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Journal of Structural Geology 53 (2013) 15e26

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Journal of Structural Geology

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Sheath fold morphology in simple shear

Jacqueline E. Reber*, Marcin Dabrowski, Olivier Galland, Daniel W. SchmidPhysics of Geological Processes, University of Oslo, 0316 Oslo, Norway

a r t i c l e i n f o

Article history:Received 17 December 2012Received in revised form19 April 2013Accepted 6 May 2013Available online 14 May 2013

Keywords:Sheath foldSlip surfaceEye-patternSimple shearShear sense indicatorBulk strain indicator

* Corresponding author. Present address: UTIG, UniTel.: þ1 512 471 0376.

E-mail address: reber@ig.utexas.edu (J.E. Reber).

0191-8141/$ e see front matter � 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.jsg.2013.05.003

a b s t r a c t

Sheath folds are highly non-cylindrical structures often associated with shear zones. We investigate theformation of sheath folds around a weak inclusion acting as a slip surface in simple shear by means of ananalytical model. We present results for different slip surface orientations and shapes. Cross-sectionsperpendicular to the shear direction through the sheath fold display closed contours, so called eye-structures. The aspect ratio of the outermost closed contour is strongly dependent on the initial slipsurface configuration. The center of the eye-structure is subject to change in height with respect to theupper edge of the outermost closed contour for different cross-sections perpendicular to the shear di-rection. This results in a large variability in layer thickness across the sheath fold length, questioning theusefulness of eye-structures as shear sense indicators. The location of the center of the eye structure islargely invariant to the initial configurations of the slip surface as well as to strain. The values of theaspect ratios of the closed contours within the eye-pattern are dependent on the strain and the cross-section location. The ratio (R0) of the aspect ratios of the outermost closed contour (Ryz) and theinnermost closed contour (Ry0z0) shows values above and below 1. R0 shows dependence on the slipsurface shape and orientation but not on the number of involved contours. Using R0 measurements todeduce the bulk strain type may be erroneous.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Sheath folds are strongly non-cylindrical structures (e.g., Ramsay,1979; Skjernaa, 1989), which are often associated with shear zones(e.g., Carreras et al., 1977). According to Ramsay and Huber (1987),non-cylindrical folds classify as sheath foldswhen the opening angleof their cone is < 90�. The study of sheath folds goes back severaldecades (e.g., Quirke and Lacy, 1941). They are of interest to fieldgeologists as they are considered to give information on strainmagnitude (e.g., Minnigh, 1979), bulk strain type (Alsop andHoldsworth, 2006) and shear direction (Fossen and Rykkelid, 1990).

Most of the studies on sheath folds are based on field observa-tions (e.g., Alsop and Holdsworth, 2006; Alsop et al., 2007;Henderson, 1981; Skjernaa, 1989; Srivastava, 2011). However, theirformation mechanisms have been studied with laboratory (e.g.,Cobbold and Quinquis, 1980; Rosas et al., 2002; Marques et al.,2008; Reber et al., 2013), numerical (e.g., Mandal et al., 2009;Vollmer, 1988), and analytical (Reber et al., 2012) models. Severaldifferent sheath fold formation mechanisms have been suggested.They can either form in simple shear (e.g., Carreras et al., 1977;

versity of Texas, Austin, USA.

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Fossen and Rykkelid, 1990; Minnigh, 1979) or through constrictionand flattening (e.g., Ez, 2000; Mandal et al., 2009; Nicolas andBoudier, 1975). In case of flattening or constriction, a pre-existingcurvilinear fold is deformed and further amplified, which leads tothe formation of a sheath fold. In a simple shear dominated envi-ronment, sheath folds can either form from pre-existing structures,such as buckle folds, which get passively amplified during defor-mation (e.g., Cobbold and Quinquis, 1980), or through perturbationof the simple shear flow by either a rigid inclusion, such as a boudin(Marques and Cobbold, 1995; Marques et al., 2008; Rosas et al.,2002), a weak inclusion acting as slip surface (Exner andDabrowski, 2010; Reber et al., 2012, 2013) or a rigid corrugatedbasement (Cobbold and Quinquis, 1980).

Even though sheath folds are three-dimensional structures, theyare commonly observed in the field on two-dimensional cross-sections. Cross-sections perpendicular to the elongation directionof the cone exhibit strongly deformed layers and closed contours,which can be described as “eyes”. In many cases, field observationsof sheath folds are restricted to one cross-section per fold.

Alsop and Holdsworth (2006) developed a morphological clas-sification of the eye-patterns based on the aspect ratio of theoutermost (Ryz ¼ y/z) and innermost (Ry0z0 ¼ y0/z0) closed contour(Fig. 1a). By taking the ratio of these two measurements, theyintroduce R0 ¼ Ryz/Ry0z0. In cases where the innermost contour ismore flattened compared to the outermost (R0 > 1), they termed the

Fig. 1. a) Sketch of sheath fold. u: opening angle of the cone, y and z: width and height of the outermost closed contour, y0 and z0: width and height of the innermost closed contour.b) Eye-pattern classification after Alsop and Holdsworth (2006).

J.E. Reber et al. / Journal of Structural Geology 53 (2013) 15e2616

eye-pattern “cat’s-eye-fold” (Fig. 1b). In cases where the innermostclosed contour is less flattened than the outermost contour (R0 > 1),Alsop and Holdsworth (2006) termed the eye-pattern “bull’s-eye-fold”. They also measured the layer thickness (t) along the y- and z-axes as an additional parameter supporting their classification.Based on an extensive empirical study they related the observedeye-patterns to bulk strain types, where they linked cat’s-eye-foldsto simple or general shear and bull’s-eye-folds to pure shear.

Fossen and Rykkelid (1990) used sheath folds to determine theshear direction in a shear zone. They observed that if the sheathfolds show limbs of different thicknesses and the closing directionof the fold is known, then the shear sense can be determined. Forfolds with a thinned inverted limb and a closing direction towardsthe observer, they obtain a “top-towards-observer” sense of shear.

Even though sheath folds are used as bulk strain and shear senseindicator, little is known about the relation between the amount ofstrain, the bulk strain and the resulting shape of the fold for differentformation mechanisms. The existing classifications are based onempirical studies and have not been tested or validated with a me-chanical model. Here, we investigate the effect of the strain and slipsurface configurations on the shape of sheath folds in simple shear.As cause for the fold formation we use a weak inclusion acting as aslip surface and study the fold development by means of ananalyticalmodel.We analyze systematically the impact of the strain,the cross-section location, and the initial slip surface shape andorientation on the resulting fold patterns in cross-sections paralleland perpendicular to the shear direction. In the first part of thisstudy, we investigate the overall morphology of sheath folds whilewe concentrate in the second part on the internal eye-structure ofthe folds.We examine the effect of the contour resolution on the eyepatterns. We apply the classifications of Alsop and Holdsworth(2006) and Fossen and Rykkelid (1990) on the calculated sheathfolds and test their applicability and robustness for different initialconditions. Note that we term all structures sheath folds that showeye-patterns in cross-sections perpendicular to the shear directioneven though theyare in some cases not sheath folds according to thedefinition of Ramsay and Huber (1987) but early sheath fold-likestructures with an opening angle > 90�.

Fig. 2. Model setup for the analytical model. a and b describe the main axes of theelliptical weak inclusion. q is the angle of the initial inclusion orientation. The graylayer illustrates a passive marker layer.

2. Method

2.1. Analytical model

We use a three-dimensional, analytical model to simulatesheath folds in simple shear. The model is based on an adapted

external Eshelby solution (Eshelby, 1959) for incompressible linearviscous materials in the limit of an elliptical and inviscid inclusion.The inclusion is a two-dimensional object, which acts as a slipsurface, embedded in a three-dimensional homogenous matrix.During the deformation it behaves passively, i.e. it deforms andstretches, depending on its orientation with respect to the shearplane. The sheath folds form at both tips of the stretching slipsurface. We vary the initial aspect ratio of the inclusion (a/b), wherea and b denote the length of the main axes of the ellipse, the initialangle (q) with respect to the shear plane (Fig. 2), and the amount ofshear strain (g). The model is positioned in a Cartesian coordinatesystemwhere shear takes place in x-direction in the x-y plane. Thesame model was used by Reber et al. (2012), who investigated thepossibility of a slip surface as sheath fold initiator. A detailed modeldescription can be found in Exner and Dabrowski (2010).

We analyze the sheath folds with cross-sections that are eitherparallel to the shear direction in the x-z plane or perpendicular tothe shear direction in the y-z plane (Fig. 1a). We concentrate ouranalysis on the areas where closed contours (layer group 3, Reberet al., 2012) are visible. Contours outside the so-called “eye-struc-ture” are strongly deformed (layer group 2, Reber et al., 2012), butare not considered in this study. We visualize only half of thestructure in the y-direction, as it is mirror symmetric (green areaFig. 3a).

All length scales in the models are normalized by the verticalaxis length (a) of the inclusion for the case where q ¼ 90� and thenkept the same for all other cases (q ¼ 0�, 135�). The dimensionlesslocations of the cross-sections along the x-axis are given from thecenter of the inclusion (center of the model).

Fig. 3. a) Sheath fold cross-section perpendicular to shear direction, g ¼ 10, cross-section location x ¼ 10, q ¼ 90� , and a/b ¼ 2.67. The shear direction was bottom towards theobserver. Green area shows where closed contours, building the eye-structure, can be found. b) Close up of the green area illustrated with 2, 3, 5, and 10 contours. c) Definition of d1and h. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 1Names and parameters of the different models.

a/b ¼ 0.375 a/b ¼ 1 a/b ¼ 2.67

q ¼ 0� H_0.375 H_1 H_2.67q ¼ 90� V_0.375 V_1 V_2.67q ¼ 135� O_0.375 O_1 O_2.67

J.E. Reber et al. / Journal of Structural Geology 53 (2013) 15e26 17

2.2. Visualization

Tomake the effect of the deformation visible in the homogenousmatrix, we track passive marker surfaces during the deformation(Fig. 3a). For each configuration of interest (q, a/b, g, section loca-tion), we use a backwards-in-time integrationmethod to obtain theinitial z-coordinate for each grid point in a given y-z section. Theinitial z-coordinates are used to correlate the grid points to x-ymarker planes introduced prior to the deformation. In comparisonto approaches where multiple marker surfaces are discretized,tracked forward in time, and finally intersected, the backwards-in-time integration is superior because it provides uniformly spaceddatasets in the deformed state, facilitating the shape analysis.

To find the outermost closed contour bounding the eye-structure (green area in Fig. 3a) we sample a vertical line in thez-y plane (y¼ 0, mirror plane) with 5000 points per unit length. Fora chosen initial configuration (q, a/b) and a value of strain (g) wedetermine contours forming closed eye-structures. For this purposewe place a contour line through each of the sampling points.Whenever both the start and end points of the contour have a y-value of 0 themarker line forms a closed contour and lies within thegreen area in Fig. 3a. By locating all closed contours it is easy todetermine the outermost. We use the center of the innermostclosed contour as an approximation for the center of the eye-structure, and define the ratio d0 to locate the center of the eyestructure with respect to the outermost closed contour:

d0 ¼ d1h

(1)

where d1 is the distance from the top of the outermost closedcontour to the center of the eye-structure and h is the height of theoutermost closed contour (Fig. 3c). When d0 <0.5, the center of theeye-pattern is in the upper half of the outermost closed contour.Values of d0 >0.5 represent eye-structures with a center in thelower half of the outermost contour area.

To analyze the internal structure of the sheath fold weconcentrate on the area inside the outermost closed contour (greenarea Fig. 3a). Again, using the backwards-in-time integration to

obtain the initial z-coordinates, we can choose the number anddistribution of the contours building the eye-structure. We decidedto have an equally spaced layering before the deformation. After thedeformation, the contours are not equally spaced within an eye-structure. Fig. 3b shows the eye-structure visualized with 2, 3, 5,and 10 contours. Note that these four figures are different visuali-zations of the same cross-section of one sheath fold.

3. Results

We perform a series of simulations where we test the effect ofusing three different values of the initial orientation angle (q ¼ 0�,90�, and 135�) and the aspect ratio (a/b ¼ 0.375, 1, and 2.67) of theslip surface (Table 1). The values for the final shear strain are be-tween g ¼ 0.5 and 15 (Table 2).

In cross-sections parallel to the shear direction cut through thecenter of the slip surface in y-direction (Fig. 4), folds originate fromthe tip of the slip surface. All simulations lead to the formation offolds, however, they vary in shape and size. Fig. 4 shows cross-sections for a constant final strain g ¼ 10.

For q ¼ 0�, the folds are small and do not propagate far into thematrix (Fig. 4a, d, g). The different values of a/b have only a minimaleffect on the shape of the resulting structures. Because the slipsurface is initially horizontal it keeps its initial angle and shapeduring the deformation.

For q ¼ 90� (Fig. 4b, e, h), the value of a/b has a dominant effecton the fold shape. For a slip surface elongated in the z-direction (a/b ¼ 0.375), the deflection of the passive layers is so small that it canonly be detected with a high contour resolution (inset is a close upand has 7 timesmore contours than the full section). For a/b¼ 1, thefolds can also only be resolved with a high contour density. Note

Table 2Variables and symbols used in this study.

AR Aspect ratio of the closed contourRy0z0 Aspect ratio of the innermost closed contourRyz Aspect ratio of the outermost closed contourR0 Ryz/Ry0z0d0 Center of the eye-structurel Extent of area where sheath folds occurreq Equivalent radiusxs Scaled cross-section locationN Number of contours in the eye-structureg Straina/b Aspect ratio of the slip surfaceq Initial orientation of the slip surface

Fig. 5. Extent of the area where eye-patterns are visible in cross-section perpendicularto shear direction plotted against strain for the different models.

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that the insets in both figures (Fig. 4b and e) show the samestructure as the corresponding full section, but a larger number ofcontours was used for the visualization (see Fig. 3b). For a/b ¼ 2.67,well-developed folds can be observed. The slip surface rotated from90� to 3.65� with respect to the shear plane (see black lines rep-resenting orientation and length of slip surfaces at g ¼ 10).

For q ¼ 135�, the fold resulting from a slip surface with a/b¼ 0.375 is not prominent (Fig. 4c). The inset of Fig. 4c shows againthe same structure as the full section but resolved with morecontours. For a/b ¼ 1 (Fig. 4f) and a/b ¼ 2.67 (Fig. 4i), well-developed folds can be observed. The slip surface, which had aninitial orientation of 135�, shows an angle of 6.8� with respect to theshear plane after a shear deformation of g ¼ 10 (Fig. 4).

3.1. Overall sheath fold shape

The following analysis is based on cross-sections perpendicularto the shear direction in the y-z plane.We investigate the structureson sections between the center of the inclusion (0) and a normal-ized distance of 20 in the x-direction. Note that the outermostclosed contours in the different cross-sections do not necessarilybelong to the same folded surface (compare to Reber et al., 2012) asthe entire sheath fold structure is built of several stacked cones. Theextent of the area where closed contours are visible is considerablylarger than the length of one single sheath fold defined by one layer(Fig. 4). We do not investigate the behavior of one single folded

Fig. 4. xz-cross-sections parallel to shear direction for different q and a/b values. The insetscontours are purely passive. The black line at the bottom of each column visualizes the orieorientation and a/2 of the slip surface. Note that all the length scales are normalized with a othis figure legend, the reader is referred to the web version of this article.)

interface but of the entire area where closed contours are visible incross-sections perpendicular to the shear direction.

3.1.1. Extent of folded areaIn all models, we monitor the change in total extent of the area

where overturned folds (in x-z cross-sections) and eye-patterns (iny-z cross-sections) can be seen with increasing strain (Fig. 5). Wemeasured the extent l defined by

l ¼ xmax � xmin (2)

where xmin is the location of the first and xmax is the location of thelast cross-section showing closed contours. The steps in the curvesare due to the coarse sampling of the cross-sections. All curvesshow an approximately linear relation between l and g. Whenq ¼ 0�, the sheath folds evolving from the slip surface are small andshow only little growth with increasing strain. The first discernibleeye-pattern appears at a strain of g ¼ 3 to 3.5. However, theappearance of the first eye-pattern is not only dependent on thestrain magnitude but also on the layer resolution. For both q ¼ 90�

and q ¼ 135� l depends on a/b. Independent of q, we can observe

are close-ups of part of the full structure visualized with more contours. Note that thentation and half of the length of the slip surface at g ¼ 10. The red lines give the initialf the inclusion for models with q ¼ 90� . (For interpretation of the references to color in

J.E. Reber et al. / Journal of Structural Geology 53 (2013) 15e26 19

that l is larger for a/b ¼ 2.67 than for a/b ¼ 1 or a/b ¼ 0.375. As weconsider only cross-sections up to a distance of 20 we limited themaximum observable length to approximately 12, which is anunderestimate of the real extent of the folded area.

3.1.2. Aspect ratio e outermost closed contourEye-pattern can be characterized by two features that are in-

dependent of the number and distribution of contours involved inthe structure: the aspect ratio of the outermost closed contour (Ryz)and the position of the center of the eye-structure. The accuracy oftheir location, however, is dependent on the number of contours. Todetermine the location of the outermost closed contour and thecenter of the eye pattern as precisely as possible we used a highcontour resolution (see 2.2. visualization).

Fig. 6. Aspect ratio measurements of the outermost closed contour (Ryz) plotted for differenRyz measurements where the color indicated the value. (For interpretation of the references

Fig. 6 shows the aspect ratio of the outermost closed contour fordifferent q and a/b values. The color code gives the value of Ryz fordifferent strains and cross-section locations. Vertical columns ofcolored squares show Ryz values for one strain magnitude repre-senting different cross-sections along the x-axis. For q ¼ 0�, valuesof Ryz increase with an increase of a/b (Fig. 6a, b, c). For a fixed g, Ryzgenerally decreases from cross-sections close to the tip of the slipsurface towards the apex of the folded area. As the slip surface doesnot rotate or stretch for this initial condition all data points plot atlow x values. The extent of the area where sheath folds occur isshort and spans amaximumof 4 cross-sections. Ryz values of sheathfolds originating from a slip surface with q ¼ 90� are shown inFig. 6d, e, and f. Where the sheath folds are well developed (Fig. 6eand f), Ryz decreases systematically from the slip surface tip to the

t cross-section locations (�) and strains (g) for each model. The colored dots representto color in this figure legend, the reader is referred to the web version of this article.)

J.E. Reber et al. / Journal of Structural Geology 53 (2013) 15e2620

apex of the folded area. The sheath folds forming from a slip surfacewith a/b ¼ 0.375 are rather small and not well developed. In somecross-sections close to the slip surface, Ryz differs significantly fromthe values of the neighboring sections. This is due to the closeproximity of the cross-section to the slip surface, which influencesRyz. Fig. 6g, h, and i show Ryz values for the sheath folds withq ¼ 135�. The triangular area below the measurements where noclosed contours are visible (models with q ¼ 90� and 135�) gives aminimum estimate of the length of the slip surface at each strain.Sheath folds that developed from slip surfaces which rotated dur-ing the deformation (q ¼ 90�, 135�) show overall smaller values ofRyz (note the difference in color bar) than sheath folds forming fromslip surfaces with q ¼ 0�. Ryz also shows less variation along the x-axis.

3.1.3. Center of the eye-patternThe center of the eye-structure is not at a constant positionwith

respect to the outermost closed contour throughout the entireextent of the folded area (Fig. 7, red arrows). The center is located inthe lower half (d0 > 0.5) of the areamarked by the outermost closedcontour (gray area, Fig. 7) in cross-sections close to the slip surface.In contrast, in cross-sections further away from the slip surface thecenter is located closer to the upper edge of the outermost closedcontour (d0 < 0.5). The value of d0 depends therefore on the cross-section location. Note that the outermost closed contours in thedifferent cross-sections do not belong to the same folded interface.

To investigate the behavior of d0 with increasing strain, we plotd0 versus the cross-section location for the model V_2.67 (Fig. 8).While in Fig. 8a the data points are shown, the values of the x-axisare normalized in Fig. 8b, such that the x-axis is scaled as follows,

xs ¼ ðx� xminÞ=l (3)

Thedata of Fig. 8a collapses toone curve. Curves for lower strains,however, show a slightly different shape. Fewer data points wereavailable for these curves as the extent of the folded areas wererelatively short at small strains. Fig. 8b shows that d0 is dependent onthe cross-section location, but almost independent of strain.

We measure d0 for all simulations (Fig. 9). The gray areas showthe spread of the data within each simulation due to differentstrains. Data from the models with q ¼ 0� are shown in Fig. 9a. Theratio a/b has only a small effect as all the data plot close together.

Fig. 7. 3D sketch of sheath fold, g ¼ 7, q ¼ 90� , a/b ¼ 2.67. Gray area shows shape of outerindicate the center (d0) of the eye-structures. Note that the sketch is exaggerated by a factorfolded interface over the entire length of the fold. (For interpretation of the references to c

Fig. 9b shows the data from the models with q ¼ 90�. The light grayarea (a/b ¼ 0.375) has a different shape compared to the results ofthe other two models (a/b ¼ 1 and 2.67). This is because only twodata points were available for each strain as the sheath folds wereshort and not well developed (see Fig. 5). This result is, therefore,not reliable. The results from themodels with q¼ 135� are shown inFig. 9c. Compared to the Fig. 9a and b, the maximum d0 values arelower in Fig. 9c. In Fig. 9d, the results of all models are plotted. Thevalues of d0 show only a small dependence on a/b. The dependenceon the initial q value is bigger. All models show values above andbelow d0 ¼ 0.5, which indicate a variable layer thickness withincross-sections belonging to the same folded area.

3.2. Morphology of the eye-structure

To study the internal eye-structure of the sheath folds, i.e. theeffect of multiple contours and how they behave with increasingstrain, we investigate eye-structures containing 100 closed con-tours (N ¼ 100, g ¼ 7, model V_2.67). N1 labels the outermostclosed contour and N100 the 100th and innermost closed contour(Fig. 10a). We measure the aspect ratio (AR) for each of the 100contours in every cross-section and plot the results versus thecorresponding equivalent radius (req, Fig. 10b). The equivalentradius is the radius of a circle with the same area as a rectangledefined by the y and z dimensions of the closed contour. Theoutermost closed contour, having the largest equivalent radius,plots to the right of the plot of Fig. 10. Contours closer to the centerof the eye have smaller equivalent radii and plot towards the left ofthe plot. Note that the lines are interpolated from 100 discrete datapoints. We obtain one curve for each cross-section showing thevariability of AR. In Fig. 10c, the values are scaled with themaximum equivalent radius and the minimum AR, which corre-sponds to Ryz. The scaled AR values show a variation of almost 0.6 incross-section close to the tip of the slip surface while cross-sectionsin the middle or towards the apex show a smaller variation (min-imum of 0.16). We could not collapse the data into one curve, whichindicates that the location of the cross-section has an influence onthe values of AR, and in particular on R0. Data points that areplotting at the left edge of the plot of Fig. 10c represent the inversevalue of R0 as the innermost closed contour (Ry0z0) is scaled to theoutermost (Ryz).

most closed contour. Only half of the sheath fold is shown in y-direction. Red arrowsof 10 in x-direction and that the outermost closed contour does not belong to the sameolor in this figure legend, the reader is referred to the web version of this article.)

Fig. 8. a) Dependence of d0 on cut location for different strains of one sheath fold (q ¼ 90� and a/b ¼ 2.67). The small circles show the data points. b) The same data set as in a) butd0 plotted against scaled distance (x-axis) for different strains.

J.E. Reber et al. / Journal of Structural Geology 53 (2013) 15e26 21

To examine if AR behaves similarly within a given cross-sectionfor increasing strain, we investigate the cross-section closest to thetip of the slip surface in model V_2.67 (Fig. 11). The strain isincreased from g ¼ 3 to 15. Each curve represents the scaled ARmeasurements for one strain value. With increasing strain thevariability of AR increases and the curves could not be collapsed.Figs. 10 and 11 show that AR for a given initial slip surfaceconfiguration is strongly dependent on strain and cross-sectionlocation.

We study the effect of the slip surface configurations (q, a/b) onR0 (Fig. 12). Measurements for R0 are shown for different cross-section locations and strains for all the investigated models with

Fig. 9. d0 measurements plotted versus the normalized distance for different q and a/b valuesthe initial q angle.

N ¼ 100. Fig. 12a, b, and c show the results for the models withq¼ 0�. The values of R0 increase away from the tip of the slip surface.For a/b¼ 2.67, the R0 values are higher than for a/b¼ 1 or 0.375. Themajority of the R0 values are below 1. In some cross-sections,however, values above 1 can be observed. Fig. 12d, e, and f showdata from the models with q ¼ 90�. All R0 measurements for this setof models show values below 1. The well developed sheath folds ofV_1 and V_2.67 show that R0 first increases away from the tip of theslip surface and then decreases towards to the apex. Fig. 12h and ishow well-developed sheath folds originating from a slip surfacewith initial orientation of q ¼ 135�. The majority of the measure-ment points in figures g and h show R0 values above 1. In Fig. 12i,

. a) q ¼ 0� , b) q ¼ 90� , c) q ¼ 135� , d) data form all models, the gray scale corresponds to

Fig. 10. Model V_2.67. a) Illustration of analyzed eye-pattern exemplary for N ¼ 8.N1 ¼ outermost closed contour, N8 ¼ 8th closed contour. b) Aspect ratio (AR) versusequivalent radius (req) for N ¼ 100. The individual lines correspond to different cross-sections of one sheath fold at g ¼ 7. c) AR normalized with Ryz and plotted againstnormalized equivalent radius.

Fig. 11. Normalized aspect ratio versus normalized equivalent radius for the cross-section closest to the slip surface of model V_2.67 presented for different strains.

J.E. Reber et al. / Journal of Structural Geology 53 (2013) 15e2622

approximately a third of the measurements show R0 values equal orabove 1. Fig. 12 shows that R0 is dependent on both q and a/b.

Fig. 13 shows R0 plotted against the cross-section location forfour different, well developed sheath folds. Each curve representsone vertical column in the plots of Fig. 12 at g ¼ 10, i.e. R0 mea-surements on cross-sections along the x-axis. Model O_1 showsvalues above and below 1 while the other three sheath folds showonly values below 1. R0 initially increases with increasing x, andthen decreases. The decrease is not visible for V_2.67 as the foldstructure was not investigated further than x ¼ 20.

So far we used N ¼ 100 for the analysis of the internal structure.We showed that the AR values for 100 contours change within onecross-section (Fig. 10). The number of the sampled contours has,therefore, an impact on the aspect ratio of the innermost sampledcontour. Because the number of layers composing sheath folds innature is variable, and mostly less than 100, we need to investigatethe effect of the number of contours on AR and ultimately on R0. Todo so, we vary the number of closed contours within the eye-structure between 2 and 100 (Fig. 14). We use three cross-sections and sample them with 10, 5, 3, and 2 contours. The solidlines in Fig. 14 show AR values plotted against equivalent radiusnormalized by Ryz and the maximum equivalent radius for N ¼ 100(see Fig. 10) and are used as reference lines. Note that Ryz and themaximum equivalent radius are those of the outermost closedcontour, and are thus independent of the number of contours. If weexamine an eye-structure that contains 10 contours in each cross-section (small black crosses AR values of the individual contours),we obtain the aspect ratio of the innermost closed contour Ry0z0 (redcross), which is close to the value we get for N ¼ 100. If we build aneye-structure with 5 contours (small black circles) Ry0z0 (red circle)is smaller. If we decrease the number of contours to 3 (black di-amonds), Ry0z0 (red diamond) decreases again. For a classification ofthe eye-pattern according to Alsop and Holdsworth (2006) a min-imum of two closed contours is needed. If we examine an eye-pattern with only 2 contours Ry0z0 (red square) decreases oncemore; this value of Ry0z0 is the minimum that can be obtained. Theeffect of the number of closed contours on Ry0z0 is bigger for cross-sections close to the slip surface. The light gray vertical linesillustrate the maximum error that can be expected due to thesampling of either 100 or 2 contours.

In Fig. 15, we show the effect of the number of contours on R0 formodel V_2.67 for varying strains and cross-section locations. Eachcolored square represents one R0 measurement, while the colorgives its value. We calculate R0 using 100 contours (Fig. 15a) and 2contours (Fig.15b). For a constant g value, R0 first increases and thendecreases again with increasing x. For the case with N ¼ 2, the R0

values are slightly higher than for the case with N ¼ 100 butotherwise no striking differences can be observed. Some of the datapoints close to the slip surface resulted in R0 values of 1 as Ryz andRy0z0 had approximately the same values. These cross-sections arestrongly influenced by the slip surface and we do not considerthem.

Fig. 16 shows the evolution of R0 with increasing x for modelV_2.67 at g ¼ 7. The gray curve shows the result for N ¼ 2 and theblack line shows the result for N¼ 100. Both curves have essentiallythe same shape. The decrease of the R0 value due to the sampling ismaximal 0.05. For both cases, all the R0 values are below 1.

4. Discussion

Ourmodels show that sheath folds or sheath fold-like structuresdevelop for all tested slip surface orientations and sizes. In all testedcases, we could observe three-dimensional structures that exhibiteye-patterns in cross-sections perpendicular to the shear direction.Some of the structures discussed in this paper, however, would not

Fig. 12. R0 plotted for different strains (g) and cross-section locations (�) for different initial slip surface orientations and shapes.

Fig. 13. R0 plotted against the cross-section location for four different sheath folds atg ¼ 10.

J.E. Reber et al. / Journal of Structural Geology 53 (2013) 15e26 23

classify as sheath folds as their opening angle is larger than 90�

(Ramsay and Huber, 1987). Yet, we decided to call all structuresshowing eye-patterns in cross-sections sheath folds as they wouldeventually develop into real sheath folds with sufficient strain.

In this study we investigate a setup where the deformation issimple shear and the deformed system, except for the inclusion,homogenous. For a more complex setup such as a pure shearcomponent in the far field conditions or a mechanically layeredmatrix with the corresponding bending stiffness we would expectan even higher diversity of the sheath fold shapes (compare toReber et al., 2013).

We chose the outermost closed contour for the analysis of theoverall sheath fold shape. Layers displaying omega-shapes (Reberet al., 2012) or double vergent folds (Alsop and Carreras, 2007)take part in the distinctive sheath fold shape, but we do notconsider them in our analysis. Contours belonging to the samefolded interface can exhibit different shapes in different cross-sections, in agreement with the results of Reber et al. (2012). Theextent l of the area where eye-patterns are visible in cross-sectionsis much larger than the actual cone length of a single sheath fold

Fig. 14. Effect of number of closed contours on value of Ry0z0 evaluated for one sheath fold (V_2.76, g ¼ 7). Results for three cross-sections are shown. The solid lines shownormalized aspect ratio for N ¼ 100 plotted against normalized equivalent radius. The red signs show values of the innermost closed contours (Ry0z0) depending on the number ofsampled contours. The small black signs show the location of the sampled contours for N ¼ 10, 5, 3, and 2. The gray vertical lines show the maximum error in case the structure issampled only with 2 contours instead of 100. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

J.E. Reber et al. / Journal of Structural Geology 53 (2013) 15e2624

defined by one folded interface (compare Alsop and Holdsworth,2012). Our results show that l is dependent on q and a/b. Smallervalues of a/b lead to smaller l.

4.1. Sheath folds as shear sense indicators

In the study of Fossen and Rykkelid (1990) the thickening ofthe fold limbs and therefore the location of the center of theeye-pattern is an important measure to determine the shear

Fig. 15. Impact of number of contours on R0 . a) Cross-section location versus strain, color shothe colors (N ¼ 2). (For interpretation of the references to color in this figure legend, the r

direction. In our study we observe that the center of the eye-structure changes its position with respect to the upper edge ofthe outermost closed contour.We see d0 values above and below 0.5in all models and we show that d0 is almost independent of thestrain. d0 is also largely independent of the initial parameters of theslip surface. It shows, however, a slight dependence on q. In cross-sections, the layers appear to be thickened or thinned depending onthe value of d0 and, therefore, on the position of the center ofthe eye-structure. For d0 > 0.5, the layers in the upper half of the

ws R0 values (N ¼ 100). b) Cross-section location versus strain, R0 values are given witheader is referred to the web version of this article.)

Fig. 16. R0 plotted against cross-section location for N ¼ 100 and N ¼ 2 at g ¼ 7.

J.E. Reber et al. / Journal of Structural Geology 53 (2013) 15e26 25

eye-structure seem to be thickened and the layers in the lower partthinned. For d0 < 0.5, the layers in the upper part of the eye-structure appear thinned while the layers in the lower partappear thickened. In cross-sections close to the slip surface, thelayers in the upper half of the eye-structure appear to be thickened,while they appear thinned in cross-sections close to the apex. Dueto the big variation in thickness in z-direction within one layer, wedid not perform a measurement similar to the one performed byAlsop and Holdsworth (2006), where they measured the layerthickness in y and z direction.

For each g we obtain d0 values smaller and larger than 0.5,depending on the cross-section location. This contradicts theconclusion of Fossen and Rykkelid (1990), who state that the shearsense could be determined based on the thinning or thickening ofthe layers observed in a cross-sections perpendicular to the sheardirection and by knowing the closing direction of the fold. Assheath folds are similar to flanking folds in cross-sections parallel tothe shear direction, this orientation might be better suited todetermine the shear sense. Especially as the potential use offlanking folds as shear sense indicators has been extensivelystudied (e.g., Grasemann and Stuwe, 2001; Grasemann et al., 2003).

4.2. Sheath folds as bulk strain type indicators

The sheath fold classification by Alsop and Holdsworth (2006)is based on aspect ratios of the closed contours. In our study, weobserve that the aspect ratio depends on the number of involvedcontours, strain, cross-section location, and initial slip surfaceconfigurations. All cross-sections show a general increase in theaspect ratios (AR) from the outermost to the innermost layer. Thiscan also be observed in natural sheath folds (Alsop andHoldsworth, 2012). Cross-sections close to the slip surface showa bigger variation in AR between the outermost and innermostclosed contour than cross-sections in the middle or towards theapex of the folded area. Neither the curves for different cross-section locations nor the curves for different strains could becollapsed for one set of initial slip surface parameters, showingthat the aspect ratios of the closed contours are dependent onstrain and the cross-section location.

We observe that the aspect ratio of the outermost closed contour(Ryz) varies along x, and that it is dependent on the initial slip surfaceorientation and shape. Alsop et al. (2007) and Alsop andHoldsworth(2012) noted that the y-z ratios are dependent on the location of the2D section innatural sheath folds,which coincideswithourfindings.Our results show that Ryz increases with increasing deformation.

This is in accordance with the results of Alsop and Holdsworth(2006). We can, however, not unequivocally correlate certain Ryzvalues to a set of initial conditions (q and a/b) or strain magnitude.

Using the aspect ratio of the inner and outermost closed con-tour, we calculate R0 for N ¼ 100 and different initial slip surfaceconfigurations and could show that R0 is strongly dependent onboth q and a/b. We observe R0 values ranging from 0.5 to 1.3. Thiscontradicts the hypothesis of Alsop and Holdsworth (2006), wholink R0 values to the bulk strain type. For instance, they predict thatsheath folds developing in simple shear exhibit mainly R0 values<1.Our results show that one third of all measured R0 values are equalor larger than 1, though our model are all in simple shear. Alsop andHoldsworth (2012) investigated a three-dimensional naturalsheath fold, which they cut into serial cross-sections. They observecat’s-eye-folds in cross-sections over the entire length of the fold.This result does not necessarily contradict our findings as weobserve a similar situation in two thirds of our simulations. Alsopet al. (2007) state that natural sheath folds can show variable R0

values along the x-axis of the fold, which they attribute to theviscosity ratio and mechanical influence of the layering in thematrix. Using a homogenous matrix, we see in our results already asignificant variation in R0 values along the x-axis, and expect aneven larger variation in a mechanically layered matrix (compareReber et al., 2013). In comparison to Alsop and Holdsworth (2006)who state that R0 is constant with increasing deformation we seethat R0 is only constant in cross-sections that keep their relativeposition with respect to the slip surface. This is, however, only truein well-developed sheath folds.

While the outermost closed contour is a clearly defined feature,detecting the innermost closed contour is in some cases difficult asit can be dependent on the scale and resolution of the observation.We sample a sheath fold with a maximum of 100 contours and aminimum of 2 contours observing that R0 between the two casesdiffers only slightly. Our results imply that R0 is a relatively stablemeasurement and is not subject to large changes depending on thedefinition of the innermost closed contour. Even though R0 is stableand easy to measure, the contradiction between our results and thefindings of Alsop and Holdsworth (2006) suggests that R0 should beused with care to infer bulk strain type. To investigate this matterfurther, the effect of pure shear on the shape of the eye-patternsshould be analyzed in detail.

5. Conclusion

All the tested slip surface orientations and shapes lead to theformation of sheath folds or sheath fold-like structures in simpleshear. Their sizes and shapes are strongly dependent on the initialparameters and on strain. The extent of the area, over which eye-pattern can be observed in cross-section perpendicular to theshear direction, is dependent on the initial slip surface configura-tion. The location of the center of the eye-pattern, expressed by theparameter d0, is dependent on the cross-section location in x-di-rection. d0 can have values smaller and larger than 0.5. The resultingapparent thickening and thinning of layers in y-z-cross-sectionscannot be used to infer the shear direction. d0 is almost a strainindependent measurement. No specific initial slip surface config-uration could be linked to values of the aspect ratio of the outer-most closed contours (Ryz). The variation of the aspect ratios ofcontours within one eye-structure is dependent on the cross-section location and on the strain. Cross-sections located close tothe slip surface show a bigger variation than cross-sections in themiddle of the fold or close to the fold’s apex. R0 is dependent on q

and a/b. Depending on the initial settings of the slip surface, R0

values below and above 1 can be observed in models deformed insimple shear. The number of sampled contours, however, has a

J.E. Reber et al. / Journal of Structural Geology 53 (2013) 15e2626

minimal effect on R0. Using R0 measurements to deduce the bulkstrain type may be erroneous.

Sheath folds are highly complex three-dimensional structures.This study and other recent work on sheath folds (Alsop andHoldsworth, 2012; Reber et al., 2012, 2013) show that it is notsufficient to use two-dimensional eye-patterns to deduce kine-matic information. Future studies on sheath folds should thereforenot only concentrate on the analysis of eye-patterns but insteadfocus on the three-dimensional nature of the structures.

Acknowledgments

This work was supported by a Center of Excellence grant fromthe Norwegian Research Council to PGP. We thank the editor G. I.Alsop and the reviewers B. Grasemann and F. W. Vollmer forconstructive and helpful comments.

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