Three-dimensional folding of an embedded viscous layer in pure shear

A thick-plate analysis is carried out to first-order in interface slope for three-dimensional folding of a single linear viscous layer of thickness H and viscosity η embedded in a uniform viscous medium with viscosity η₁, Layer and medium are subject to a basic state of homogeneous pure shear with principal directions of strain-rate x and y lying in the plane of the layer. The solution is explicitly carried out for a fold perturbation in interface shape that is symmetric with respect to the principal strain-rate axes, f cos (lx) cos (my), but it is shown to apply to an arbitrary perturbation in the x,y plane. The growth rate of the perturbation is found to be {A figure is presented} are the principal strain-rates of the basic state. The wavenumber of the most rapidly growing perturbation, λ_d, is the same as that obtained for a cylindrical perturbation (m = 0) in a basic state of plane strain ε{lunate} ̄_yy = 0). For maximum rate of shortening parallel to x, the cylindrical fold form with axis normal to x, m/l = 0, grows most rapidly for any ratio of ε{lunate} ̄_yy/ ε{lunate} ̄_xx < 1. If ε{lunate} ̄_yy = ε{lunate} ̄_xx all fold forms grow at the same rate and in particular, an 'egg-carton' fold form is not preterentially amplified. The velocity field for the perturbing flow consists of a poloidal field, which solely determines the growth of the fold form, and a toroidal field, with non-zero component of vorticity about the axis normal to the layer. The latter is required to satisfy both of the two independent shear traction continuity conditions at interfaces. There is no coupling between the two fields. Three-dimensional fold forms in the Appalachian Plateau province in western Pennsylvania are described.