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

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Since , xz yz   by the last equations of the Hooke’s law (24). In order to find derivatives from displacements along the coordinate z , the system of differential equations of Cauchy (23) is considered. From it, with known strains , , xz yz z    , displacements, and derivatives , z z u u x y     at the boundary, decision variables , , y x z u u u z z z       are obtained. By integrating these expressions we find 0 z  , stresses , xz yz   are known owing to (20), we find strains

displacements

, , x y z u u u on the boundary

z z h   , where z h is a difference step along the coordinate z .

, , xz yz z    the differential equilibrium equations (22) are

In order to define stresses on the boundary

z z h  





2   u





x 

xy

, from the second

integrated. From the first equation we obtain





 

xz

x

2

t

x y

t



 



2 u    xy   y







z 

x 



2   u





 

yz

yz

xy

, and from the third one

. Thus, as a result of integrating



xz  





t

z

2

2

z

x y

y

t

t



 



 





these equations, we find the values of quantities z z h   . The process of calculations according to the scheme mentioned above goes on. To check the calculations, the well-known solution of rock mass vibrations around a single source located at a depth z H   is used. The functions of this solution for the coordinate 0 z  are taken as initial functions , , x y z u u u , , xz yz z    depending on coordinates , x y , and time t . The task is, using these data, to restore the position of the source, its intensity, stresses, strains, and displacements in the vicinity of the source. Some calculated curves are shown in Fig. 2. , , x y z u u u , , xz yz z    on the layer

Fig. 2. Dependence of displacements , x y u u on coordinates on the surface of the buried source, calculated according to the proposed scheme. The obtained calculated values are as test values. It makes it possible to use this program for calculating other (real) situations. 3. Equations A non-destructive testing method for determining the structure of the medium has been proposed. An algorithm for finding the stress-strain state of the structure according to data of measurements of displacements and stresses on its surface has been developed (Cauchy problem).