PSI - Issue 33

K. Mysov et al. / Procedia Structural Integrity 33 (2021) 365–370 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

369

5

and

n  are the coefficients from the decomposition

(

)

1 1 a b y b a + + −  1 1

  

1 

 

2

 

( ) y

( )

2

1 = −

,

(2.15)

y

U y

1 



=

n n

(

)

1 1 a b y b a + + − 1 1

0

n

=

2

( ) , n A x  is the regular part. Derived integral equation (2.13) is transformed into infinite system of algebraic equations which is solved with the help of reduction method. Then by using (2.10) and (1.2) both unknown displacement ’s jump over crack’s tips and unknown rotation angle could be found. 3. Numerical results One of the important mechanical characteristics of crack in cone is stress intensity factor (SIF). It has next form Popov (1992): ( ) ( ) 1 1 2 1 1 0 0 lim , lim , r a r a N a r r a r r       + → − → −     = − = −     (3.1) After substitution of formula for the stress the SIF can be written as ( ) 2 1 1 2 1 0 0 1 1 1 lim 1 ln x n G d N x y dy x y x     + →− − = −   −  = −  − −  −        , (3.2) which after integration by parts takes form ( ) 1 1 0 1 1 1 lim 1 x y G d N x y y x    + →− − =−     − = − −    −        (3.3) After limit calculation and substitution of (2.15) the final form of SIF calculation was obtained ( ) ( ) ( ) 1 1 1 1 0 2 1 n n n G a b b a N U y     + = − + − − =  (3.4) The next parameters were used 7 N = , 10 2 80 10 / / G g cm s =  , 3 7.86 / g cm  = , 10 a cm = , 2 b a = , l b a = − , / 3   = , 5 / 3.19 10 / c G cm s  = =  , 3 0.1 V cm = , 0.786 m V g  = = , ( ) 2 2 arcsin / 2 39.3 J m g cm  = =  , overlays of volume and mass respectively. Having SIF in form (3.4) one can calculate it for different crack positions inside the cone. For example, on Fig. 2 it is visible that the SIF is higher when the crack tip 1 r a = is closer to lower face of the cone where rotation is applied. Another important task, from the point of view of mechanical applications, is finding the cone’s eigen frequencies. With this aim the method proposed in Grinchenko (1981) was applied. There were found several first eigen frequencies of the cone for different crack’s linear position (Table 1). 10 2 2 g cm s  800 10 / M =  , 1 / 2 1/ 2 a a l = + − , 1 / 2 1/ 2 b a l = + + , / 2   = , 2 1.5 l c    = = , here V and m are the

Table 1. First eigen frequencies for different conical crack ’s linear positions inside cone. 1 a 2 , 1,...,4 i i l i c    = = 12 1.609 2.440 3.197 4.002 4.814 14.5 1.609 3.308 3.932 3.982 4.583 17 1.609 3.308 3.922 3.992 4.608