Technique
3.7 A Comparison of Results Obtained byDifferent Fits
3.7.1 Overview and Experiment Details
The consequences from choice of fitting terms on final results depends on the stress fields being fitted. Obviously if the crack tip is racing along curved paths under mixed mode loading, the full span of all the terms is necessary to hope to capture the essence of the crack tip stress field. In simpler situations such as a stationary crack loaded in pure mode I, it would be expected that fewer terms should suffice in which case the fitting mathematics should return coefficients near zero for inactive degrees of freedom if used. However, with real fringe patterns from real experiments, it may be of interest to observe what actually happens on a case by case basis.
In this section fitting results are presented from a drop weight tower test of a commercial grade 6Al-4V titanium alloy. A pre-cracked plate with nominal thickness of 0.5 inch is impacted at 9 m/s in three point bend. Lower span is 9 inches. The overall in-plane dimensions of the plate specimen are 10 inches by 4 inches. The crack consists of fatigue crack extending 2millimeters ahead of a 1.25 inch notch created by wire EDM. The crack is centered below the load point to obtain nominal mode I loading (Figure 3.11). The distance between the two diffraction gratings, ∆, is 30 mm.
All fits were performed using full searches, i. e., crack tip position and tangent were searched in turn until both converged. Comparison is made between a fit usingKId,KIId , and higher order terms over data outside the three-dimensional zone, a fit of theKId term only applied to data outside the three-dimensional zone, and theKId term fit only to data inside the three-dimensional zone making use of the conversion factor discussed in section 3.6.3. Interframe time for the image sequence is 5µs. All elapsed times are from camera trigger, or when the specimen is first impacted by the drop weight tower tup.
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❅❅❅❅❅❅❅❅❅❅❅
✁ ❆❆✁
❅❅❅❅❅❅❅❅❅❅❅❅
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Load
EDM Notch
z ✛Fatigue Crack
Figure 3.11: Loading configuration for mode I drop weight tower test.
3.7.2 Comparison of Stress IntensityValues
A plot of KId verses time from the three fits is given in Figure 3.12. This comparison has three noteworthy features. First, at initiation (dashed vertical line) as chosen, all three fits provide values ofKId which are in close agreement.
Second, up to initiation the fits over data inside the three-dimensional zone using the conversion factor agree very well to the full all-term fit. After initiation it diverges, as should be expected considering that the conversion factor as used is valid for stationary cracks only.
Third, theKId fit outside the three-dimensional zone does not agree well with the full fit until initiation, after which agreement is excellent. Since the full fit does not indicate the presence of significant mode II components in the crack tip field (Figure 3.13), this disparity must be due to the contribution of higher order terms. Figure 3.14 shows the fringe pattern image at 90µs, just before initiation, when the disagreement between the full term fit and leading term fit is maximum.
3.7.3 Comparison of Fit Error
The fit errors verses time for the three fitting term/data selections are plotted in Figure 3.15. As expected, the most general fit produces the least fitting error. Often the error for the fit over data
Figure 3.12: KId versus time from three different fitting term / data set combinations.
inside the three-dimensional zone will jump up at initiation, but in this particular experiment it does not.
Error can be influenced greatly by the number of data points fit, even though error is normalized to be per point. An extreme example would be that of a few random data points fit without error by many terms.
3.7.4 Comparison of Crack Tip Locations
Finally crack tip location for each fit is taken to be the location estimated during digitizing refined by the change in crack tip location found during crack tip searching. Crack position versus time for the three fits is given in Figure 3.16. The crack tip data is not as smooth as hoped for from an objective location method. Part of this location noise is due to error introduced in indicating the fixed reference point during digitizing. The more precise the fixed point, the more difficult it is to see due to the optical grating shear and adjacent fringes. A second source of error prior to initiation is that early in the loading there is very little fringe data available for fitting.
Figure 3.13: KIId versus time from all-term fit.
The apparent location of the crack tip can also move forward prior to initiation due to crack tip blunting, which is the forward motion of the crack tip stress fields as yielding occurs at the crack tip.
In any case, the crack tip location data is helpful for determining initiation and sufficient to estimate crack velocities. The crack tip searches/error minimizations are primarily intended to determine fit coefficients objectively.
3.7.5 Comments
The above plots are just a few items for consideration and comparison. Other items of possible interest may include coefficients of higher order terms, corrections of digitizer-indicated crack tip location found by crack tip searches in both directions normal and tangent to the crack plane, as well as changes of crack plane angle. All this data can be useful for checking fringe numbers and other procedural operations, for choosing and justifying fitting terms and search procedures, and finally for understanding the crack mechanics observed. Interpretation is somewhat of an art form
Figure 3.14: CGS fringe pattern at 90µs after impact. (Beam diameter = 50 mm)
requiring understanding of both theory and implementation particulars, such as how the error is defined, how the searches work, and how data is filtered.