Finite amplitude folding of single layers: elastica,bifurcation and structural softening

The amplification of viscous single-layer folds, from infinitesimal amplitudes up to finite amplitudes and large strains, is investigated analytically. Analytical solutions for finite amplitude folding of viscous layers valid for large viscosity contrasts and for post-buckling of elastic columns are shown to be identical. The failure of the classical, exponential amplification solution for folding is quantified using a nonlinear amplification equation similar to the Landau equation. The evolution of fold amplitude–strain for single layers with different initial amplitudes and viscosity contrasts essentially depends on a single parameter rather than three parameters as commonly assumed (strain, initial amplitude and viscosity contrast). This single parameter is constructed by scaling the strain with the crossover strain, which is the specific value of strain at which the linear solutions fail. Scaling the strain with the crossover strain yields a collapse of all amplitude evolution paths for different initial amplitudes and viscosity contrasts onto a single amplification path. Analytical solutions for the evolution of the layer-parallel deviatoric stress within the layer during folding are presented showing a decrease of the layer-parallel deviatoric stress with increasing fold amplitude. All stress–amplitude evolution paths for different initial amplitudes and viscosity contrasts can be collapsed onto a single stress–amplitude evolution path, if the amplitude is scaled by the crossover amplitude. The decrease in stress is proportional to a decrease in effective viscosity of the layer during folding. This decrease in effective viscosity represents structural softening, because the true, Newtonian viscosity of the layer remains constant.