PSI - Issue 5

U. Zerbst et al. / Procedia Structural Integrity 5 (2017) 745–752 U. Zerbst et al./ Structural Integrity Procedia 00 (2017) 000 – 000

747

3

            2 r J K f L E

2



(1)

with the function f(  L r ) being a plasticity correction function based on a cyclic  L r defined by

2 

.

(2)

app

L  

r

 

0

In Eqn. (2), the parameter  app is the applied cyclic stress (referring to the gross cross-section) and  0 is what the authors call a reference yield load for which parametric equations are provided by Madia et al. (2014). The f(  L r ) function is taken from the R6 routine (2009), where it is given there for monotonic loading. The highest analysis option is:

-1 2

  

  

2

E

1

L

 



 f L



(3)

 



ref

r

r

2 E



  

ref

ref

ref

with the stresses  ref , respective  ref , and strains  ref , respective  ref , referring to the stabilized cyclic stress-strain curve. Physically short cracks are characterized by the gradual development of the crack closure effects during crack propagation. Crack closure is usually described by a crack closure parameter U =  K eff /  K with  K eff being K max – K op and  K being K max – K min (K max and K min are the upper and lower K values in the loading cycle and K op is its value at crack opening). Whilst U is independent of the crack size, a, at the long crack stage and can be determined, e.g., by the NASGRO approach (2000) it becomes a function of the crack size at the (physically) short crack propagation stage. An option for determining its gradual decrease with crack growth is provided by the observation that this is mirrored by the so-called cyclic R curve which describes the crack size dependency of the fracture crack propagation threshold  K th (a)     th th,eff SC K a K    

LC U a 1 U 1

.

(4)

th,LC K K     

th,eff

The indices SC and LC stand for “short crack” and “long crack”.  K th,eff is the intrinsic, closure free fatigue crack propagation thres hold which only depends on the crystal lattice of the material, see the overview in Zerbst et al. (2016). Fig. 3 shows the cyclic R curve along with the Kitagawa-Takahashi diagram.

Fig. 3: Kitagawa-Takahashi diagram (left) and cyclic R curve (right) as two comple mentary approaches.

Both approaches are complementary, however, in the light of the cyclic R curve, El Haddad’s approximation of the Kitagawa Takahashi diagram (El Hadda et al., 1979) requires a slight modification in that a term a* has to be added to fulfill the requirement that the minimum threshold,  K th,eff , at the beginning of crack propagation (i.e., at d) is correctly determined (Zerbst & Madia, 2014).