CALCULATION OF THE BEST-FIT HOMOGENEOUS STRAIN FROM A DISPLACEMENT FIELD USING DISPLACEMENT STRETCH PLOT AND STRAIN PROBE THEORY

Studies of glacial flow, continental deformation, slope creep, and analog experiments use the positions of survey markers at successive intervals in time to determine displacement fields. Relationships between displacements and strain are given by continuum mechanics (Mase 1970; Ramsay 1967; Elliott 1970; Means 1976; Robin 2019).

A "displacement stretch plot", implemented in FORTRAN by Vollmer (in Means 1983) for analog experiments, is quantified here. The plot is created by calculating stretches and final orientations of the line segments defined by combinations of n markers. For i in [1..n], initial positions are [x_i, y_i], and final positions are [x'_i, y'_i], giving c = n!/(2(n-2)!) line segments. For k in [1..c] the initial lengths are given by d_k² = dx_k² + dy_k², final lengths by d'_k² = dx'_k² + dy'_k², stretches S_k = d'_k/d_k, and final angles α'_k = atan(dy'_k/dx'_k). A symmetric polar plot of the stretches defines the estimated strain ellipse. The best-fit ellipse is calculated by minimizing the sum of the squares of the radial residuals (Fitzgibbon et al. 1996; Halır and Flusser 1998), and solved for strain axes and dilation, Δ.

Robin (2019) proposed a "strain probe" method to estimate the deformation tensor U, and displacement gradient tensor, E, from the initial and final positions. For finite strain, U can be decomposed into a spin tensor R and a symmetric deformation tensor. Left decomposition gives the ellipse in a Lagrangian reference frame, the spin tensor R, rotation angle ω, and symmetric tensor U_L. Eigenvectors of U_L give the principal strains. A coefficient of determination, C_D on [0..1], is calculated to evaluate the solution.

Data from an analogue experiment by Schweinberger and Fueten is used to compare results. Displacement stretch plot: A = 1.0914, B = 0.9749, R = 1.1194, ϕ = 28.155°, Δ = 0.0640, C_D = 0.9996. Strain probe: A = 1.0943, B = 0.9680, R = 1.1304, ϕ = 28.623°, Δ = 0.0592, C_D = 0.9909. The methods are implemented in EllipseFit 3.7.0.