Issue 61

F. Ferrian et al., Frattura ed Integrità Strutturale, 61 (2022) 496-509; DOI: 10.3221/IGF-ESIS.61.33

( GE ’) and the material fracture toughness K IC =  ( G c E ’), E ’ being the Young’s modulus under plain strain conditions. The two unknowns are represented by the critical (failure) stress  f , implicitly embedded in the stress field and the SIF functions, and the critical crack increment  c . This latter quantity results a structural parameter, since dependent on both material properties and geometric characteristics. The behavior will be addressed more in details in Section 3.     d 2 2 0           c c y c I IC c K a a K (2) Considering just the former equation of system (2), the Point Method (PM) can be defined [14, 16]. According to this criterion, fracture takes place when the stress equals the tensile strength  c at a critical distance  c = l ch / (2  ), where l ch = ( K IC /  c ) 2 is the well-known Irwin’s length. Thus, according to TCD, the crack advance is a material property. Cohesive Crack Model Let us now consider the CCM implementing a Dugdale type cohesive law (Fig. 1).  

 c

G c

v c

v

Figure 1: Dugdale’s cohesive law.

According to this model, a process zone of length a p is present ahead the crack/notch tip, where the cohesive stress keeps constant and equal to  c : a p increases with the external load  , finally reaching the critical value a pc when  is maximum, i.e.  =  f . To achieve a pc and  f , two different conditions must be considered. The former is a stress requirement: the global SIF K I has to vanish at the fictitious crack tip, such to eliminate the stress singularity. The superposition principle allows to exploit the SIFs due to the external loading K I  and the cohesive stresses K I  c , so that: 0      c I I I K K K (3) The latter is an energy condition: crack nucleates when the crack tip opening displacement (CTOD) v attains its critical value v c = G c /  c . In formulae, thanks again to superposition:      c c v v v v (4) where v  and v  c are the CTODs related, respectively, to  and  c . They can be computed by a straightforward application of Paris’ equation as:

  ,

IP K P a

2

p a

  , 

d

v

K a

a

(5)

I

E

P

'

0

  ,

IP K P a

2

a

d

p

, K a  c I  c

v

a

(6)

 c

E

P

'

0

498

Made with FlippingBook - Online Brochure Maker