MilAMinMillion A Minute

Folding and boudinage of viscous layers

One or more stiff viscous layers embedded in a weaker matrix are shortened or extended parallel to the layers. Under shortening, small deflections of the interfaces amplify into folds, fastest at the dominant wavelength predicted by Biot and Fletcher; under extension, power-law layers neck into pinch-and-swell structures and boudins. Everything below runs in your browser when you press Run: the box is remeshed with Triangle at every step, the Stokes equations are solved with 7-node Crouzeix–Raviart elements (with Picard iterations for power-law rheology) and the interfaces are advected with the computed velocities, the same scheme as in the Folder tool this page is based on.

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Perturbation

Growth-rate curve

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What this shows

The model is a rectangular box of viscous fluid with one or more embedded layers of the same fluid, only more viscous by the chosen ratio. Both layer and matrix can be linear viscous or power-law (stress exponent above 1), with the viscosity ratio defined at the background strain rate. The box is deformed horizontally at a constant logarithmic strain rate under free-slip pure shear: the walls move in or out, and material slides along them freely. The layer interfaces start with a small perturbation of the chosen type, the types of the original Folder: a sine, white, red or fractal noise (all box modes with random phases, at equal amplitude, amplitude 1/n or n−(H+½) with the Hurst exponent H), a step, a triangle wave or a bell, each scaled to the chosen amplitude. The matrix above and below the layer stack is set explicitly, since the free-slip walls act back on the layers when they are close.

In each time step the domain is triangulated from the current interface positions, the Stokes equations are solved (with Picard iterations updating the effective viscosity for power-law materials, each step starting from the previous solution, and the strain rate entering the viscosity floored at 1% of the background rate so that hinges stay finite) and the interfaces are moved with the computed velocity field (a second-order Heun step, two solves per step), then redistributed along their arc length when advection has stretched them. Shortening makes interface deflections grow into folds; extension of a power-law layer localizes into pinch-and-swell and boudins, while a linear viscous layer just thins. Of the fields on offer, the perturbing velocity is what remains after the background pure shear is subtracted, the part the layer adds. The amplitude A quoted next to the time slider is, for every interface, half its peak-to-peak deflection, averaged over all interfaces, divided by its initial value A0. The dominant wavelength shown under the perturbation controls comes from the analytical small-amplitude theory for a single layer: Fletcher (1977) for linear folding, Fletcher (1974) for power-law folding, Johnson & Fletcher (1994) for linear thinning and Pollard & Fletcher (2005) for power-law necking; Biot (1961) gives the classic thin-plate estimate. The solver reproduces these rates, which is checked in the build.

This page is a browser version of Folder: Adamuszek, M., Dabrowski, M. & Schmid, D. W. (2016), FOLDER: A numerical tool to simulate the development of structures in layered media, Journal of Structural Geology 84, 85–101, doi:10.1016/j.jsg.2016.01.001. The original MATLAB tool, including power-law rheologies, multilayer stacks and boudinage, is available at github.com/dwschmid/folder (BSD 3-Clause). This page is a re-implementation, not a port: the effective viscosity of a power-law material is one value per element, taken at its centroid and updated by Picard iteration rather than the original's integration-point rheology and Newton iteration; the fractal noise is a spectral synthesis with the given Hurst exponent rather than the original's correlated random field; and the Stokes element writes the viscous term as 2μ ε(u):ε(v), which equals the original's deviatoric form for exactly incompressible fields.

References