PSI - Issue 40

A.I. Chanyshev et al. / Procedia Structural Integrity 40 (2022) 97–104 Chanyshev A.I. at al. / Structural Integrity Procedia 00 (2022) 000 – 000

101

5

0 y  functions 1 ( ) f x , 2 ( ) f x , 3 ( ) f x , 4 ( ) f x be represented in the

Let us consider a specific example. Let on

2 2 a x H p p x H   2 ( ( ) 2 2 2

2

)

2

 

 

2 a x

p

,

,

,

forms

2 p xa H x H  (

(1 2 )     x



xy  

  

u

 

y

x

2 2

)

2 x H

2  

2



 

 

2

pH

a

.

1 2    

u

 

y

2 x H

2  

2



Substituting these functions into (16) and (17), we obtain that

2

pa

( ) 2 pz z    

( ) z  

const

(18)



,

.

const

z iH 

Here, there is a pole of the first order at point z iH   , which is obvious from (18). The solution (18) corresponds to the case when at depth y H   there is a cavity of the radius r a  loaded with pressure p . Stress and displacement fields in a formula view are presented below

            

2 x y H p pa x y H x y H p pa x y H x y H pa x y H            2 2 2 2 2 2 2 2 2 ( [ ( ( [ ( 2 ( ) , [ ( ) ]

2 ) ,



  

x

2 2

) ]

2 ) ,

  



y

2 2

) ]

xy    

(19)

x

2

2

(1 2 ) 

,

u

px pa 

  



x

2 x y H   (

2

)

y H 

2

2

(1 2 )(      

) y H pa

.

u



y

2 x y H   (

2

)



From (19) we can see that if there are stress-free cavities in the stress field, then they can be found from the condition:

x     xy xy y

0.

 



The dynamic setting of the Cauchy problem gives us the solution of the proposed method for diagnosing product condition. Let's imagine that there is the Earth surface in the form of a half-space with equation 0 z  (Fig. 1) (here z is a coordinate of point in XOYZ space) Postonen and Dolgirev (2020), Alpatov and Zhalilova (2020), Bikov and Chelyadinova (2019). The surface is stress-free, which means that.

0 zx zy       . z

(20)

Displacement measurements mean that functions of three variables are set on the Earth surface

( , , ) x x u u x y t  ,

( , , ) z z u u x y t  .

( , , ) y y u u x y t  ,

(21)

The task is to determine according to (20), (21) the stress-strain state of the Earth surface, its defectiveness at the