EVALUATING THE EFFECTIVENESS OF FLINN'S K-VALUE AND LODE'S RATIO USING SYNTHETIC STRAIN DATA

Lode's ratio (ν) and Flinn's k-value are the most commonly used parameters for characterizing the shape of ellipsoids. Both parameters typify the shape of an ellipsoid by utilizing ratios of the lengths of the principal axes. Lode's ratio is defined from -1 to 1, whereas Flinn's k-value is defined from 0 to infinity; the values ν = -1 and k = infinity characterize perfectly prolate ellipsoids; ν = 0 and k = 1 characterize plain strain ellipsoids; and ν = 1 and k = 0 characterize perfectly oblate ellipsoids. For oblate, plain strain, and prolate ellipsoids, there is an exact correlation between k and ν; however, this is not true for any other ellipsoid. In fact, for a given k the variation in ν can be up to 0.54, which is 27% of the total range of possible values of ν. The range for a given k or ν varies depending on the value of ν or k that is held constant, respectively. ν has a larger range for low k-values, i.e. flattening strains, while k has a larger range where ν has low, positive values, i.e. nearly plain strain.

To test whether one of these parameters is more effective at characterizing strain for different strain geometries, we produced synthetic data for a series of ellipsoid shapes. For a given ellipsoid, the axial ratio (R) and the angular orientation (φ) were determined for three mutually perpendicular sections. Using these data as mean-values, we generated a random, normal-distribution dataset for each R and φ and the best-fit ellipsoid was determined from these datasets. Repeating this process (typically 100 times), we were able to calculate the standard deviations of the k and ν-values of the best-fit ellipsoids. ν and k have the same magnitude in general flattening strains, i.e. 0 < k < 1; 0 < ν < 1. There is an approximately linear relationship between standard deviation (σ) versus k and ν. The ν vs. σ relationship demonstrates larger σ as ellipsoids approach perfectly oblate shapes while the k vs. σ relationship demonstrates larger σ as the ellipsoid approaches plain strain. This suggests that Flinn's k-value is a more effective parameter for nearly oblate strains while Lode's ratio is a more effective parameter for general flattening strains that are nearly plain strain.