PSI - Issue 23

I.S. Nikitin et al. / Procedia Structural Integrity 23 (2019) 125–130 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

127

3

e      e e e e , ( ) / 2 T    e v v . Here v – “macroscopic” velocity of medium particles, e e - the elastic strain tensor and accordingly to the Hooke's law: ( : ) 2 e e     e I I e  . Appropriate conditions for γ and ω can be formulated as   2 2 / 1 s F       γ τ τ , 0   while 0 n   and 0 n   τ = while 0  .

The final equation is the motion equation:   v σ

Based on the chosen normal direction in the Cartesian coordinate system the full system of equation is:

, 22 ( 2 ) ( ) k k v v H        , , 2,2 22

, k k v v         , i i 2 ii

, ij j v    , i

i

2

, i j i j v v      , 2 ( ij

)

, i j  , 2

(    

 

2, v v j

H

,2 )   j

22 ( )  

j 

,



, j i

j

j

2





 2 2 / s 

τ

F

1 ( ) ( H v v H H       

) ( ) ( ) 

 

 

,

j

j

j

j

2

22

2,

,2

22

2    τ k

, ( 2 ) ( 2 ) ( ) ( ) k k v v H H           , 2,2 22

1,2,3 i j  ,

  , ,

2 2  

.

k k

In the layered medium the only system of slip-detachment planes with the normal n is available. The block medium consists of parallelepiped elastic elements with three possible slip-delamination planes. These planes are defined with normals ( ) s n , s =1,2,3. In this case the non-elastic strain tensor is:

3

(  e n  

( ) s γ γ n

( ) 0 s s   γ n ,

) / 2

( ) s   

( ) s

( ) s

, ( )

s

1



( ) s        e n ω ω n ( ) s ( ) s ( ) s ( ) / 2

( ) s n n , ( ) 

( ) ( ) s s s   ω n

( ) ( ) s s

If three normals to slip-detachment planes are oriented along coordinate axis of the Cartesian system then ( ) s s j j n   . As for layered media, we rewrite the main system into the suitable form based on the assumption that s j j  is the number of the slip-detachment plane:

( ) l v H                   , , k k , j j 2 ( ) jj jj l j v

, ij j v    , i

, i j j i v v     , , , ( ) ij

s i j j  , i j  ,

s j j  , i j  ,

( ) j        ( ) i , i j v v , j i (    ) ( ij i j

) ( ) H   

,





jj