PSI - Issue 65

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

64

9

Note . The equation (26) coincides with the definition of the principal values of the stress tensor. One of these values (the constant one) corresponds to the normal stress at the boundary of the hole, while the other (the variable one) represents the value of the tangential stress. The expression under the square root in (27) coincides with the expression for max  . We insert (24) into (27), then, after squaring and summing, we find:

2

    

  

2 2 A

2

2 8 (2

2   

b b

16

)  





x

y

.

(28)

2    xy



 

 

 

2  

2 2

2   

2 3

2

1

(  

)

(

)

By turning  to zero in (28), we obtain a circle of radius a with the equation 2 2

2 2 2 x y yH H a     , where

2 0 a H b    . This circle is loaded with uniform pressure modulus, parameters A , H , b are related to the boundary conditions 3 4    ,  is the Poisson's ratio. Similarly, a solution to an initially anisotropic medium can be derived by determining its loading conditions and structure. 2 2 2 2 / ((1 ) ) A a  p   , where 2  is the shear

4. Conclusions

We has obtained the solutions to the problem of determining the stress state of an initially anisotropic elastic half space under conditions of plane strain for cases of simple and multiple roots of the characteristic equation. This solution is based on displacement values on the surface. It has been demonstrated that such a solution exists and is unique. The distribution of stresses and displacements throughout the rock mass has been obtained for the case of multiple roots of the characteristic equation (initially isotropic medium). It has been shown that under certain boundary conditions on the surface, it is possible to discover cavities within the rock mass, their geometry, and pressure values at their boundaries.

Acknowledgements

The study was funded by a grant from the Russian Science Foundation (project No. 22-17-00188).

References

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