EVALUATION OF TRANSTENSION AND TRANSPRESSION AT 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 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 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, tan1 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.

10xy We repeat the simple shear calculations for the

D 010 second model, in which the shear plane is parallel to the (8) 001 deformed dike. In this case, the modeled dike in the deformed state has a plunge of 43° and plunge direction where xy is the shear strain. In the Seven Gables of 211° with respect to north. This orientation is plotted outcrop, the dike is offset 42 cm in a left-lateral sense on the stereonet in Figure 12b. across an initially 10 cm wide step (Fig. 9), making The misfit in the model results, given in Table 1, is calculated as xy 4.2 . | | = × 100. Eq. (5) is used to calculate the orientation of the dike in the deformed state for each of the two models. Compared to using the first kinematic model, the For the first model, in which the shear plane is parallel predicted orientation of the dike in the second model to the step-bounding faults, has greater misfit in the plunge, but lesser misfit in the 1 4.2 0 plunge direction. The total misfit for the first shear zone x 0.5947 0.1593 0.7880 0 1 0 model is 52%, compared to 64% for the second shear zone model. While both models produce results with 001 (9) significant misfit to the outcrop measurements, the first 0.5947 2.6570 0.7880 model is slightly better at predicting the deformation than the second model. We convert x back to plunge and plunge direction values relative to geographic north in order to compare Transtension and transpression the model results to the field measurements. It is Transtension and transpression build upon the necessary to first calculate the magnitude, : simple shear calculations by incorporating pure shear 2221/2 (10) (diagonal components) into the deformation matrix xxxxx y z 2.8345 (Tikoff and Fossen, 1993):

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xy kk12 k1 0 ln(/) k12 k D 00k (14) 2 00k3

Whether Eq. (14) represents transtension or transpression is determined by the values assigned to k1, k2, and k3, which in turn depends on how the shear plane is defined. According to the field measurements (Fig. 9), the step in the Seven Gables outcrop widens to 1.8 times its initial width during deformation. In addition, the length of the dike within the step increases to 3.3 times its initial length, and the dike’s thickness decreases to 0.4 times its initial thickness. Interestingly, the observed deformation features within the step indicate transtension when assuming the first model geometry and transpression when assuming the second model geometry. For the first model, in which the shear plane is Figure 13. Strain ellipses calculated from the transtension/transpression deformation parallel to the step-bounding faults, the widening of the matrices for the (a) first and (b) second step during deformation represents extension kinematic models. Figures also show the perpendicular to the shear plane. Thus, the deformation orientations of the shear planes (dashed is best characterized as transtensional. We estimate lines) in each model and of the modeled using the decreased thickness of the dike during dikes in the deformed state. deformation. The trend of the deformed dike diverges from the X-axis by an angle of 31°. By determining the dike in the direction parallel to the shear plane, and X-component of the dike shortening, we estimate that k2 0.4 , which represents the contraction of the dike k1 1 (1 0.4)cos31 0.49 . In addition, we estimate perpendicular to the shear plane. To maintain constant that k2 1.8 based on the increased step width volume, kkk3121/ ( ) 0.8 . In this case, the product perpendicular to the shear plane. Then, assuming of the unit normal X and the inverse deformation matrix constant volume during deformation, D-1 indicates that the pole to the dike in the deformed state has a plunge direction of 202° and a plunge of 43° kkk3121/ ( ) 1.1. As in the simple shear calculations, we multiply the with respect to geographic north. Again, this orientation unit normal X and the inverse deformation matrix D-1 to is plotted in the stereonet in Figure 12b. model the orientation of the pole to the dike in the The results for both models are improved by using deformed state assuming transtension. We repeat the transtension or transpression over simple shear (Table 1 frame-of-reference conversion (Eq. 10-13) to determine and Fig. 12). For the first model, the misfits in the that the deformed dike has a plunge direction of 191° plunge and plunge direction decrease by 6% and 18%, and a plunge of 31°, with respect to geographic north. respectively. For the second model, the misfit in the This orientation is plotted on the stereonet in Figure plunge remains at 39% while the misfit in the plunge 12a. direction decreases by 20%. For the second model, in which the shear plane is Figure 13 shows the final orientation of the dike in parallel to the deformed dike, the decreased width of the X-Y plane for each of the two models. Overlain on the dikes are the strain ellipses calculated from the the dike within the step indicates contraction T perpendicular to the shear plane, which is consistent eigenvalues and eigenvectors of DD , where D is the with the definition of transpression. Again, we use Eq. deformation matrix employed in each model (Tikoff (14) to define the deformation matrix, but edit the and Fossen, 1993). The major and minor axes of the components for the different model geometry. In this ellipses represent the maximum and minimum principal elongations (Means, 1976, p. 140). case, k1 3.3 , which captures the elongation of the

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Figure 14. (a) Model geometry and (b) boundary conditions for the finite element model.

of planar overlapping node sets that obey the Coulomb criterion for fault slip (Coulomb, 1773) with a Mechanical model To be consistent with the outcrop measurements, the southwestern fault is 1.1 To further explore deformation in the fault step, we m long and the northeastern fault is 2.2 m long. construct a mechanical model of the deformation based The finite element model assumes plane strain and on the outcrop measurements. Mechanical models use thus is two-dimensional. The kinematic analysis the laws of conservation of mass and momentum and indicates that there may be a component of out-of-plane constitutive laws defining material behavior to calculate deformation in the Z-direction. However, if the faults the resulting kinematic relations. Treatment of a fault are very long in the Z-direction, then the plane strain step from a mechanical perspective avoids many of the assumption is not unreasonable. An exposed fault common assumptions (e.g., constant volume, surface with the same orientation as the faults in the homogeneous deformation, shear zone geometry) Seven Gables outcrop, is located ~100 m away to the inherent to purely kinematic studies (Ando et al., 2004; southeast and is ~40 m long in the Z-direction (Fig. Brankman and Aydin, 2004; Bürgmann and Pollard, 10a, b). Since the Seven Gables outcrop faults appear to 1992; Bürgmann et al., 1994; Duan and Oglesby, 2006; be in the same set as the exposed fault surface, they Harris and Day, 1993; Okubo and Schultz, 2006; may also be long in the Z-direction compared to the Parsons et al., 2003; Pollard and Segall, 1987; Reches, step width. We use this observation to justify the plane 1987; Segall and Pollard, 1980, 1983b). strain assumption. We use Abaqus CAE, a commercial finite element The boundary conditions are imposed in two steps modeling software package, to investigate the (Fig. 14). The first step brings the model to a lithostatic deformation within the contractional step in the Seven pressure of 250 MPa. This is the average of the Gables outcrop. The model geometry, shown in Figure estimated range (100-400 MPa) for the lithostatic 14, is consistent with the conceptual model for pressure during deformation in Bear Creek based on deformation within the step (Fig. 8). The dike is distinct amphibole geobarometry (Ague and Brimhall, 1988). In from the host granodiorite, so that the two rock types the second step, a displacement boundary condition is are defined by different material properties (Table 2), introduced that produces a bulk contraction of 3% taken from the experimental rock mechanics literature across the model. The faults are oriented such that the (Leeder and Pérez-Arlucea, 2006; Morrow and contraction occurs at an angle of 35° to the fault planes. Lockner, 2006). We use an elastic-plastic constitutive This angle is based on field observations of apparently law based on the von Mises yield criterion for perfect conjugate faults with an internal angle of 70°, plasticity (von Mises, 1913). The faults are composed suggesting that the most compressive principle stress is

Table 2: Material properties for finite element model Young’s Modulus Poisson’s Ratio Yield Stress Aplite 60 GPa 0.2 377 MPa Granodiorite 74.8 GPa 0.2735 377 MPa

Stanford Rock Fracture Project Vol. 24, 2013 H-13 oriented at an angle of 35° from the fault planes. In which corresponds exactly to the outcrop addition, the orientation of wing cracks, which measurements. The modeled step increases in width by propagate along a curvilinear path toward the most 1.3 times, compared to 1.8 times in the field, but the compressive stress (Mutlu and Pollard, 2008), has been dike attains less than half the measured offset. The reported to range from 15-35° (Martel and Pollard, reason for the offset discrepancy is that the model uses 1989) to as high as 50° (Bürgmann and Pollard, 1992) the true thickness of the dike (4 cm) rather than the counterclockwise to the fault planes in Bear Creek. We apparent thickness (10 cm) at the outcrop surface. In therefore take 35° to be a reasonable angle between the order for the faults to terminate at the edges of the dike, faults and the orientation of the most compressive the fault overlap in the model is reduced to ~4 cm, stress. rather than ~10 cm as measured at the outcrop, which The model results are shown in Figure 15 and greatly reduces the modeled offset. The modeled dike compared to the outcrop measurements in Table 3. The rotates 41° and lengthens by a factor of 2.4, compared principal plastic strains (Fig. 15) are the eigenvalues of to 44° rotation and a lengthening factor of 3.3 in the strain tensor, which is the non-rotational component outcrop. Overall, the model provides a reasonable first of the deformation gradient tensor (Eq. 2). Thus, approximation to the deformed features observed in the variations in the principal strains indicate that the Seven Gables outcrop. A more detailed analysis deformation gradient tensor also varies and the utilizing a variety of constitutive laws is provided in a deformation is heterogeneous. The model does a good companion paper (Nevitt et al., in prep.). job of reproducing the change in thickness of the dike The mechanical model shows that plastic strain is (0.4 times its initial thickness during deformation), strongly localized with a sharp transition from large

Figure 15. Results from the mechanical model for (a) maximum principal plastic strain and (b) minimum principal plastic strain. The colored contours and white lines indicate the magnitude and directions, respectively, of the strain. The smallest white lines, located outside the step region, indicate the lowest magnitude strains between 0 and 0.08.

Stanford Rock Fracture Project Vol. 24, 2013 H-14 Table 3: Comparison between field measurements and mechanical model results Dike Step widening Dike lengthening Dike rotation Offset thinning Outcrop 0.4 1.8 3.3 44° 42 cm Model 0.4 1.3 2.4 41° 19 cm magnitude strain within the step to no significant strain have clear terminations within the outcrop. The second immediately across the fault traces outside the step model geometry uses the orientation of the deformed (Fig. 15). Both the minimum and maximum principal dike to define the shear plane. Again, the deformed dike plastic strains reach their greatest magnitudes along a has clear terminations, as it abuts against each fault. narrow zone that cuts diagonally through the step, The contradiction between the geometry of natural fault consistent with the second kinematic model in which steps and that assumed by the kinematic model suggests the shear plane is parallel to an internal fault. This zone that the deformation matrix should be used with caution is slightly asymmetric in the mechanical model due to in contexts similar to this. the unequal lengths of the fault segments on either side Although the kinematic analysis neglects many of the step. The magnitude of both principal strains important aspects of the deformation, it does reveal an decreases to zero within approximately 6 cm of the interesting perspective on the terminology used to deformed dike in the center of the step, providing an describe fault steps. Contractional steps are commonly unambiguous measure of heterogeneous deformation. referred to as sites of transpression in the structural In addition to the principal plastic strain values, Figure geology literature (; Elliott et 15 includes the orientations of the maximum and al., 2009; McClay and Bonora, 2001; Miller, 1994). minimum principal plastic strains. These results This study, however, demonstrates that whether or not a indicate that the plastic strain ellipsoid varies in contractional step is consistent with transpression orientation throughout the step, providing a second depends on the definition of the shear plane orientation measure of hetereogeneous deformation. The maximum within a step. If the shear plane is defined as being principal plastic strain axis rotates to become nearly parallel to the step-bounding faults in the Seven Gables parallel to the deformed dike, while the minimum outcrop, the deformation apparently fits the principal plastic strain axis rotates to become nearly transtensional characterization. On the other hand, orthogonal to the deformed dike. transpression is the apparent fit when the shear plane is defined by an internal fault. A central tenet of both Discussion kinematic and mechanical deformation studies is that the arbitrary choice of coordinates should not affect the The kinematic and mechanical models each make magnitudes or directions of the principal strains. Here, several important assumptions that require evaluation. this choice, as imposed by the choice of shear zone Perhaps the most tenuous assumption in the kinematic geometry, leads to a very different deformation matrix models is that of homogeneous deformation; that is, and different principal strains (Fig. 13). each component of the deformation matrix D at one Furthermore, definitions of transtension and point in the deformed body is exactly the same as the transpression suggest there is no extension or corresponding component of D at any other point. This contraction parallel to the shear plane (Sanderson and assumption is inconsistent with three observations from Marchini, 1984). This clearly is not the case for the the Seven Gables outcrop (Fig. 9): (i) the foliation Seven Gables outcrop, regardless of how the model strengthens gradationally toward the center of the step; geometry is defined. For the first model (shear plane (ii) the fault slip decreases to zero at the fault tips; (iii) parallel to the step-bounding faults), the dike the degree of dike thinning is non-uniform throughout experiences significant shortening in the direction the step. These spatial variations in deformation likely parallel to the shear plane. For the second model (shear had an important influence on the dike rotation. This plane parallel to an internal fault), significant extension aspect of the deformation cannot be captured using a occurs in the direction parallel to the shear plane. single deformation matrix and assuming homogeneous Therefore, invoking transtension or transpression deformation throughout the step. kinematics in contexts similar to this may neglect an Another assumption inherent to the kinematic important component of deformation parallel to the analysis is that the shear planes extend indefinitely in shear plane. the x-y plane. This assumption is not consistent with the Like the kinematic model, the mechanical model is step in the Seven Gables outcrop for either of the two also underlain by significant assumptions, including model geometries previously defined. For the first plane strain, the orientation and magnitude of the kinematic model geometry, the shear plane is defined imposed load, the choice of continuum and contact by the orientation of the step-bounding faults, which constitutive laws, and the material properties. The use

Stanford Rock Fracture Project Vol. 24, 2013 H-15 of plane strain in modeling the Seven Gables outcrop is orientation of the modeled shear plane within the step justified by the occurrence of faults near the outcrop may change the classification of the step from that are very long in the Z-direction (addressed in transtensional to transpressional, or vice versa. Thus, Section 6). The load orientation in the model used here these terms may not be appropriate descriptors of is based on field observations of apparently conjugate extensional and contractional steps, respectively. faults and wing cracks; however, the magnitude of the The mechanics-based finite element model provides load is arbitrarily chosen to optimize the model results. a more satisfactory analysis of the deformation within The choice of constitutive law(s) is particularly the step. The model results provide a suitable match to ambiguous in the Bear Creek field area, since the outcrop measurements and indicate that plastic deformation is interpreted to have occurred under strain localizes along a zone that cuts diagonally brittle-ductile conditions. In this paper, the model uses a through the step. This is consistent with the second simple elastic-plastic constitutive law as a first shear zone model (shear plane parallel to internal fault) approximation of the rock behavior. A companion considered in the kinematic modeling. In addition to paper (Nevitt et al., in preparation) builds upon the capturing the rotation of the dike, the mechanical model mechanical model presented here and tests a variety of captures the dike thinning, ductile fabric distribution, possible constitutive laws. Changes in the constitutive and step widening. An important difference between the behavior have a significant impact on the model results. kinematic and mechanical models is that the In addition, the material properties that parameterize the mechanical model accounts for heterogeneous constitutive laws are taken from experimental rock deformation, as documented by the spatial variation in mechanics literature. Conducting experiments on the plastic strain. Because fault steps observed in the polymineralic rocks, such as granodiorite, is field are characterized by strongly heterogeneous challenging at high temperature-pressure conditions due deformation, the mechanical models are a better choice. to partial melting and mineral interactions (Ji and Xia, They also provide a platform to examine the underlying 2002). In addition, strain rates used in experiments are physics of the process, including the causative tectonic usually several orders of magnitude faster than natural stress, the frictional contact conditions on the faults, strain rates. It is therefore necessary to extrapolate and the constitutive properties of the deforming rock. experimental data to natural conditions, which adds uncertainty to the material properties used in Acknowledgments mechanical models. We are grateful for the field assistance provided by On the other hand, a mechanics-based model W. Ashley Griffith, Libby Ritz, Betsy Madden, overcomes many of the limitations of the kinematic Meredith Townsend, Mark McClure, Michael Pollard, analysis by allowing heterogeneous deformation, and Nick Tokach. In particular, we would like to thank requiring fault slip to decrease to zero at the fault tips, W. Ashley Griffith, who introduced J.M. Nevitt to the and allowing non-uniform thinning of the dike. In Bear Creek field area and to the Seven Gables outcrop. addition, the mechanical model enables an accurate Funding for this research was provided by the Stanford description of the shear zone and fault geometries, and Rock Fracture Project and a Stanford McGee Grant. the incorporation of rock properties. Ultimately, J.M. Nevitt is supported by a NSF Graduate Research mechanical models can be used to deduce the Fellowship. relationships between deformation and the causative tectonic stresses, a connection that kinematic models cannot achieve. References

Conclusions Ague, J.J., and Brimhall, G.H., 1988, Regional variations in bulk chemistry, mineralogy, and the compositions of This research makes use of both kinematic and mafic and accessory minerals in the batholiths of mechanical models to investigate deformation within a California: Geological Society of America Bulletin, v. fault step located in the Bear Creek field area. The 100, p. 891-911. kinematic modeling allows us to experiment with two Ando, R., Tada, T., and Yamashita, T., 2004, Dynamic possible shear plane orientations and also to test evolution of a fault system through interactions between different formulations for deformation (simple shear, fault segments: Journal of Geophysical Research-Solid Earth, v. 109. transtension/transpression). The results of both Aydin, A., and Nur, A., 1985, The types and role of stepovers kinematic models have considerable misfit, and neither in strike-slip , in Biddle, K.T., and Christie- provides a convincing advantage over the other. Thus, Blick, N., eds., Strike-Slip Deformation, Basin we advocate that future investigations utilize Formation, and Sedimentation, Volume 37: Tulsa, OK, mechanical models. An interesting result of the USA, Society of Economic Paleontologists and kinematic analysis, however, is that changing the Mineralogists, p. 35-44.

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