Issue 75

R. Ince et alii, Fracture and Structural Integrity, 75 (20YY) 435-462; DOI: 10.3221/IGF-ESIS.75.30

Figure 2: a) A typical quasi-brittle beam b) P-CMOD curve c) FPZ formation behind the crack

To fully characterize this multiphase behavior, cohesive crack models, which take into account the cohesive nature of the FPZ, simulate quasi-brittle structural members using the finite element method and the boundary element method. On the other hand, equivalent elastic fracture models, which consider only the behavior up to the peak load of the structural member, aim to predict the crack extension at the peak load (  a c ), which is proportional to the FPZ depth, as illustrated in Fig. 2c [2–4]. It is emphasized that none of the above-mentioned fracture models employ a single fracture parameter, such as fracture toughness or fracture energy, contrary to classical LEFM. Among the equivalent elastic fracture models, the two-parameter model (TPM) [2] predicts only the unstable crack growth at peak load in quasi-brittle structures by utilizing two critical quantities corresponding to the critical crack length ( a c = a 0 +  a c ): the fracture toughness ( K s Ic ) and the crack tip opening displacement value ( CTOD c ), which describes the blunting at the crack tip well, as defined above:

a

a c Y c    



s

c



K

(1)

d     

Ic

Nc



a

4

a

 

  

a

a

  

 

Nc c

c

c

0 ,







c 

c 

CTOD

V

M

(2)

c

1

E

d

a

d

 

c

Here,  Nc refers to the strength value computed according to the classical strength of materials theory for unnotched specimens, and it is commonly referred to as nominal strength. E in Eqn. (2) represents Young’s modulus of the material. In Eqns. (1) and (2), the normalized functions Y , V 1, and M are based on crack lengths ( a 0 and a c ) and the specimen depth ( d ). These functions can be computed from crack analysis by employing numerical methods such as the finite element method (FEM) for the specimen considered or may be found in LEFM books for simple specimen types. As an example, the normalized functions employed for beam specimens with span/depth=2.5, which are also used in this study, can be given below:

 3 4 1.83 1.65 4.76 5.3 2.51 3 2 1 2 1                2

  

(3)



Y

0.68

2

3

  V 

0.65 1.88 3.02 2.69      





(4)

1

2









1

2

2

   

   

a

a

a

a       a c

 

     

  

a c

0

0

0 0





, M c a  

c 

   

1.081 1.149 



(5)

1

d

a

a

c

c

c

438