Boudinage and two-stage folding of oblique single layers under coaxial plane strain: Layer rotation around the axis of no change (Y)

Highlights

• Oblique single layers exposed to high plane strain show two-phase folding.

• Fold asymmetry decreases and wavelength increases with layer inclination.

• Boudinage of single oblique layers is symmetric if finite plane strain is high.

• Both folded and boudinaged oblique layers show striking thickening.

Abstract

Scaled analogue modelling using non-linear viscous plasticine as rock analogue has been carried out to investigate the influence of varying layer obliquity on the geometry of single-layer folds and boudins under bulk coaxial plane strain. Two deformation series with different viscosity ratio between layer and matrix (m = 18 and 82) were carried out. The initial angle between shortening axis and competent layer (θ_Z(i)) was gradually changed from 0° to 90° by multiples of 11.25°. Shortening at θ_Z(i) < 30° was accommodated by layer thickening and two-stage folding. Low-wavelength F1-folds were refolded by large-wavelength homoaxial F2-folds, both with similar degree of tightness. With increasing layer obliquity, the number of folds and the degree of F2-fold asymmetry decrease, and F1- and F2-folds approximate in size. Although bulk shortening was high (e_Z = −70%), the rotated long limbs of asymmetric folds are free from boudinage. Boudins, however, developed by combined necking and tensile fracturing at θ_Z(i) > 60°. Because of the high finite strain and related layer rotation, these boudins are not asymmetric (as expected) but symmetric. Folds and boudins like those produced in the present study occur in salt rocks and in crystalline basement deformed at deeper structural levels.

Introduction

Buckle folds and boudins are common deformation structures in rheologically stratified rocks, which develop on different spatial and temporal scales at various structural levels and in various geotectonic settings. The basic driving mechanism for buckle folding and boudinage is the difference in effective viscosity - and thus in strain rate - between the competent layer and the incompetent matrix (Ramberg, 1955, 1959). If bulk strain is coaxial, there is a difference between folding and boudinage concerning the attitude of the layer with respect to the principal strain axes. At very low fold limb dip (<15–20°, Chapple, 1968), the layer is oriented subparallel to the principal shortening axis (Z) and the strain rate in the layer should be approximately equal to the strain rate imposed at the boundaries. As the limbs rotate due to buckling of the layer, less shortening occurs inside the layer, compared to the matrix. Increasing fold amplitude is associated with a decrease of the layer-parallel deviatoric stress that is proportional to a decrease in effective viscosity of the layer resulting in structural softening (Schmalholz, 2006).

Structural softening is also related to the formation of pinch-and-swell structures (Schmalholz and Mancktelow, 2016), although the layer does not significantly rotate if bulk strain is coaxial and the initial angle (θ_Z(i)) between the principal shortening direction (Z) and the layer is 90°. The driving mechanism for boudinage is the transfer of stress from the matrix to the layer. If the competent layer is brittle, tensile-fracture boudinage develops by successive ‘mid-point’ fracturing of the layer segments, until the layer is reduced to boudins, all of which are shorter than some critical length, i.e. the length for which the strength of the layer is higher than the tensile stress at the mid-point (Ramberg, 1955; Lloyd et al., 1982; Bai et al., 2000). The formation of drawn boudins is related to viscous necking. With increasing layer-parallel stretch, pinches or cusps are growing along the layer/matrix interface, with their tips pointing into the competent layer (e.g. Ramberg, 1955; Goscombe et al., 2004; Schmalholz et al., 2008; Schmalholz and Mancktelow, 2016; Schmalholz and Duretz, 2017). Numerical simulations for initial random geometrical perturbations show that pinches and individual boudins do not need to develop simultaneously (Schmalholz and Maeder, 2012). Although the deformation geometry of a competent layer is entirely different for layer-parallel shortening and extension, the initial stages of both folding and necking can be mathematically described using the same theory (e.g. Fletcher, 1974; Smith, 1977; Peters et al., 2016).

Given a rheologically stratified rock is deformed by dislocation (power-law) creep, the geometry of folds and/or boudins produced in the deformed layer(s) depends on various parameters (Hudleston and Treagus, 2010, and references therein): (1) thickness of layer(s), (2) shape and distribution of initial perturbations along the layer(s); (3) viscosity ratio between layer(s) and matrix; (4) stress exponent of the matrix and – more important – of the layer(s); (5) single or multilayer; (6) anisotropic viscosity of matrix and/or layer; (7) magnitude of bulk finite strain; (8) bulk strain rate; (9) bonding between matrix and layer(s); (10) bulk finite strain geometry (flattening, plane strain, constriction); (11) orientation of layer(s) with respect to the principal strain axes. Only few of these parameters can be accurately determined from naturally deformed rocks.

The situation is relatively simple for cases where a single competent layer is oriented perpendicular to one of the principal strain axes (X > Y > Z) under bulk coaxial plane strain (Fig. 1). Single fold and single boudin axes will develop in the field of shortening and in the field of elongation, respectively. A large number of experimental and numerical models were produced under such conditions, where the layer is oriented either perpendicular to the long axis, X, or perpendicular to the short axis, Z, of the finite strain ellipsoid. This holds for classical plane-strain folds, with the fold axis parallel to the intermediate Y-axis (e.g. Biot, 1961; Biot et al., 1961; Ramberg, 1961; Hudleston, 1973; Cobbold, 1975; Abbassi and Mancketelow, 1992; Mancktelow, 1999; Schmalholz, 2006), and for plane-strain boudins, with the boudin axis (or neck line) parallel to the Y-axis (e.g. Ramberg, 1955; Neurath and Smith, 1982; Price and Cosgrove, 1990; Kidan and Cosgrove, 1996; Schöpfer and Zulauf, 2002; Zulauf et al., 2003; Treagus and Lan, 2000, 2004; Schmalholz et al., 2008; Abe and Urai, 2012; Marques et al., 2012; Samanta and Deb, 2014; Samanta et al., 2017; Dabrowski and Grasemann, 2019). The shape of boudins varies significantly and can be used to estimate the viscosity ratio among the boudin, inter-boudin and matrix material.

Apart from bulk coaxial plane strain, nearly all the theory and modelling of folds and boudins are based on a layer (or layers) oriented perpendicular to one of the principal strain axes. Physical experiments with oblique layers are difficult because the ends of the layer(s) tend to slide along the boundaries and parts of the layer can be pulled out of the sample. Moreover, oblique loading of the ends introduces additional perturbation components because the planar boundary is no longer parallel to the axial plane of the developing folds (Schmalholz and Mancktelow, 2016).

If the layering is oblique to all three far-field principal strain rates, the directions of maximum shortening rate in the competent layer vary continuously with time and the base strain rate on which a buckling or necking instability would be superimposed is therefore also changing (Flinn, 1962). A schematic drawing of coaxial plane strain of a competent layer in incompetent matrix, with various layer orientation but parallel to the Y-axis, is shown in Fig. 2.

Price (1967) has shown that asymmetric folds are possible under plane strain if the competent layer is inclined to the Z-axis (see also Frehner and Schmalholz, 2006). The rigid layer rotates slower than the surrounding matrix causing an asymmetry to develop. Cobbold et al. (1971) carried out experiments using multilayers consisting of non-linear, lubricated plasticine, which had been deformed coaxially with the shortening direction at 0°, 10°, 20°, and 30° to the layering. In cases where the layering was oblique to the shortening direction, asymmetric folds developed. Watkinson (1976) deformed a regular multilayer of non-linear modelling clay, set in a softer uniform matrix. The shortening direction wat set at 10° to the layering. Also in this case, asymmetric folds developed. It has further been shown by Treagus (1973) that the same dominant wavelength of a cylindrical waveform, as described by the Biot-Ramberg equation, would arise for Newtonian materials under plane strain with the layer oblique to the principal shortening axis (Z), but parallel to the Y-axis. To first order, the initial low-amplitude buckles are symmetric (Treagus, 1973), but become asymmetric during progressive folding. The same results have been obtained by Reches and Johnson (1976), but for elastic materials. Anthony and Wickham (1978) used finite-element modelling to simulate asymmetric folding of a competent, linear-viscous layer inclined up to 20° to the Z-axis under coaxial plane strain.

At higher degree of inclination between the competent layer and the Z-axis (θ_Z(i) > 45°) asymmetric boudinage is expected (Fig. 2). Also in this case, the asymmetry results from the fact that the rigid layer rotates slower than the surrounding matrix. Abe and Urai (2012) showed how boudins may rotate in pure shear deformation. According to their simulation results, the sense of rotation correlates with the shape of the blocks: rhombic blocks with a positive angle between block and surface and layer normal direction rotate clockwise, rhombic blocks with negative angles rotate anticlockwise and trapezoidal blocks with a different sign of angles for both ends do not rotate significantly. Numerical simulation of oblique boudinage was also carried out by Komoróczi (2014). By varying the cohesion of the competent layer, pinch-and-swell structures, drawn and torn boudins were simulated.

Oblique boudinage, however, may also occur in other settings. Mandal and Khan (1991) used theory and experiments to demonstrate that competent layer-segments, separated by oblique fractures in a ductile matrix undergo rotation under layer-normal shortening. Mandal et al. (2000) produced oblique boudins in brittle multilayers inside a linear-viscous matrix under layer-normal shortening. Oblique boudins have also been obtained by numerical modelling focusing on non-coaxial deformation and rotation of boudins in shear zones (Passchier and Druguet, 2002; Dabrowski and Grasemann, 2014). The same holds for physical analogue modelling of asymmetric boudinage carried out by Hanmer (1986) and Goldstein (1988). Asymmetric boudins, however, may also occur under coaxial conditions as has been shown by numerical modelling (Abe and Urai, 2012).

The published results focusing on deformation of oblique layers listed above are concerned either on boudiange or on folding. Experiments and numerical modelling, which consider both oblique layers on the one hand and folding and boudinage on the other hand, are still lacking. In the present paper, the results of a systematic study of the impact of layer obliquity on the deformation geometry in coaxial bulk plane strain are presented. Our studies focus on a single layer, which rotates around the intermediate Y-axis of the strain ellipsoid meaning that the layer shows initially a different inclination with respect to the principal strain axes X and Z, respectively.