Finite amplitude folding: transition from exponential to layer length controlled growth

Abstract

A new finite amplitude theory of folding has been developed by the combined application of analytical, asymptotic and numerical methods. The existing linear folding theory has been improved by considering nonlinear weakening of membrane stresses, which is caused by the stretching of the competent layer during folding. The resulting theory is simple and accurate for finite amplitude folding and is not restricted to infinitesimal amplitudes, as is the classical linear theory of folding. Two folding modes relevant to most natural settings were considered: (i) both membrane and fiber stresses are viscous during folding (the ‘viscous’ mode); (ii) membrane stresses are viscous whereas fiber stresses are elastic (the ‘viscoelastic’ mode). For these two modes, the new theory provided a nonlinear, ordinary differential equation for fold amplification during shortening and an estimate for crossover amplitude and strain where the linear theory breaks down. A new analytical relationship for amplitude versus strain was derived for strains much larger than the crossover strain. The new relationship agrees well with complete 2D numerical solutions for up to threefold shortening, whereas the exponential solution predicted by the linear theory is inaccurate by orders of magnitude for strains larger than the crossover value. Analysis of the crossover strain and amplitude as a function of the controlling parameters demonstrates that the linear theory is only applicable for a small range of amplitudes and strains. This renders unreliable the large strain prediction of wavelength selection based on the linear theory, especially for folding at high competence contrasts. To resolve this problem, the new finite amplitude theory is used to calculate the evolution of the growth rate spectra during progressive folding. The growth rate spectra exhibited splitting of a single maximum (predicted by the linear theory) into two maxima at large strains. This bifurcation occurred for both deformation modes. In contrast, the spectra of the cumulative amplification ratio (current over initial amplitude) maintained a single maximum value throughout. The wavelength selectivity is found to decrease at large strains, which helps explain the aperiodic forms of folds commonly observed in nature and the absence of long dominant wavelengths for high competence contrast folding. Calculation of the cumulative amplification spectra for different initial amplitude distributions, ranging from white to red noise, showed that the initial noise has a strong influence on the amplitude spectra for small strains. For larger strains, however, the cumulative amplification spectra were similar despite the strong difference in the initial noise.