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

102

6

0 z  , the presence of cavities, solid inclusions, and so on.

depth

Fig.1. Flying device measuring 3D displacement above the Earth surface.

To solve the problem, we use the equilibrium equations:

2

y u





xy x y    z    

y      yz



2

2

x u

z u

xy               x xz x y z

yz               xt z x y z

2 ,

2 .

2 ,

(22)



t

t

t



Cauchy relations:

u

u

u







1 2 z         u u y z     y

u



  

  

  

  

u

u

u

1 2

1 2







y

y

y

. z

(23)

z 



,

,

,

,

,











x 





y 





x

x

x

yz

xy

xy

z



x

x y

x y

y



 

 



Hooke's law:



1

1









yz

.

(24)





…,

,

,

x 

x        y z

y 

y        x z

yz

2



0 z  . From (21) it

The plan of problem solving is as follows. Let displacements (21) are set on the surface

0 z  deformation xy  is set, and then, owing to (24), the stress xy  is set. Further, 0 z  the displacement x u is known, it follows that x  is set. And, as displacement

follows that on the surface considering that on the surface

y u is known, we define strain y  . At known , , x y z    , stresses

, x y   are found from the first two equations (24):









2 E E           2 1 z x y

2 E E           2 1 z x y

(25)

,

.

x 







y

At known stresses , , x y z    , strain z  is obtained from the third equation (24)         2 3 2 2 2 1 3 2 1 z x y z E E E                    .

(26)