Shear Zone Kinematics: Displacement and Vorticity Numbers

Recently I have been making calculations to estimate the displacement along Shear Zone 2 (SZ2). This requires integrating the shear strain across the width of the zone and accounting for the vorticity number for a particular sample, or more generally, the average vorticity number for the shear zone. Shear strain is the ratio of length change due to deformation compared to the original length from applied shear stress (a force over an area causing slippage/deformation that is parallel to the imposed force). It is important to realize that stress causes strain and not the other way around. This integral includes strain measurements.

As I mentioned, the bounds of integration are the width of the shear zone, which is this case, is 0 to 300 cm (3 m). Shear zones are incredibly variable in the field, and thus there is no general function that can actually be integrated in the traditional, exact, mathematical sense. The best that can be done is to plot the shear strain against width of the zone and calculate the area under the resulting curve. This is as accurate as can be accomplished and is thus what I did in order to calculate the displacement. I used the area I calculated (essentially via a Riemann sum) and accounted for different vorticity numbers from 0 to 1 in intervals of 0.1.

Vorticity numbers, as a reminder, signify the rotational component of deformation. A vorticity number of 1 represents simple shear, which is complete and only lateral displacement (picture a force pushing the top of a deck of cards and the consequential shape that deck makes). A vorticity number of 0 representing pure shear, which is complete flattening (the force is applied directly normal to the object). A vorticity number in between 0 to 1 represents general shear and has both pure and simple shear components.

Thus, if the shear zone experienced solely simple shear, the displacement would be the maximum it could be, which is the total area under the shear strain vs. width curve. If the zone experiences only pure shear, it would have no displacement (no lateral components of shear stress were applied). If the shear zone experienced general shearing then there would be less displacement than with simple shear but more than pure shear, and this depends on the degree of general shear (accounted for in the calculations by the vorticity number < 1).

As of now, I am in the process of calculating the exact vorticity numbers for my samples in order to average them across my shear zone. Until I complete these calculations, I made various estimates of the displacement using a variety of vorticity numbers. The maximum displacement estimate is ~54 m (simple shear, vorticity number of 1), while the lowest is 0 m (pure shear, vorticity number of 0). This calculation will of course become more accurate after I apply my calculated, specific vorticity numbers.