EVALUATION OF TRANSTENSION AND TRANSPRESSION AT FAULT STEPS: COMPARING KINEMATIC AND MECHANICAL MODELS TO FIELD DATA Josie Nevitt, David Pollard, and Jessica Warren Department of Geological and Environmental Sciences, Stanford University, Stanford, CA 94305 e-mail: [email protected]
models. In addition, the mechanical model provides a Abstract means to investigate the underlying physics of the problem, including the governing constitutive laws, Deformation within fault steps contributes to many causative tectonic stress states, and frictional contact important geologic phenomena, including earthquakes, boundary conditions on the fault. In contrast, the mountain building, and basin development, and has kinematic model is constrained only by the geometry of previously been investigated using both kinematic and the structure in the final state and the assumption of mechanical models. This paper provides a direct constant volume. The comparison between kinematic comparison of these modeling techniques in the context and mechanical models presented here should compel of a meter-scale contractional fault step located in the future investigators to use the latter when considering Seven Gables outcrop (Bear Creek field area, Sierra deformation within fault steps. Nevada, CA). The Seven Gables fault step contains locally foliated granodiorite and a stretched and rotated dike, which serve as three-dimensional Keywords: deformation markers. Kinematic models in previous fault step, transtension/transpression, deformation studies have assumed one of two possible shear plane matrix, finite element model orientations: (i) shear plane parallel to the step- bounding faults; (ii) shear plane parallel to an internal Introduction fault, which is oblique to and connects the step- Faults can be discontinuous at many scales and bounding faults. This study presents kinematic models often exhibit en echelon geometries, characterized by for the Seven Gables fault step using each of these sub-parallel fault segments that are either left- or right- geometries. Kinematic modeling is accomplished by use stepping (Aydin and Schultz, 1990; Segall and Pollard, of the deformation matrix, which is first formulated for 1980, 1983b; Wesnousky, 1988). Deformation within simple shear and then for transtension/transpression. fault steps plays a significant role in both long- and The components of the deformation matrix are based on short-term fault processes, including fault coalescence outcrop measurements and the assumption of constant and lengthening, and earthquake rupture nucleation and volume. Both models result in dike orientations with termination (Cowgill et al., 2004; Harris et al., 1991; significant misfit (model 1: 28% total misfit; model 2: Harris and Day, 1993, 1999; Harris et al., 2002; Kase 44% total misfit) compared to the dike measured in and Kuge, 1998; King and Nabelek, 1985; Oglesby, outcrop. An interesting result of the kinematic analysis 2005; Sibson, 1985; Wesnousky, 2006; Zhang et al., is that the contractional step may be classified as either 1991). Thus, an improved understanding of deformation transtensional or transpressional, depending on which within fault steps will shed light on how fault structures model geometry is used, suggesting that these terms evolve through time, with consequent benefits for may not be appropriate descriptors of deformation seismic hazard analysis. within fault steps. The ambiguity of the kinematic The nature of deformation within steps depends on results motivates the use of a mechanics-based finite the relationship between the step geometry and the element model of deformation in the Seven Gables fault sense of slip (Fig. 1a). A step with the same sense as step. The results of this mechanical model indicate that that of the fault slip (e.g., left step along a left-lateral plastic strain localizes along a narrow zone that runs fault) results in extensional deformation, such as open diagonally through the step (consistent with the cracks at the meter scale (Fig. 1b) (Flodin and Aydin, orientation of the shear plane in the second kinematic 2004; Kim et al., 2000; Kim et al., 2003, 2004; Martel model). The mechanical model provides additional et al., 1988; Segall and Pollard, 1980, 1983b) and the insights into the heterogeneous nature of deformation development of a pull-apart basin at the kilometer scale within the step, including the spatial variability of (Fig. 1c) (Aydin and Nur, 1985; Mann et al., 1983; plastic strain, slip gradient along the faults, and non- Westaway, 1995). In contrast, a step with an opposite uniform dike thinning. The ability to characterize sense to that of the fault slip (e.g., right step along a heterogeneous deformation represents a significant left-lateral fault) results in contractional deformation advantage of the mechanical model over the kinematic structures, including ductile fabric (Fig. 1d) or pressure
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Figure 1. (a) Schematic diagram relating the geometry of extensional and contractional steps along left- lateral faults to block diagrams of transtension and transpression. (b) Photograph of a small-scale extensional step between left-lateral faults with opening-mode cracks surrounded by an alteration halo (pale yellow discoloration) in the Bear Creek field area. (c) Digital elevation model (DEM) of a large-scale extensional step between right-lateral faults in southern California. (d) Photograph of a small-scale contractional step containing mylonitic foliation between left-lateral faults in the Bear Creek field area. (e) DEM of a large-scale contractional step along the right-lateral Coyote Creek Fault, southern California, which resulted in the uplifted Ocotillo Badlands. solution seams at the meter scale (Bürgmann and extension in the vertical direction, such that Pollard, 1992, 1994; Peacock and Sanderson, 1995) and deformation is accommodated through other push-up ranges at the kilometer scale (Fig. 1e) (Aydin mechanisms (i.e., development of foliation or pressure and Nur, 1985; Westaway, 1995). Differences in solution seams). secondary structures for the small- and large-scale steps Extensional and contractional steps are sometimes may be related to the confinement during deformation. referred to as being sites of “transtension” and For example, kilometer-scale contractional steps at the “transpression,” respectively (De Paola et al., 2008; surface are unconfined in the vertical direction, which ; Elliott et al., 2009; Miller, facilitates the development of push-up ranges. For 1994). These terms were first introduced to describe meter-scale fault steps that form at depth, however, the deformation associated with oblique divergence and three-dimensional confinement discourages preferential convergence of tectonic plates (Harland, 1971). In
Stanford Rock Fracture Project Vol. 24, 2013 H-2 or set of faults, linking the bounding faults. In the absence of a well-developed or well-exposed internal fault, Westaway (1995) suggests using the plane connecting the fault tips in the deformed state as the shear plane orientation. As fault steps are often examined from a kinematic perspective (Barnes et al., 2001; Cembrano et al., 2005; De Paola et al., 2008; Pluhar et al., 2006; Wakabayashi et al., 2004; Westaway, 1995), determining the appropriate shear plane orientation will benefit future studies of fault steps that rely on a correct understanding of the shear plane geometry for the interpretation of kinematic indicators (e.g., microstructures). In this study, we have identified and mapped a meter-scale contractional fault step that can be used to study the relationship between shear plane orientation and deformation. This fault step, located in the Lake Edison granodiorite in the Bear Creek field area (Sierra Nevada, CA), contains a leucocratic dike and ductile fabric that serve as deformation markers. The relatively small scale and excellent outcrop exposure allow a thorough kinematic analysis using the deformation matrix. We test the ability of simple shear Figure 2. (a) Photograph of a contractional fault and transtension/transpression to reproduce the rotation step located in Bear Creek with an internal fault; of the dike using the two models for shear plane (b) Outcrop map; (c) Zoomed-in view of internal orientation (step-bounding faults vs. internal fault). The fault; (d) Zoomed-in view of step-bounding fault. kinematic analysis is complemented with a mechanics- kinematic terms, transtension and transpression based finite element model of the deformation within describe strike-slip deformations that differ from simple the step, and the assumptions inherent to both the shear due to a component of extension or contraction kinematic and mechanical models are evaluated. orthogonal to the shear plane (Dewey et al., 1998; Fossen and Tikoff, 1998; Sanderson and Marchini, 1984). Thus, knowledge of the shear plane orientation is necessary to identify a region as transtensional or transpressional. Previous kinematic studies of fault steps can be classified into two groups, based on two different models for shear plane orientation. The shear plane orientation in the first model, which we interpret to have been used by Cembrano et al. (2005) and De Paola et al. (2008), is equivalent to the orientation of the step- bounding faults. This may be the most intuitive orientation, since faults are generally modeled as parallel to the shear planes in kinematic analyses (e.g., Cladouhos, 1999). Furthermore, block diagrams of transtension and transpression (Fig. 1a) are visually consistent with a fault step in which the shear plane is defined by the orientation of the step-bounding faults. The shear plane orientation in the second model, proposed by Westaway (1995), is equal to that of an “internal fault,” which connects and is oblique to the step-bounding faults. Fault steps, both at the meter scale Figure 3. Infinitesimal vectors dX and dx, located (Fig. 2) and at the kilometer scale (Madden and Pollard, at positions X and x, lie along a material line in 2012; Manaker et al., 2005; Sieh et al., 1993; the initial and deformed states, respectively, and Westaway, 1995), may include a discrete internal fault, u is the displacement vector (after Malvern, 1969, pp. 155).
Stanford Rock Fracture Project Vol. 24, 2013 H-3 Deformation gradient tensor and the therefore admits heterogeneous deformation. That the deformation matrix material line segment is infinitesimal and that homogeneity of deformation only applies in the The deformation matrix is a tool commonly used in infinitesimal neighborhood of the particle does not structural geology to study the kinematics of restrict the magnitudes of the components of F – they deformation. It is related to the deformation gradient may describe finite deformation. tensor, which is used in continuum mechanics to Because analytical solutions for heterogeneous quantify finite strain. Here, we provide a brief finite deformation in fault steps are not available, we discussion of both tensors to illustrate their relationship use the finite element method (FEM) for the mechanical to each other and highlight the important differences in model. This method discretizes the model domain into their application. many small elements, each characterized by a different To introduce the deformation gradient tensor, we deformation gradient tensor. While only a numerical first consider the deformation in the neighborhood of a approximation of the continuum, the FEM admits particle initially located by position vector X with heterogeneous deformation, user-defined constitutive components X (,,)XYZ and located after relations, and requires prescribed boundary conditions deformation by position vector x with components and contact conditions. x (,xyz ,), as shown in Figure 3. In the initial state, a In the structural geology literature, a special case of material line segment extends from the particle and is the deformation gradient tensor has been referred to as coincident with the infinitesimal vector dX with the “deformation matrix” and is given the symbol D (Flinn, 1979; Fossen and Tikoff, 1993; Tikoff and components dX dX,, dY dZ . During deformation, Fossen, 1993). The application of D departs from that the material line is displaced, stretched, and rotated to of F in several ways. D has been used to study lie along the infinitesimal vector dx with components deformation not in the neighborhood of a particle, but dx dx,, dy dz . This deformation may be represented within finite volumes ranging up to the kilometer-scale by the equation (e.g., Tikoff and Teyssier, 1994). This usage implicitly ddxFX (1) assumes that deformation is homogeneous for kilometer-scale regions, DD (,,,)XYZt, though they
may contain geologic structures (e.g., folds and faults) where F is the deformation gradient tensor defined as follows (Malvern, 1969, pg. 156): that attest to the heterogeneity of the deformation. In addition, D is generally given as an upper triangular matrix (e.g., Flinn, 1979; Fossen and Tikoff, xxx 1993; Tikoff and Fossen, 1993): XYZ k1 xy xz yyy F D 0 k 2 yz XYZ (2) (3) 00k3 zzz This is accomplished using the QR algorithm (Strang, XYZ 2006, pg. 364), which factors an asymmetric, non- orthogonal matrix into an orthogonal matrix, Q, and an This is a Lagrangian formulation, because the upper triangular matrix, R. components of F are partial derivatives with respect to Another difference from the usage of F is that the the initial material coordinates. The deformation initial and current vectors are given in non-differential gradient tensor can also be derived in an Eulerian form when using D: formulation, in which the partial derivatives are taken xDX (4) with respect to the current coordinates (Malvern, 1969, pg. 157). The components of F must be real and F must This necessitates homogeneous deformation. The be non-singular (i.e., detF 0 ), but there are no material line is finite, so no longer represents restrictions on the magnitudes of the partial derivatives deformation in the neighborhood of the particle. In (Flinn, 1979). addition, it necessitates that the “tails” of X and x are The initially straight infinitesimal line segments located at the origin and the “heads” are located at the remain straight, so Eq. (1) is consistent with position of X and x. homogeneous deformation in the neighborhood of the Note that Eq. (4) is used to model the deformation particle. In the context of continuum mechanics, the of a line. To model the deformation of a plane, X is components of F may be functions of the spatial defined as the unit normal (pole) to the plane and Eq. coordinates and time, FF (,,,)XYZt, which (4) is replaced with (Fossen and Tikoff, 1993):
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Figure 4. Block diagrams illustrating the different types if deformation represented by the various Figure 5. Geologic map of the Bear Creek field area components of the deformation matrix (after Tikoff in the Mount Abbot quadrangle, central Sierra and Fossen, 1993). Nevada, CA (modified from Lockwood and Lydon, 1975). Kmr – Mono Recesses quartz monzanite; Kle 1 (5) xXD – Lake Edison granodiorite; Kl – Lamarck The diagonal terms of D represent extension or granodiorite; Kj – granitic rocks of uncertain contraction parallel to the Cartesian coordinate axes, affinities; J Tr – Metavolcanic rocks; Tt – Olivine while the off-diagonal terms represent shear trachybasalt; Q – alluvium. deformation. Figure 4 illustrates the types of deformation represented by each component of D. deformed fault step in the field example of the Seven Because matrices are generally non-commutative Gables outcrop. (Strang, 2006, p. 25), using the deformation matrix to characterize general deformation that involves Bear Creek field area combinations of pure shear, simple shear, and/or The fault steps discussed in this paper are located in volume change is not straight-forward, and was the Bear Creek drainage of the central Sierra Nevada, developed by Tikoff and Fossen (1993) as the “unifying CA (Fig. 5), within the Lake Edison biotite-hornblende deformation matrix.” We experiment with this granodiorite of late Cretaceous age (88±1 Ma) deformation matrix in following sections to determine (Lockwood and Lydon, 1975; Tobisch et al., 1995). The how well certain formulations of deformation (e.g., structural development of fractures, faults and shear simple shear, transtension) represent the naturally zones in the Bear Creek field area has been the subject
Figure 6. Schematic diagram of the deformational history of Bear Creek (after Bürgmann and Pollard, 1994).
Stanford Rock Fracture Project Vol. 24, 2013 H-5 Pollard, 1986; Griffith et al., 2008; Kronenberg et al., 1990; Martel et al., 1988; Pennacchioni and Zucchi, 2012; Segall et al., 1990; Segall and Pollard, 1980, 1983a, b). Between 85-79 Ma, shortly following pluton crystallization, a set of joints developed that strike predominantly ENE and contain quartz, chlorite and epidote mineral fill (Martel et al., 1988; Segall et al., 1990; Segall and Pollard, 1983a). Individual joints, ranging from ~0.5 m to nearly 100 m long, generally consist of multiple subparallel segments that form an en echelon geometry (Segall and Pollard, 1983a). In the ~5 Ma following joint nucleation, a change in the stress regime caused the joints to reactivate as left-lateral strike-slip faults (Segall and Pollard, 1983b). Due to the faults’ inherited en echelon geometries, the field area contains abundant examples of extensional and contractional steps, where the faults are left- and right-stepping, respectively (Bürgmann and Pollard, 1992, 1994; Segall and Pollard, 1980, 1983b). The two step types are characterized by distinct styles of deformation. Extensional steps contain opening mode structures, such as splay cracks and rhombochasms (Fig. 1b), while contractional steps typically contain a locally strong mylonitic foliation (Fig. 1d). The occurrence of contemporaneous brittle and ductile structures led Bürgmann and Pollard to conclude that the faults were likely active near the brittle-ductile transition (Bürgmann and Pollard, 1992, 1994) . The temperature of deformation has been estimated through microstructural analysis to be 300- Figure 7. Three examples of contractional 450°C (Christiansen and Pollard, 1997) and as high as steps that stretch and rotate dikes. In Figures >500°C (Pennacchioni and Zucchi, 2012). The (b) and (c), dikes at locations marked “A” are broken and offset when cross-cut by faults, variability of temperature estimates may be related to while dikes at locations marked “B” are the location of the samples considered in those studies, stretched, thinned, and rotated through since the emplacement of the Mono Creek Granite is contractional steps. theorized to have produced a temperature gradient across the older Lake Edison granodiorite (Bürgmann of numerous papers over the last three decades and is and Pollard, 1994). summarized in Figure 6 (Bürgmann and Pollard, 1992, 1994; Christiansen and Pollard, 1997; Davies and
Figure 8. Conceptual model for the deformation of dikes within contractional steps.
Stanford Rock Fracture Project Vol. 24, 2013 H-6 Leucocratic dikes, ranging from aplite to pegmatite, are abundant throughout the field area and serve as kinematic markers when offset across a fault or through a step (Fig. 7) (Pennacchioni and Zucchi, 2012). Outcrops that contain dikes displaced through contractional steps provide an opportunity to investigate the three-dimensional deformation within steps. The scale of the contractional steps containing dikes ranges from decimeters to tens of meters, with the larger examples occurring near the contact with the Mono Creek Granite (see Fig. 5). We have found that the contractional steps containing dikes generally share three characteristics (Fig. 7): (i) the granodiorite within the step develops a mylonitic foliation that parallels the deformed dike; (ii) the volume of the dike remains approximately constant within the step; and (iii) the fault tips deflect outward from the step. A conceptual model for the deformation of dikes within fault steps is presented in Figure 8. Initially, the dike has a uniform thickness and is cross-cut by two joints that terminate near their intersection with the dike. As the joints slip and become small faults, the step region becomes a site of contraction and left- lateral shear. This initiates the counterclockwise rotation of the dike and the development of foliation within the step. With increasing deformation, the local foliation intensifies and the fault tips deflect outward to accommodate the material moving into the step region.
Seven Gables outcrop The Seven Gables outcrop (referred to by Pennacchioni and Zucchi (2012) as the Middle East Fork or MEF), located on the south bank of East Fork Creek near the base of Seven Gables Mountain (Fig. 5), includes an example of a contractional step that stretches and rotates a dike. The fault system associated with this step includes ~15 individual fault segments that vary in length from approximately 1 to 4 m. While most of the right-stepping discontinuities between left- lateral faults are characterized by a mylonitic foliation in the granodiorite, one step also includes a deformed dike (Fig. 9). The Seven Gables outcrop is glacially polished, Figure 9. Orthorectified photograph (a) and which provides a beautiful surface exposure, but limits annotated map (b) of the Seven Gables Outcrop. the three-dimensional perspective of the deformation. To overcome this, we used well-exposed nearby faults (Fig. 10a, b) and core samples (Fig. 10c) to determine times its initial length and rotated 44° counter- the orientations of the structures in the Seven Gables clockwise, as viewed on the outcrop surface. The dip of outcrop in three dimensions. The contractional step is the dike increases from 25°outside the step to 61° defined by two sub-parallel left-lateral faults that trend within the step, and the true width of the dike decreases 259° and dip 76°, as measured just outside the step from 4.2 cm outside the step to 1.7 cm within the step. (Fig. 9). The faults contain slickenlines with a rake of The combined result of the stretching and thinning of 10° in the direction 259°. The dike is offset 42 cm the dike corresponds to the volume of the dike through the right step between the two left-lateral remaining approximately constant (within 10%) within faults. Within the step, the dike has been stretched to 3 the step region.
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Figure 10. Additional data used to constrain the orientation of structures in the Seven Gables outcrop: (a) exposed fault surface near the outcrop; (b) slickenlines on the exposed fault surface in (a), (c) slickenlines on a slip surface from a core sample collected on the eastern fault in the Seven Gables outcrop (see Fig. 9 for location). Assuming that the orientation and thickness of the refer to as the “shear zone.” The orientation of the pole dike outside the step is representative of the initial state, to the dike in the initial and deformed states is these observations indicate that the dike underwent determined for the two kinematic models of fault steps rotational deformation about a non-vertical axis within used in previous literature: (i) shear plane parallel to the the contractional step. One viable axis of rotation is step-bounding faults; (ii) shear plane parallel to an determined by fitting a small circle with the shortest internal fault. possible radius to the orientation data (strike and dip) for the pole to the dike in the initial and deformed states Shear plane parallel to step-bounding (Fig. 12). This indicates that the dike rotated 98° about faults an axis plunging 55° in the direction 180°. Rotational deformation within fault steps has primarily been The first model geometry assumes that the shear studied using paleomagnetic data and has consequently plane is parallel to the step-bounding faults. As shown been limited to rotation about a vertical axis (Berger, in the stereonet in Figure 11a, the planes defining the 2007; Pluhar et al., 2006). This study thus provides new shear zone are as follows: the fault plane (259°, 76°); insights into how planar objects deformed within plane 2, which is perpendicular to the fault and contains contractional steps undergo rotation about a non- the slickenlines (114°, 17°); and plane 3, which is vertical axis. perpendicular to the first two planes (351°, 80°). As measured in the Seven Gables outcrop, the pole to the Kinematic models dike in the initial state has a plunge of 65° and plunge direction of 244°. To convert the azimuth to the model reference frame, we calculate the angle between the Defining the shear plane reference frame positive Y-axis and the plunge direction of the pole to The deformation matrix is formulated such that the the dike: initial 244 169 075 , as illustrated in X-coordinate is parallel to the shear plane. It is Figure 11a. therefore necessary to convert the outcrop orientation To determine the plunge ( ) relative to the shear data, which is relative to geographic north, to a zone model, it is necessary to consider the apparent dip reference frame in which the X-coordinate is aligned of plane 2 in the dip direction of the dike. The apparent with the shear plane and the Y- and Z-coordinates are dip, aligned with two additional orthogonal planes. strike of plane 2 and the dip direction of the dike, and Together, these three planes define a cube-shaped block the true dip, , of plane 2. In the deformed state, the identical to the blocks shown in Figure 4, which we pole to the dike relative to geographic north has a
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Figure 11. Geometry of the two model shear zones tested in the kinematic analysis.
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Figure 12. Stereonet plots of the results from the (a) first and (b) second kinematic models. plunge of 29° and a plunge direction of 200°. x 0.5038 0.8384 0.2079 , respectively. model reference frame: 200 169 031 . In this case, the apparent deformed Shear plane parallel to an internal fault dip of plane 2 in the plunge direction of the dike is The second shear zone model assumes that the shear arctan sin(200 114 ) tan17 17 . Thus, the app plane is parallel to an “internal fault,” as proposed by plunge of the dike within the model reference frame in Westaway (1995). The Seven Gables outcrop (Fig. 9)
the current state is deformed 29 17 12 , shown in lacks a clearly defined internal fault, but the deformed cross-section B-B’ (Fig. 11a). dike links the fault tips and is assumed to provide a The next step is to represent the orientation data, good proxy for the orientation of the shear plane. The and , as a unit vector to use in Eq. (5). We use the three orthogonal planes that define the shear zone, following formula to carry out this conversion (Pollard shown in Figure 11b, are as follows: the deformed dike and Fletcher, 2005, p. 69): (290°, 61°); plane 3, which has an azimuth rotated 90° clockwise from that of the deformed dike and includes x sin cos cos cos sin (7) the pole to the deformed dike (200°, 88°); and plane 2, This equation assumes a right-handed coordinate which is perpendicular to the first two planes (105°, system with the z-axis directed up where is measured 29°). As in the first model, we determine the orientation clockwise from the positive Y-axis and is measured of the dike relative to the model reference frame in the from the horizontal plane. Using Eq. (7), the vector initial and deformed states. representations of the unit normals (poles) to the dike in The plunge direction of the dike relative to the the initial state and in the deformed state are model reference frame in the initial state is and X 0.5947 0.1593 0.7880 initial 244 200 044 (Fig. 11b). Using Eq. (6),
Table 1: Kinematic model results for poles to deformed dike compared to outcrop measurements (reported as plunge direction and plunge amount in a geographic reference frame) Shear Zone Model 1 Shear Zone Model 2 Outcrop measurement 200°, 29° 200°, 29° Simple shear 182°, 33° (error: 41%, 11%) 211°, 43° (error: 25%, 39%) Transtension/transpression 190°, 31° (error: 23%, 5%) 202°, 43° (error: 5%, 39%)
Stanford Rock Fracture Project Vol. 24, 2013 H-10 the apparent dip of plane 2 in the dip direction of the The direction cosines for each component of the dike in the initial state is resultant vector are equal to the ratios of the components to the magnitude of the vector (Pollard and app arctan sin(244 105 ) tan 29 20 . Thus, the plunge of the pole to the dike in the model reference Fletcher, 2005, pg. 69): cos x x (xx ) / 0.2098 frame in the initial state is initial 65 20 45 , as cos x (yx ) / 0.9374 shown in cross-section C-C’ (Fig. 11b). In the deformed y (11) state, the orientation of the dike is the same as that of cos z x (zx ) / 0.2780 plane 1 (the shear plane) and the pole is coincident with The plunge direction and plunge of the pole to the the positive Y-axis. Using Eq. (7), we calculate the unit dike in the model reference frame are extracted from normal vectors that represent the poles to the dike in the the direction cosines as follows: initial and deformed states to be cos X 0.4912 0.5087 0.7071 and x 010, tan 1 x 13 (12) cos respectively. y 1 (13) sin cosz 16 Simple shear As in Eq. (7), the above equations assume a right- Simple shear often is used to idealize deformation handed coordinate system with the z-axis directed up. within shear zones (Davis, 1983; Herren, 1987; To transform these values to the geographic reference Ramsay, 1980; Warren et al., 2008; Wernicke, 1985). frame, we reverse the calculations done in Section 5.1. For this reason, we are interested in testing its accuracy The plunge direction is 169 13 182 with in reproducing the three-dimensional rotation of the respect to geographic north. Using Eq. (6), the apparent dike within the Seven Gables outcrop. Simple shear is dip of plane 2 in the orthogonal cross-section of the defined as constant-volume, non-coaxial (i.e., dike in the deformed state (as modeled by simple shear)
irrotational lines are not parallel to the instantaneous is app arctan sin(182 114 ) tan18 17 . Thus, strain axes), plane strain that occurs by shearing along a the plunge of the deformed dike is 16 17 33 with series of closely-spaced discrete shear planes (Ramsay, respect to horizontal. The plunge and plunge direction 1969, pg. 83). of the pole (182°, 17°) to the dike in the deformed state, The deformation matrix for simple shear in three as modeled by simple shear, is plotted on the stereonet dimensions is in Figure 12a.