PSI - Issue 28

2200 F. Conrad et al. / Procedia Structural Integrity 28 (2020) 2195–2205 F. Conrad, A. Blug et al. / Structural Integrity Procedia 00 (2019) 000–000 The arguments � � are transformed using equation (1). As known from the J-integral (see Roesler et. al. (2016)), displacement fields can be treated similar to electromagnetic fields. In particular, the divergence �⃗ is a coordinate independent measure for crack contour. It can be defined as �⃗ ≡ �� � �� � �� � �� � � � � � � � � (8) As demonstrated below, the rotation �⃗ of the displacement field �⃗ can be used for crack tip localization: �⃗ ≡ � (9) The rotation exhibits maxima and minima near the crack tip because of the direction of the displacement vectors changes. Thus, the ‘gradient’, i.e. the slope within the rotation field perpendicular to crack-growth direction, should be a good candidate to detect crack tip position. 4. Comparison of DIC and uniaxial ACPD crack-depth results Fig. 3 shows full-field DIC results obtained under a maximum load of 25 kN of a crack growth experiment with a CC specimen (Ti6246, see Table 1) with an optical resolution of 4.5 µm/pixel and 80 x 40 ROIs. The evaluation time was about 150 ms per full-field evaluation. On the left, displacement � � � parallel to the direction of force is given in strained coordinates so that the discontinuity of the crack overlaps with the crack in the camera image. The crack length is measured from the end of the starting notch of the specimen coinciding with the left border of the displacement field. At the end of the crack-starter notch, the crack is opened by a width � of 58.6 µm, which appears as vertical discontinuity in the displacement field. Towards the crack tip, this discontinuity decreases continuously to zero. The displacement in x-direction is not given because it is almost zero. Therefore, the divergence �⃗ corresponds to the size of the discontinuity (Fig. 3b). One straight forward option to measure crack length is to measure crack tip position by a simple threshold ∆ε within the divergence. However, the value of the threshold is somewhat arbitrary as smaller thresholds lead to increasing crack lengths. In addition, the value might depend on the load. Therefore, the crack tip position was alternatively measured as the maximum gradient in the rotation rot u�⃗ according to equation (9), indicated by the green arrow in Fig. 3c. The physical motivation behind the rotation is that the displacement vector u�⃗�� �� changes its direction in the ambience of the crack tip. 6

Fig. 3: Full-field analysis superimposed to the camera image of a uniaxial measurement at a CC-sample under load in vertical direction. (a) Displacement in y-direction with crack opening displacement � . (b) Divergence with threshold ∆ for crack tip detection. (c) Rotation of the displacement field. The green arrow marks the gradient whose absolute measure is maximized for crack length measurement in Fig. 4b.