PSI - Issue 36

Nataliya Yadzhak et al. / Procedia Structural Integrity 36 (2022) 401–407 Nataliya Yadzhak, Oleksandr Andreykiv, Yuri Lapusta / Structural Integrity Procedia 00 (2021) 000 – 000

405

5

(

) 2 2 IIt   

 

2 + − max IIIt 

2

2

1

R







−

−

dl dN

( / )0 II III

( / ) II III

max

max

max

IIth

IIIth



(8)

.

=



IIt  

− −

( / )

II III fc

IIIt

The initial and final conditions ( ) ( )0 0, 0 II III N l l + = = ;

(

)

( ) , II III N N l N +  =

.

(9)

l

=

(

) + 

(

) + 

II III

II III

added to the previous equation, lead to the mathematical model (8) – (9) for approximate lifetime determination of structural elements, which can be modelled by thick cracked plates under mixed mode II and III loading. As compared to the initially obtained model (7), (9), the model (8) – (9) is easier to use and requires the common material characteristics for mode II and mode III. At the same time, it allows us to effectively describe the crack max IIIt  . The corresponding formulas for their determination have been earlier proposed by the authors based on the critical crack tip opening criterion and representation of the shear problems in the harmonic functions. It has to be noted that these formulas take into account the relative load level of the plate i ip   that is crucial to correctly model the CTOD and further the crack growth rate of small cracks. After substitution of the CTOD formulas for mode II growth kinetics under the condition of mixed mode II and mode III loading. The application of the model envisages the necessity to determine the CTODs max IIt  ,

2 II l   

(3 4 ) −

(10)





IIt

2

8 (1 ) −

1 ( / −

)

  

II  

IIp

IIp

and for mode III

2 l       − − (1 ) 1 ( / III III IIIp

,

(11)





IIIt

2

2

)

IIIp

the model (8) – (9) can be reduced to the following form:

(

)

2

 

 

2 ( ) ( D l +

2

2

2

1

)

R

III D l



−

−



−



dl dN

( / )0 II III

( / ) II III

max

max

II

IIth

IIIth



,

=

(12)

D l D l



− −

( / )

II III fc

II

III

(

)

( ) 0

0, N l =

;

( ) +  , N N l N = II III

,

l

l

=

=

(

)0

(

) + 

(

) + 

II III +

II III

II III

2 II  

2 (1 )       − − 1 ( / III III IIIp

(3 4 ) −

.

,

where

D

D

=

=

II

III

2

2

8 (1 ) −

1 ( / −

)

2

)

  

II  

IIp

IIp

IIIp

Thus, the relation (12) represents the sought model aimed at the mathematical description of small crack propagation and subsequent determination of residual lifetime of thick cracked plates under the simultaneous action of mode II and mode III loading. 4. Model validation In order to verify the correctness and effectiveness of the model, a specific problem is considered. Let us consider a thick plate (Fig. 1) containing a small central crack and is subjected at infinity to the simultaneous action of mode II II  and mode III III  loading. The plate is ma nufactured from steel 65Г (analogue to AISI 1066) that has the