PSI - Issue 59

5

Ivan Shatskyi et al. / Procedia Structural Integrity 59 (2024) 246–252 Shatskyi et al. / Structural Integrity Procedia 00 (0000) 000 – 000

250



1

 0

.

 G( ) lim z

( ,0, )[ ]( r z U

, ) r z dr







y

y



0





Having calculated this integral taking into account the formulas (5), we get

h 3

z 2

z



  

  

  

  

z G( )

K

K K

K









.

N

M N

M

h E 2

2

(2 )

h

According to the energy criterion, steady crack progression at the most dangerous points will begin if

2

   

   

|

|

K

K



  

  

2 N

M

M

(9)

.

( ) max G z

 (3 )

3

2

K

K



 









N



2 h E

h

h

(2 )

z

Finally, the development of a crack as a through object can be estimated by the average value of the energy flow to its tip (Wynn and Smith (1969)):

 h

2

   

   

1

K



  

  

2 N

M

(10)

.

( ) G z dz

2

G

K















2 h E

2

h

h

(2 )

h



Let us compare the above criteria. Obviously, all expressions (8), (9), (10) coincide for the case of a plane stress state ( 0 0,   N M K K ). Since 3   , the local force criterion (8) gives a more conservative estimate of the lower limit load than the local energy criterion (9). Since the maximum value of the energy flux is always greater than its average value, the lower limit load calculated by the local criterion (9) will always be less than the upper destructive load found by the integral criterion (10). 3.2. Limit state analysis Using the analytical results for the coefficients of force intensity and moments (6), (7), we find the characteristic values of the limit loads of uniform bending for a plate with a rectilinear crack with and without taking into account the contact of the edges. According to criterion (8)

3 1

3 1





0

0

0

0

|

|

(0,500...0,533)

|

| m m   

0,333

m

m

m

m





;

(11)

,

*





according to criterion (9 Ошибка! Источник ссылки не найден. )

2 (3 ) 1   

3 1

0

0

0

0

|

|

(0,707...0,689)

m

m

m

;

(12)





,

|

|  

(0,577...0,509)

m

m

m



*

 



and according to criterion (10 Ошибка! Источник ссылки не найден. )

1





1

0

0

0

0

|

|

(1,414...1,333)

m

m

m

|

| m m    

(1,000...0,882)

(13)





m

,

.

*

