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
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