AUTOMATIC CONTOURING OF TWO-DIMENSIONAL FINITE STRAIN DATA ON THE UNIT HYPERBOLOID AND THE USE OF HYPERBOLOIDAL STEREOGRAPHIC, EQUAL-AREA AND OTHER PROJECTIONS FOR STRAIN ANALYSIS

Two-dimensional strain data are plotted on a cartesian R_f-ϕ graph (Dunnett, 1969) or a polar strain graph (Elliott, 1970). A logarithmic scale is related to the deviatoric natural strain, ϵ (Nadai, 1950). Yamaji (2008) showed that a two-dimensional unit hyperboloid H² provides a unifying parameter space. Points on H² are x = (x₀, x₁, x₂)^T, with origin C. If strain is represented by (ρ, ψ) = (ln R, 2ϕ), then an ellipse is x = (cosh ρ, sinh ρ cos ψ, sinh ρ sin ψ)^T. Reynolds (1993) gave equidistant, equal-area, orthographic, gnomic and stereographic azimuthal projections to map H² to the x₁x₂ plane. The equidistant is the Elliott graph, it preserves radial distance so ϵ is undistorted. Wheeler (1984) discussed the orthographic. The equal-area distorts ϵ but preserves area for comparing densities. The stereographic is conformal. Curves of equal distance from C remain circles with strain, so for a symmetrical distribution the centroid of the projected data is the best-fit ellipse. Projecting H² onto a surface whose axis is parallel to x₀ gives a family of cartesian graphs. The equidistant is the R_f-ϕ graph. The centroid in none of these graphs is a good estimator of the best-fit ellipse.

Elliott (1970) hand-contoured strain data on the polar graph to bring out indications of pre-strain fabrics. It is desirable to have a method that is rapid, reproducible, and based on the underlying geometry of the data, rather than the projection. H² provides a measure of distance directly related to strain, d_H = cosh^-1(-a ⸰ b), analogous to a great-circle distance on a sphere. By analogy with methods for spherical orientation data (Diggle and Fisher, 1985; Fisher et al., 1987; Vollmer, 1995), contouring strain data can be done by back-projecting a grid onto H² using inverse functions. The distances from each node to each data point x_k are summed to determine the node value, f_ij, using a weighting function, w_k, with parameter κ , based on the cumulative distribution function for H² (Jensen, 1981). To account for sample size, n, κ is replaced with a normalized parameter: κ_n = κ/n^⅓, by analogy with the spherical case (Fisher et al., 1987). The f_ij values are contoured as percentages of the maximum f_ij value. A computer progam, EllipseFit, that implements these methods is freely available.