PSI - Issue 54

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

194

7

The Tiedemann equation is modified by replacing (P mean -1) with n (equation 6). ௗ ௔ ൌ ݍ ή݊ ௥ ൅ ௔ ೙೚೟೎೓ ௗ

(6)

Figures 7 a and b show the relative crack depth a/d as a function of the radius coordinate of the normal vector n for all investigated crack initiation sites with the fitted modified Johnson equation (equation 5) and Tiedemann equation (equation 6), respectively. The fit of the Johnson equation (figure 7a) shows a bad correlation coefficient of only R 2 =0.69 and underestimates the measured values for short cracks. For longer cracks the Johnson equation overestimates the optically measured crack depths. In contrast, the fitted Tiedemann function (figure 7b) shows a very good agreement with the crack sizes measured optically on the fracture surface over the entire range of values. This also reflects in a good regression coefficient of R 2 =0.923. Compared to the Johnson equation, the Tiedemann equation can achieve a much stronger curvature, which results in a much better fit.

a

b

0.6

0.6

y 0 = 1.303 R 2 = 0.69

a/d = 0.168*n 0.422 + 0.0118 R 2 = 0.923

0.5

0.5

0.4

0.4

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°

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

0

1 2

3 4

5 6

7 8

9

0

1 2

3 4

5 6

7 8

9

radius coordinate of the normal vector n

radius coordinate of the normal vector n

Fig. 7. Relative Crack depths measured on the fracture surface as a function of the radius coordinate of the normal vector n for all investigated crack initiation sites with fitted functions: (a) Johnson equation; (b) Tiedemann equation.

The investigations clearly show that the multiple potential drop measurement combined with the geometric model of Hartweg and Bär (Hartweg und Bär 2019), can be used to locate the position of the incipient crack on the specimen circumference, and also allows to determine the depth of cracks in round specimens. Using the Tiedemann function, it is possible to calculate the crack depth of short and long fatigue cracks and, accordingly, crack propagation rates on notched round specimens in addition to the determination of the crack initiation lifetime. This allows the investigation of all 3 phases of fatigue life with the same method on a single specimen. 4. Conclusions The investigations have shown that the multiple potential drop measurement in combination with the simple geometrical model by Hartweg and Bär (Hartweg und Bär 2019) is a valuable tool for the investigation of crack initiation and the propagation of short and long cracks on round specimens in fatigue experiments. Form the results the following conclusions can be drawn:  The multiple potential drop measurement allows the determination of the crack size in round specimen independent of the location of the crack initiation site on the circumference of the specimen.  The use of the radius coordinate of the normal vector n gives better results than the mean potential P mean .  By using the radius coordinate of the normal vector n the influence of a heating of the specimen on the crack depth measurement is suppressed.