusions (Folorunso 2013, Polyansky et al. 2013, Zuza et al. 2022). The situation is further complicated by the fact that the mediums may be locally inhomogeneous and initially anisotropic, characterized by the corresponding set of 21 elastic constants. Traditionally, to solve mining problems, previously developed formulations of the first, second, and third boundary value problem (Muskhelishvili 1977, Ilyin 2009, Haberman, 1987) are used, where either the Cauchy stress vector or the displacement vector is specified along the entire boundary of the rock mass, or the Cauchy stress vector is set on one part of the boundary and the displacement vector on another (mixed boundary value problem). In geophysics larger scales are investigated, where the structure of a large-sized rock mass and its loading conditions remain unknown (Wang et al. 2012, Epov et al. 2019). In our paper, in contrast to the preceded works, we solve the problem where at the boundary of a half-space, in addition to the condition that the surface is stress-free, the displacements are considered to be known from observations (e.g., space images). It is shown that such a problem for an initially anisotropic half-space has a solution and it is unique. For simplification, we considered the case of plane strain. The solution of the problem proposed in this work seems to be actual and has an applied significance 2. Problem statement Let there be a rock mass, as depicted in Fig. 1, and the following conditions are simultaneously specified on its boundary:

0, y xy    

0;

(1)

( ), x y u f x u x    ( ),

(2)

here, condition (1) indicates that the boundary 0 y  in Figure 1 is free from stresses, while condition (2) represents the effect of unknown loads applied to the sides of the examined area and the impact of pressure exerted on the boundaries of the cavities located in the bulk. The objective of the problem is to determine the loads acting on the sides of the examined area, identify the positions of the loaded cavities within it, as well as any rigid inclusions. The problem is solved in the framework of the theory of elasticity in the case, when the material of the half space under plane strain is an initially anisotropic body, and the Hooke’s law is:

11 a a a a a a a a a                   12 12 22 13 32 x x y y x y xy x y     

2 , 2 , 2 , xy xy xy   

13

.

(3)

23

33

1 2





, i j u u 

,

ij  

(4)

, j i

where

i u are displacements.