Microstructural evolution of an overlay coating on a single-crystal nickel-base superalloy
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From Eqs. [11] and [15], we observe that 10/a,oy _ f= 1 +~(E-E0)
[17]
Similarly, we obtain from Eqs. [13] and [16] the relation 6/z07 Ar = -- - AE
e0
[18]
Hence, in addition to predicting the linear correlations of f - 1 vs JF - E0 and of Ar against AE, the present theory dictates also that the slopes of these two straight lines (dependent variables: f - 1, Ar; independent variables: /~ - E0, AE) have a ratio of - 5 / 3 . While one would expect that Eq. [6] and the assumption of weak anisotropy would lead to various linear correlations between plastic and elastic anisotropic parameters, this slope ratio of - 5 / 3 , universal in the sense that it is independent of K, % and k2, is not an obvious consequence. The article of Stickels and Mould ul includes data by which we can determine whether this predicted slope ratio of " 5 / 3 is corroborated by their steel samples. The 35 samples of Stickels and Mould have an f value that runs from 0.66 to 2.73. Since the present theory pertains to weakly anisotropic sheets, we consider only those 24 samples with f -< 1.50. Following the original work, we express E and AE in 106 psi (or 6.895 GPa). For the 24 samples, we find that f = 0.48 /~ - 13.96; the uncertainty in the slope is 0.04, and the correlation coefficient is 0.93. For the Ar vs A E plot, we seek the slope m so that the equation Ar = mAE best fits the data. We obtain m = - 0 . 3 0 --+ 0.02, and the correlation coefficient is -0.95. The predicted slope ratio of - 5 / 3 compares favorably with the empirically determined value of - 1 . 6 --- 0.2. From Eqs. [12] and [14], we observe likewise that the two straight lines which relate f to W400 and Ar to W440, respectively, are predicted to have a slope ratio of -Q~/-0/28. The existence of such "universal" slope ratios in the present modeling reflects purely the effect of material symmetry, namely, that which pertains to a weakly orthotropic sheet texture resulting from the preferred orientation of cubic crystallites. The predicted universal slope ratios could serve as a means for the corroboration or refutation of the present phenomenological theory, which is based on Hill's quadratic yield criterion. It will be interesting to see whether the more refined micromechanical theories in the literature (e.g., that of Davies et al.16]) also predict the existence of such universal slope ratios and, if they do, to compute the predicted values of these ratios. Hill's quadratic yield function has six material parameters. If a material has a yield function that assumes the form given in Eq. [6] or [7], plasticity experiments are required for the determination of only two parameters, namely,/3 -= 3K/z03,and Y02= 3/z0k2. Further information required for the evaluation of Hill's parameters (cf. Eqs. [8] and [9]) is embodied in the texture coefficients W400, W420, and W44o, which can be ascertained by X-ray or ultrasonic techniques. It would be interesting to see whether Eq. [6] could adequately serve as the initial yield criterion for some METALLURGICALAND MATERIALSTRANSACTIONSA
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