Finite-amplitude folding of an isolated, linearly viscous layer under compression and imbedded in a medium of lower viscosity is analyzed theoretically by using a variational method to derive finite difference equations which are solved on a digital computer. The results depend on a single physical parameter, the ratio of the fold wavelength, L, to the “dominant wavelength” of the infinitesimal-amplitude treatment, Ld. A useful range of physical parameters is covered by the computation of three folds, with L/Ld = 0,1, and 4.6.
Significant differences in fold shape are found among the three folds; folds with higher L/Ld have sharper crests. Folds with L/Ld = 0 and L/Ld = 1 become fan folds at high amplitude. A description of the shape in terms of a harmonic analysis of inclination as a function of arc length along the folded layer makes evident this systematic variation of shape with L/Ld and shows that the fold shape at high amplitude is relatively insensitive to the initial shape of the layer. This method of shape description is proposed as a convenient way of measuring the shape of natural folds.
The infinitesimal-amplitude treatment does not predict fold-shape development satisfactorily beyond a limb-dip of about 5°. A proposed extension of the infinitesimal treatment continues the wavelength-selection mechanism of this treatment up to a limb-dip of 15°; after this stage the wavelength-selection mechanism no longer operates and fold shape is mainly determined by L/Ld and limb-dip.
Strain-rates and finite strains in the medium are calculated for all stages of the L/Ld = 1 and L/Ld = 4.6 folds. At limb-dips greater than 45° the planes of maximum flattening and maximum flattening rate show the characteristic orientation and fanning of axial-plane cleavage. At a limb-dip of about 65° an important change in the style of deformation occurs. The medium ceases to move into the crestal regions of anticlines and starts to be extruded from inside the folds. As a result, the planes of maximum flattening and flattening rate change from an antifanning to a fanning orientation, and the longitudinal stresses in the layer change from compressive to tensile.
Most natural folds have sharper crests than those computed for the dominant-wavelength fold; natural fan folds such as the L/Ld = 0 and the L/Ld = 1 folds that develop into at high amplitude are rare. f hese features indicate that most natural folds may have followed nonlinear theological laws.