PSI - Issue 4

Stefan Kolitsch et al. / Procedia Structural Integrity 4 (2017) 95–105 Stefan Kolitsch/ Structural Integrity Procedia 00 (2017) 000 – 000

102

8

Fig. 4. Schematic plot of the endurable stress range Δ σ depending on the crack length (Kitagawa-Takahashi diagram)

The horizontal axis represents the crack length a and the vertical axis the endurable stress range Δ σ . The stress range is limited by the endurance limit on one hand and by the long crack threshold Δ K th,lc on the other hand. This diagram gives the safe design region for stress range and crack length combinations below these curves, highlighted by the red area in Fig.4. The transition from the endurance limit to the long crack threshold (red line) was first calculated by El-Haddad et al. (1979). In addition, the effective threshold is displayed in Fig. 4 and represents a lower bound for fatigue crack growth. Between the long crack and the effective threshold, the crack growth threshold depends on the build-up of crack closure. The displayed blue transition between the effective and the long crack thresholds was presented by Tabernig et al. (2000). Maierhofer et al. (2014) proposed a modified description of the cyclic resistance curve by a summation of exponential functions, each of them characterizing a different closure mechanism:

      

   

  2 1 i

     l

a

       K K K K (

v

.

(13)

) 1

exp

 



i

th

th,eff

th,lc

th,eff

i

To determine the threshold values and the cyclic resistance curves several fracture mechanics experiments in a four point bending support with a single edge notched bending (SENB) specimen are conducted for different stress ratios. Additionally, not only the endurance limit but also the long crack threshold depends on the stress ratio; the higher the stress ratio, the lower is the threshold. To consider this behavior, a linear dependence of the threshold on (1 - R) is assumed. As the crack tip loading is a function of the total crack length a = a 0 +  a , whereas the build-up of crack closure depends only the crack extension  a , the endurable stress range can then be calculated depending on the initial defect size a 0 , the crack extension  a and the stress ratio R by Eq. 16 (Kolitsch et al. 2016):





      R

   

 K a R ,

Δ





th

 a R a MIN , ,

.

(14)

,

  





0

e





Y

  a a







0

For a simplified illustration of the behaviour described by Eq. 14, the endurable stress range is calculated for a surface crack with a 0 → 0 and a geometry factor Y = 0.8. The resulting Kitagawa-Takahashi diagram is displayed for the different materials in Fig. 5 at stress ratios R = 0.1 and R = 0.7.