PSI - Issue 54

Jürgen Bär et al. / Procedia Structural Integrity 54 (2024) 188–195 Author name / Structural Integrity Procedia 00 (2019) 000–000

192

5

Figure 4a shows the result of a least square fit of the Johnson equation with y 0 as fitting parameter on all measured data independent of the crack initiation site. The crack depth a was measured on the fracture surface as shown in figure 2. Since the potential values at final fracture can be distorted by plastic deformation, the measured values at failure of the specimens were not considered in the evaluation. These data points are shown in the graphs as open symbols. The fitted Johnson equation shows a poor agreement especially for higher crack depths which reflects in a low regression coefficient of only R 2 =0.72. The fit with the Tiedemann equation in figure 4b shows a much better agreement especially for longer cracks and exhibits a sufficient regression coefficient of R 2 =0.904. Since the notch depths a notch differed only insignificantly, a mean value of the notch depth was used for fitting both functions.

a

b

0.6

0.6

0.362 + 0.0118

a/d = 0.4*(P mean -1)

y 0 = 5.958 R 2 = 0.72

R 2 = 0.904

0.5

0.5

0.4

0.4

P01-0° P02-0° P03-0° P04-0° P05-0° P06-0° P07-15° P08-15° P09-30° P10-30° P11-60°

0.3

0.3

P01-0° P02-0° P03-0° P04-0° P05-0° P06-0° P07-15° P08-15° P09-30° P10-30° P11-60°

0.2

0.2

relative crack depth a/d

relative crack depth a/d

0.1

0.1

0.0

0.0

1.0

1.5

2.0

2.5

3.0

3.5

1.0

1.5

2.0

2.5

3.0

3.5

mean Potential P mean

mean Potential P mean

Fig. 4. Relative crack depths a/d measured on the fracture surface as a function of the relative potential P mean for all investigated crack initiation sites with fitted functions: (a) Johnson equation; (b) Tiedemann equation.

The results show that using a mean potential a crack size measurement in round specimen independent of the crack initiation site is possible using the Tiedemann equation. In the following the geometrical model of Hartweg and Bär (Hartweg und Bär 2019) will be used to show a more suitable way to determine the crack depth in round specimen. 3.3. Geometrical model In the model suggested by Hartweg and Bär the three relative potentials P 1 , P 2 and P 3 are treated as vectors in a cylindrical coordinate system spanning a plane (figure 5a). When a crack is initiated, the three potentials rise to different degrees and the plane, and thus the associated normal vector � , tilts (figure 5b).

a

b

Fig. 5. Illustration of the geometrical model by Hartweg and Bär: (a) without crack; (b) with a crack at potential probe 2.