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

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 la yer 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 vis cous whereas fiber stresses are elastic (the 'viscoelastic' mode). For thes e two modes, the new theory provided a nonlinear, ordinary differential equ ation for fold amplification during shortening and an estimate for crossove r amplitude and strain where the linear theory breaks down. A new analytica l relationship for amplitude versus strain was derived for strains much lar ger than the crossover strain. The new relationship agrees well with comple te 2D numerical solutions for up to threefold shortening, whereas the expon ential solution predicted by the linear theory is inaccurate by orders of m agnitude for strains larger than the crossover value. Analysis of the cross over strain and amplitude as a function of the controlling parameters demon strates that the linear theory is only applicable for a small range of ampl itudes 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 amplitud e theory is used to calculate the evolution of the growth rate spectra duri ng progressive folding. The growth rate spectra exhibited splitting of a si ngle maximum (predicted by the linear theory) into two maxima at large stra ins. This bifurcation occurred for both deformation modes. In contrast, the spectra of the cumulative amplification ratio (current over initial amplit ude) maintained a single maximum value throughout. The wavelength selectivi ty is found to decrease at large strains, which helps explain the aperiodic forms of folds commonly observed in nature and the absence of long dominan t wavelengths for high competence contrast folding. Calculation of the cumu lative amplification spectra for different initial amplitude distributions, ranging from white to red noise, showed that the initial noise has a stron g influence on the amplitude spectra for small strains. For larger strains, however, the cumulative amplification spectra were similar despite the str ong difference in the initial noise. (C) 2000 Elsevier Science B.V. All rig hts reserved.