PSI - Issue 66

Domenico Ammendolea et al. / Procedia Structural Integrity 66 (2024) 396–405 Author name / Structural Integrity Procedia 00 (2025) 000–000

399

4

where  is a parameter that controls the regularization model’s behavior. In the proposed theoretical formulation (Wu, 2017), 2   ,   c 2 c c 2       , and 0 c   . According to PFM, the governing equation in weak form for the crack propagation problem is the following:

2

G

c G l

  c w H d   ' c

     







(4)

0

d

d       





0

c

c

c

c

c l

c







0 0

0

where c G is the fracture energy of the material, while H and   c w  represent, respectively, the history variable and the energetic degradation function. H and   c w  are evaluated through the following expressions:

2

ud

1 

2

f

1 2

1 2









0 

0 

t

(5)

max    , onset

,

max history

H

with

and      

onset   

0,max

0,max

E

E

p

(1 )  

  c 

  c

2

3

c

(6)

w

with Q a aa aaa     



c 

c 

1

1 2

1 2 3

c

p

(1 ) 

( ) c 

Q

 

c

In Eq. (5) t f is the tensile strength, whereas the formula to evaluate 0   represent the Rankine Criterion. In Eq. (6) the parameter 1 a depends on internal length 0 l , while 2 a and 3 a are independent from 0 l , but depend on softening curves. In the technical literature several softening laws are reported, i.e. linear, exponential, hyperbolic and Cornelissen’s ones. See (Wu, 2017) for further clarification.

3. Numerical implementation 3.1. Adaptive mesh technique

Fig. 2 depicts a schematic representation of the adaptive meshing strategy employed in the proposed model. Such a technique refines the mesh exclusively in regions where damage nucleation and propagation are expected to occur.

Fig. 2. Schematic representation of the adaptive meshing strategy.