X-Ray Rocking Curve Measurements of Dislocation Density and Creep Strain Evolution in Gamma Prime-Strengthened Ni-Base Superalloys

Abstract

A linear relationship between mobile dislocation density and creep strain has generally been assumed in various models of creep in Ni-base superalloys. No stress or temperature dependence is assumed in such relationships. The current study aims to verify this relationship by use of an X-ray rocking curve-based method on DS GTD111™, a directionally solidified (DS) nickel-base turbine blade alloy. A new measurement scheme tailored for such large-grained, DS alloys using simultaneous tilt–twist methodology is outlined. Multiple test sets of DS GTD111™, creep tested under varying boundary conditions, were measured using this technique. The resulting rocking curve full-width-half-maxima (square root of the dislocation density) were found to correlate to creep strain by a simple linear transfer function with no systematic dependence on stress or temperature. As a validation case, application of this transfer function is demonstrated for long-term test data.

Appendix

1.1 Calculation of Error Bars of FWHM

The standard error at 95 pct confidence associated with a sample of size n and standard deviation s, chosen from a finite-sized population N is given by[26]:

$$ \sigma_{\text{FP}} = \frac{s}{\sqrt n }\sqrt {\frac{N - n}{N - 1}} . $$

(AI)

To calculate the standard error of each datapoint in Figure 8 that corresponds to rocking curve measurements sampled from grains of a specific creep test bar, we proceed as follows. In this scenario, n and N refer to the number of grains sampled by the XRD measurement and the total number of grains in the test bar, respectively. Considering cylindrical test bars of 6.25 mm gage diameter D_o and average diameter of columnar grains d_o to be 2 mm, the total number of grains N comprising the cross section of the bar is given by $N = \frac{{\pi D_{\text{o}}^{2} }}{{\pi d_{\text{o}}^{2} }} \sim 10.$ On average, the number of grains spanning a longitudinal section, cut along a diameter of the test bar is $n^{\prime} = \frac{{D_{\text{o}} }}{{d_{\text{o}} }}\sim 3.$ Let us assume further that these n′ grains have orientations that are normally distributed about the ideal [001] direction within ± 15 deg. Now, the coarse RC-ψ scan spans ± 7.5 deg in ω, exactly half of the ± 15 deg range of allowed excursion. Therefore, the actual number of grains n that will be picked up by the measurement is n′ multiplied by the probability of grain orientations lying in the ± 7.5 deg range. This probability can be worked out to be 0.95 for a two-tailed normal distribution by assuming that the probability of finding grain orientations beyond the extremities, viz., + 15 or − 15 deg is very small; i.e., P(X > 15 deg) = P(X < −15 deg) = 0.0001. This calculation yields n to be equal to 3. Accordingly, the fractional coverage $\frac{n}{N}$ is $\frac{3}{10},$ which was used in the calculation of standard error using the formula listed above.