In our work on folds in rock we started looking at single-layer folding, partly because of the simpler modelling. Single-layer folding assumes a situation where a single layer of one material is embedded in a large homogeneous mass of some other material. This leads to the model of a strut on a foundation, and we looked at both an elastic strut on an elastic foundation and an elastic strut on a viscous foundation.
Both papers here tell the same story, but for different audiences.
The model that we use for the elastic strut embedded
in a viscous material is the following: the pair (u,P) satisfies
the differential equation
and the constraint condition, for every time t,
The first paper gives a derivation of this model. The first equation is a force balance, the second specifies the `shortening' of the strut as a function of time. This shortening is assumed to arise as the result of tectonic compression and is generally believed to be the main reason why rock folds in the first place. A typical function g(t) would therefore be slowly increasing in time. The function P(t) can be seen as a Lagrange multiplier associated with the constraint.
One thing we discovered in analysing the behaviour of an elastic strut embedded in a viscous material is that an initial deformation grows in amplitude if g(t) is constant or increasing. The folds develop more and more `wiggles' and the fold wavelength increases.
The equations admit a self-similar scaling, but those self-similar solutions that exist are not fit to be the long-term behaviour for solutions of the initial-value problem. As a result the solutions have an asymptotically self-similar long-term behaviour.
One of the first questions we ran into was `why do these elastic materials fold into such sharp corners'. One answer was given in terms of the overburden pressure in the rock (see the appendix of the paper for the details).
The main subject of this paper arose when we tried modelling the actual rock with a simpler real-world system: stacks of paper. Although eventually the paper folded into something looking like the photograph to the right, it went through a number of interesting intermediate stages before getting there. The first deformation was of a well-known kind, kink-banding, and in this paper we show that this phenomenon can be understood in a simple model in terms of `links', elementary structural elements. The main ingredient here turns out to be the combination of friction between the layers and the intrinsic elasticity of the layers parallel to the layer.
With a slightly different experimental setup we could measure the lateral load experienced by the stack of paper, allowing for a first comparison with experiments ...
... but the better comparison arose when we also adapted the theory to take into account the lateral compressibility of the layers, too.