PSI - Issue 66

Hendrik Baarssen et al. / Procedia Structural Integrity 66 (2024) 305–312 Author name / Structural Integrity Procedia 00 (2025) 000–000

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where F is the Chow coefficient, S c is the sum of squared residuals of the dataset, S 1 and S 2 are the sums of the squared residuals of the subsets, k p =2 is the number of coefficients in the regression and N 1 = k and N 2 = n-k correspond to the number of data points in the two subsets. The null hypothesis of the test is that both linear regressions are equal. The Chow tests follows the F-distribution with and 1 + 2 − 2 degrees of freedom, which provides the means to determine the presence of a discontinuity statistically. Figure 1 shows a visual representation of the method. (a)

(b)

Chow coefficient

Dependent variable

Independent variable, k

Independent variable, k

Figure 1. a) Schematization of a dataset with a structural break, the black dashed line represents the linear regression over the entire dataset, while the red and blue dashed lines represents the linear regressions over the subsets. b) Chow coefficients for different cut locations, where the highest Chow coefficient shows the most likely location of the structural break

In line with the Linear Elastic Fracture Mechanics (LEFM) approach, it is assumed that specimens deform linear elastically until fracture. Under this assumption, the deformation in the tensile direction, with the rigid body motion removed, can be defined as a continuous function. Once a crack forms through the searched path identified by A j , the function of the dataset is no longer continuous. This discontinuity can then be located using the Chow test. By applying the Chow test and finding the highest Chow coefficient in the area of interest, the most likely position of the crack on the path can be determined. A threshold value shall be determined for the Chow coefficient to determine whether or not the values of displacement extracted along the searched path contain a discontinuity, indicating that the searched path is crossing a crack. This threshold is based on the measurements of the Chow coefficient before the crack is initiated. In particular, an empirical cumulative distribution function of the Chow coefficients is evaluated along multiple A j , before the crack appears. A threshold is established based on the probability of occurrence corresponding to µ+3σ . Once the measurements consistently, i.e. for the current and successive images, exceed the threshold value, the path is considered discontinuous and therefore cracked. 2.2. Experimental validation The method is validated in an experimental program where a (R260) steel C(T)-specimen is cyclically loaded until failure. The specimen is prepared according to the specifications of ASTM 647 (ASTM International, 2000) with an effective width of 90mm and thickness of 15mm, the dimensions are visualized in Figure 2a. The specimens are monotonically cyclically loaded to P max =10 kN with a load ratio of 0.1 at 10 Hz in a hydraulic tensile bench, as shown in Figure 2b. Based on the CMOD measurement of a clip-on gauge mounted on knife edges, a custom script estimates the length of the fatigue crack and stops the cyclic loading after a millimeter crack growth. Then the specimen is monotonically loaded to the maximum load and the measurements are taken. This process is repeated until failure.