PSI - Issue 41

Fabrizio Greco et al. / Procedia Structural Integrity 41 (2022) 576–588 Author name / Structural Integrity Procedia 00 (2019) 000–000

581

6

Fig. 2. A schematic of a two-dimensional homogeneous body affected by a dynamically growing crack

I f     F I F f

0 (no initiation) 0 (initiation)

*

K

(8)

1    

I

f

F

K

Id

P  

0 & 0 (propagation) 0 & 0 (arrest) a a    

f f

* 1      K

(9)

P

f

F

F

K

P





ID

F

Id K and

ID K represents the dynamic crack initiation and crack growth toughnesses of the material, * K is the equivalent Stress Intensity Factors defined according to the maximum hoop stress

where

respectively. Besides,

criterion developed by Erdogan and Sih (Erdogan and Sih (1963)) as follows:

*        2 

*             * sin 2  2 





(10)

* K K K ,

*

3

2

,

cos

3 cos K

K





I

II

I

II

where [ , I II K K are the Dynamic Stress Intensity Factors (DSIFs) and  is the kinking angle. The dynamic crack grow toughness is expressed using the empirical equation defined by Kanninen and Popelar (Kanninen and Popelar (1985)):   1 IA ID m L K K a a V           (11) where, IA K is the crack arrest toughness of the material, L V is the limiting crack velocity in the material, and m is a dimensionless shape factor. 2.3 The interaction integral method (M-integral) The interaction integral method (also referred to as M -integral method) is one of the most used approaches in the framework of fracture mechanics to extract fracture variables at the crack front. In the proposed approach, the M integral method is used to evaluate the Dynamic Stress Intensity factors (DSIFs), which are essential to identify the crack initiation conditions, the direction of propagation, and the advancing velocity of the crack front. The M integral expression derives from applying the J -integral to a superimposed state made up of two admissible fields. The first involves the displacement, stress, and strain fields of the problem under investigation (actual state) for ]