REPRESENTING PROGRESSIVE FABRIC PATHS ON A TRIANGULAR PLOT USING A FABRIC DENSITY INDEX AND CRYSTAL AXES EIGENVECTOR BARYCENTERS

Crystallographic fabrics (CPOs) are presented on contoured Schmidt plots to identify clusters and girdles. Samples with differing strains can be plotted on a triangular plot to compare their relative fabric strengths and symmetries. Given the normalized eigenvalues, ε, of the orientation tensor, Ω, these indexes are defined: P = ε₁ - ε₂, G = 2(ε₂ - ε₃), R = 3ε₃. As P + G + R = 1, these define a triangular plot. End members are: uniform R = 1, girdle G = 1, and cluster P = 1. A measure of fabric strength is the cylindricity, or non-uniformity, index B = P + G = 1 - R. (Vollmer 1989, 1990).

An alternative measure of fabric strength, based on the Bingham test statistic for uniformity, is the fabric intensity index:

I = (15/2) Σ[ε - 1/3]²

which ranges from 0 to 5 (Lisle 1985, Vollmer 1990). The end members are: uniform I = 0, girdle I = 1.25, and cluster I = 5. I is a measure of the "squared distance" from uniformity, and is non-linear. Therefore a fabric density index is proposed:

D = √[(3/2) Σ[ε - 1/3]²]

D is a measure of the "distance" from uniformity, and is linear from R to P, and R to G. End members are: uniform D = 0, girdle D = 0.5, cluster D = 1. The 99% level for a test against uniformity for a sample size of 300 is D = 0.1.

A single axis, as the c-axis of quartz, may be used to characterize a fabric. An example is from mylonitic quartz CPOs collected along a 9.5 km transect across the Main Central Thrust, Himalaya (Hunter et al. 2019). The progressive fabric path forms a loop across the shear zone with D increasing into the shear zone.

The full crystal orientation may be described by three axes, such as for olivine. End members are: uniform, one cluster with two girdles, and three clusters. Three girdles are not possible. Fabric indexes can be defined using the axes eigenvector barycenter:

ε_B = (1/3)(ε_[100] + ε_[010] + ε_[001])

End members are: uniform [P_B, G_B, R_B] = [0, 0, 1], one cluster with two girdles [1/3, 2/3, 0], and three clusters [1, 0, 0]. An example is from experimentally deformed olivine (Hansen et al. 2014). The progressive fabric path forms a curve from near R towards P, along which D increases with shear strain.

An implementation with example data is provided in the free software Orient, available from the author's website: https://vollmerf.github.io