A Mathematical Theory of Finite-Amplitude Rock-Folding
Geological Society of America Bulletin, 79 (1):
47-68 (1968)DOI: 10.1130/0016-7606(1968)79[47:AMTOFR]2.0.CO;2
Abstract
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 5degrees. A proposed extension of the infinitesimal treatment continues the wavelength-selection mechanism of this treatment up to a limb-dip of 15degrees; 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 45degrees 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 65degrees 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.
Description
A Mathematical Theory of Finite-Amplitude Rock-Folding -- CHAPPLE 79 (1): 47 -- GSA Bulletin