Strain partitioning and stress perturbation around stepovers and bends of strike-slip faults: Numerical results

Strain partitioning and stress perturbation around stepovers and bends of strike-slip faults: Numerical results

Accepted Manuscript Strain partitioning and stress perturbation around stepovers and bends of strike-slip faults: Numerical results Hui Wang, Mian Li...

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Accepted Manuscript Strain partitioning and stress perturbation around stepovers and bends of strike-slip faults: Numerical results

Hui Wang, Mian Liu, Jiyang Ye, Jianling Cao, Yan Jing PII: DOI: Reference:

S0040-1951(17)30403-1 doi:10.1016/j.tecto.2017.10.001 TECTO 127639

To appear in:

Tectonophysics

Received date: Revised date: Accepted date:

11 November 2016 29 September 2017 1 October 2017

Please cite this article as: Hui Wang, Mian Liu, Jiyang Ye, Jianling Cao, Yan Jing , Strain partitioning and stress perturbation around stepovers and bends of strike-slip faults: Numerical results. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Tecto(2017), doi:10.1016/j.tecto.2017.10.001

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Strain partitioning and stress perturbation around stepovers and bends of strike-slip faults: Numerical results

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Hui Wang1*, Mian Liu2, Jiyang Ye2, Jianling Cao1,Yan Jing3

1- Institute of Earthquake Science, China Earthquake Administration, Beijing 100036

2- Dept. of Geological Sciences, University of Missouri, Columbia, MO 65211

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3-Key Laboratory of Crustal Dynamics, Institute of Crustal Dynamics, CEA, Beijing

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10085

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* Corresponding authors: Hui Wang ([email protected])

Stepovers and bends elevate deviatoric stresses and localize plastic strain around

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them 

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Highlights

New faults tend to develop around stepovers and bends tend to improve the mechanical efficiency of fault slip



Model results explain first-order strain partitioning of natural pull-apart basins and push-up ranges

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Abstract

Stepovers and bends along strike-slip faults are where push-up ranges and pull-apart basins are formed. They are also commonly where fault ruptures terminate.

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Field study and analogue models suggest that the configuration of faults plays a key role in crustal deformation around bends and stepovers, but the related mechanics of stress perturbation, strain partitioning, and fault evolution remains poorly understood. Here we present results of systematical mechanical models of stress changes and

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strain partitioning around simple stepovers and bends, using three-dimensional

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viscoelasto-plastic finite element code. Our model predicts elevated deviatoric stress around all stepovers and bends, with higher stresses around the restraining ones.

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Narrow stepovers localize strain between the fault gaps to form connecting faults, whereas wide stepovers localize strain on the tips of fault segments so the stepovers

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may evolve into subparallel faults. We explored how various configurations of

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stepovers and bends change the stress field and strain distribution, and show that these results can help explain some key differences between the pull-apart basins in the

Fault.

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Dead Sea Trough and Death Valley, and the push-up ranges along the San Andreas

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1. Introduction

Geometric irregularities of strike-slip faults play an important role in intracontinental tectonics (Cunningham and Mann, 2007; Molnar and Dayem, 2010;

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Sylvester, 1988) and earthquakes process (Harris et al., 2009; Wesnousky, 2006). Although in broad view strike-slip faults may appear as continuous, linear structures, in reality they are almost always discontinuous, often consist of stepped fault segments separated by stepovers, and their surface traces are often bent. Depending

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on the direction of fault step relative to the sense of strike-slip motion, stepovers can

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be compressive (restraining) or dilatational (releasing). Stepovers are important for assessing seismic hazard, because whether or not a fault rupture can propagate from

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one fault segment to another over a stepover will determine the total rupture length, hence the maximum size of potential earthquakes (Harris and Day, 1993; Parsons et

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al., 2003; Shaw and Dieterich, 2007). Stepovers may evolve into continuous fault

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bends as the splay faults propagate and link across the stepovers (Mann et al., 1983; Martel et al., 1988; Zhang et al., 1989). Similar to stepovers, bends can be

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compressive (restraining bends) or dilatational (releasing bends). Strike-slip bends are commonly where crustal deformation is concentrated, forming push-up ranges over restraining bends and pull-apart basins over releasing bends; they are also the sites of termination, and sometimes nucleation, of many earthquake fault ruptures (King and Nabelek, 1985; Wesnousky, 2006).

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ACCEPTED MANUSCRIPT Strike-slip stepovers and bends have been the subjects of intensive studies. Analogue models (clay or sandbox experiments) are commonly used to study long-term evolution of stepovers and bends (e.g. Connolly and Cosgrove, 1999; Cooke et al., 2013; Corti and Dooley, 2015; Dooley and McClay, 1997; Dooley and

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Schreurs, 2012; Hatem et al., 2015; McClay and Bonora, 2001; Wu et al., 2009). These models generally show that the geometry of faults along stepovers and bends, such as the width of the fault steps (gaps), the under- or overlapping of fault segments, and the sharpness of bends, largely control the style and architecture of

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crustal deformation around these structures. However, analogue models, although

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capable of reconstructing detailed strain evolution, do not give direct and quantitative constraints on the three-dimensional (3-D) stress (Dooley and Schreurs, 2012) needed

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for understanding the cause of such crustal deformation. Although theoretic analyses and numerical models have been used to study stress variations along stepovers and

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bends, most of these models are based on elastic or viscoelastic rheology. They are

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useful for simulating short-term fault behavior during seismic cycles, not for studying

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long-term crustal deformation and fault evolution (e.g. Antolik et al., 2004; Duan and Oglesby, 2005, 2006; Finzi and Langer, 2012; Harris and Day, 1993; Oglesby, 2005; Rodgers, 1980; Segall and Pollard, 1980). In this study, we investigated the mechanics of long-term crustal deformation along strike-slip stepovers and bends, using a 3-D viscoelasto-plastic finite element code (Li et al., 2009). By systematically exploring the major model parameters, especially the geometry of the stepoves and bends, we attempt to address two key 3

ACCEPTED MANUSCRIPT questions: 1) How do stepovers and bends change stress states and affect strain partitioning around them? 2) What key parameters control the localization of plastic strain in the off-fault crust, hence the initiation (or reactivation) of faults around stepovers and bends? To illustrate the basic mechanics, we tried to keep the fault

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configurations simple and generic. We show that the insights gained from these models can be helpful to explain some major differences between the pull-apart basins in the Dead Sea Trough and Death Valley, and between various push-up ranges

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2. The Finite Element Model

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along the San Andreas Fault.

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We used the 3-D viscoelato-plastic finite element code (Li and Liu, 2006; Li et

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al., 2009; Liu et al., 2010) to simulate quasi-steady state lithospheric deformation with faults. The governing equations of force balance and rheology, and the numerical

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schemes, are described by Li et al. (2009). The model domain is 600 km long and 400

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km wide, with a 20 km thick brittle (elastoplastic) upper crust and a 40 km lower crustal and upper mantle (viscoplastic) (Figure 1). The large model dimension is to minimize edge effects on the stepover or bend. We use two parameters to define the geometry of stepovers and bends: the aspect ratio given by the width of the step between two fault segments (W) over the fault depth (Df), and the offset angle (θ) (Figure 1). The offset angle can indicate underlapping (θ<0) or overlapping (θ>0). The primary strike-slip faults in the model are simulated as a 400-m thick vertical 4

ACCEPTED MANUSCRIPT layer of fault elements with lower plastic yield strength than that of the surrounding crust. In most cases the faults are 20 km deep, cutting through the upper crust. Hexahedron elements are used in the finite element code. We tested different element sizes and chose the optimal model, which includes 9000 modal points and about 7000

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Serendipity type elements. This model reproduces first-order stress and strain fields reported in previous studies (Li and Liu, 2006; Li et al., 2009; Liu et al., 2010; Ye and Liu, 2017).

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To illustrate the basic mechanics of strain partitioning and stress perturbation around stepovers and bends of strike-slip faults, we kept the model parameters simple.

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Conventional values of the elastic constants are used for the entire model domain: 8.75×1010 Pa for the Young’s modulus and 0.25 for the Poisson ratio (Turcotte and

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Schubert, 2002). For the elastoplastic upper crust, we used a high viscosity (1025 Pa s)

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to simulate the near elastic behavior before plastic yielding. Plastic creeping in the fault zone and localized plastic strain in the crust occur when stress reach the plastic

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strength of the fault and off-fault crust, respectively. For the viscoplastic lower layer,

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we used Newtonian viscosity in the range of 1019-1021 Pa s (2.0×1020 Pa s in most cases) based on the inferred values for the lower crust and upper mantle (Flesch et al., 2000; Meade and Hager, 2005; Pollitz et al., 2001; Ryder et al., 2011).

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Figure 1. Numerical mesh and boundary conditions of the finite element model. The black lines are strike-slip faults. The upper layer (light grey) represents the brittle

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(elastoplastic) upper crust; the lower layer (dark grey) represents the viscoplastic

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described in the text.

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lower crust and upper mantle. The fault geometric parameters Df, W and θ are

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A key feature of this model is plastic strain outside the faults. This is the strain the crust has to accommodate to compensate for the reduced slip on the faults because of the stepover or bends, hence it is a measure of the mechanical inefficiency of the faults. Such plastic strain also indicates where new fault may develop, or old fault reactivate. We used the Drucker-Prager plastic model, in which the plastic yield depends on both shear stress and mean normal stress (Drucker and Prager, 1952). The plastic yield criteria are characterized by the values of cohesion and internal friction. 6

ACCEPTED MANUSCRIPT Low values of cohesion (10 MPa) and internal frictional (zero) are assigned to faults in most cases to simulate steady state frictional creeping within the fault zones. Outside the faults, higher cohesion (50 MPa) and internal frictional (0.4 of frictional coefficient for most cases) are used.

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The model domain is fixed on its right side and loaded by imposing a 50 mm/yr velocity on its left side (Figure 1). The front and back sides of the model domain are free to move in the direction along the strike-slip fault but fixed in other directions.

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The top surface is free; the bottom is free slip in the horizontal direction and fixed in the vertical direction.

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The numerical simulations start with an arbitrary (zero) initial stress. The stress

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increases by the imposed boundary velocity until reaching the plastic yield of the faults and the upper crust. Further loading does not increase stress, which is capped

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by plastic strain. The system then enters a quasi-steady state with nearly constant

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stress and strain rates. By this time, the effect of initial stress is negligible. We then investigate the long-term (steady-state) stress and strain fields around stepovers and

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bends with various geometries and material properties. The major variables explored in our study are presented in Table 1.

Table 1. Model variables for numerical experiments.

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ACCEPTED MANUSCRIPT Fault geometry

Cases

Aspect ratio

Material properties

Fault

Viscosity of offset angle

(step width: fault

obliquity

Fault cohesion lower crust and

(°) depth)

(MPa)

(°)

mantle (Pa s)

2:1 (40 km:20 1

Releasing 0

2.0×1020

90

2:1 (40 km:20 0

2.0×1020

90

km)

3

0

90

km)

4

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4:1 (80 km:20 km)

0

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km)

Releasing stepove

Restraining 2.0×1020

90

10 stepove

Releasing 5

2.0×1020

90

10

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km)

stepover

2:1 (40 km:20

6

stepover

10

2:1 (40 km:20 5

Restraining

10

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6:1 (120 km:20

2.0×1020

stepover

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6:1 (120 km:20

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4:1 (80 km:20 km)

10

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km)

2

Fault type

Restraining 5

2.0×1020

90

10

km)

stepover

2:1 (40 km:20 7

0

2.0×1020

30

10

km)

Releasing stepover

2:1 (40 km:20 8

0

135

2.0×10

km)

8

20

10

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0

90

4.0×1019

10

0

90

1.0×1021

10

km)

2:1 (40 km:20 10

Releasing

km)

stepover 2.0×1019 for right block and 0

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2:1 (40 km:20 11

90

10

2.0×1021 for

km)

left block

2:1 (40 km:20 12

0

2.0×1020

30

2:1 (40 km:20 0 km)

2:1 (40 km:20 0 km)

km)

90

2.0×1020

10

2.0×1020

10

stepover

Releasing bend

0

135

2.0×1020

10

0

30

2.0×1020

10

0

90

2.0×1020

10

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2:1 (40 km:20 15

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14

30

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13

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km)

Releasing

20

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2:1 (40 km:20 16

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km)

2:1 (40 km:20

17

Restraining

km)

bend

2:1 (40 km:20 18

0

2.0×1020

135

km)

9

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3. Reference Models of Stepovers

To illustrate the basic mechanics of deformation around stepovers, we first present two simple models (cases 1 and 2 in Table 1) for releasing and restraining

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stepovers, respectively. These cases serve as the reference models for further investigation of stress and strain changes around stepovers and bends.

3.1. Releasing stepover reference model

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Figure 2 shows the model results of horizontal and vertical velocity of a simple releasing stepover (case 1), when the model has reached a quasi-steady state. The

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horizontal velocity shows a simple block motion: the left block moves coherently relative to the fixed right block; the relative motion is entirely accommodated by

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plastic creeping on the fault (Figure 2a). Around the stepover, however, is a zone of

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velocity gradient that indicates extension (pull-apart) within the stepover.

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Consequently, subsidence is predicted over the releasing stepover (Figure 2b).

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ACCEPTED MANUSCRIPT Figure 2. Model results of a simple releasing stepover, in map view (case 1 in Table 1). (a) Horizontal velocity (arrows and color contours) at the surface relative to the fixed right side of the model domain. Solid lines are fault segments. (b) Vertical

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motion at the surface, negative for subsidence.

The releasing stepover causes asymmetrical stress changes around the fault tips. To illustrate the evolution of stress and strain rates, we show the results at various

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stages when the model is loaded from an initial stress state (zero deviatoric stress everywhere). The stress state remains dominantly strike-slip during the first 5 Kyr of

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loading (Figure 3a). With further loading, it changes to that favoring normal faults

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within the releasing stepover (Figure 3b). The model reaches a quasi-steady state after ~25 Kyr (Figure 3c); the stress pattern and values remain nearly constant with further

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loading. The corresponding off-fault plastic strain rates are shown in Figure 3d-f. No

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plastic strain is predicted during the first 5 Kyr, when stress is below the plastic yield of both the fault and the off-fault crust (Figure 3d). Then the fault plane starts to yield,

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which accommodates the continued loading (Figure 3e). Eventually the imposed relative motion is entirely accommodated by creep on the fault except around the stepover, which causes plastic strain to accumulate outside the faults, mostly localized between the stepover (Figure 3f). The localized strain could initiate new faults, or reactivate old faults, to connect the stepover, probably at high angles relative to the master fault segments (θ ~ 90°) as suggested by the strain patterns.

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Predicted stress (top panels) and strain rates (lower panels) of a simple

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Figure 3.

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releasing stepover (case 1). (a-c) The maximum shear stress (color contours) and the

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three-dimensional stress state at selected nodes, presented with lower-hemisphere stereonet projection similar to that for earthquake focal mechanisms. (d-f) Plastic

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strain rate in off-fault crust, vertically averaged for the upper crust. Time is since the

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beginning of model simulation.

3.2. Restraining stepover reference model

Contrasting to a releasing stepover, a restraining stepover (case 2) impedes the relative block motion and causes contraction around it (Figure 4). While both releasing and restraining stepovers elevate shear stresses around them, the stress is

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ACCEPTED MANUSCRIPT higher and broader under similar conditions (Figure 5). Around the restraining

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stepover the stress state is compressive, favoring reverse faulting.

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Figure 4. Model results of a simple restraining stepover (case 2). (a) Horizontal velocity (arrows and color contours) at the surface relative to the fixed right side of

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the model domain. Solid lines are fault segments. (b) Vertical motion at the surface,

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positive for uplift.

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ACCEPTED MANUSCRIPT Figure 5. Predicted stress (top panels) and strain rates (lower panels) of a simple restraining stepover (case 2). (a-c) The maximum shear stress (color contours) and the three-dimensional stress state at selected nodes, presented with lower-hemisphere stereonet projection similar to that for earthquake focal mechanisms. (d-f) Plastic

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strain rate in off-fault crust. Time is since the beginning of model simulation.

The restraining stepover causes a broader distribution of plastic strain than the

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releasing stepover does under the same conditions (Figure 5d-f). Similar conclusions were also implied in analogue experiments (Cooke et al., 2013; Hatem et al., 2015).

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For example, physical models of pull-apart basins generally show a narrower zone of

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deformation in and around the stepover whereas restraining bends show a broader zone of uplift for the same stepover width and depth to detachment.

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The predicted strain localization around the restraining stepover depends on the

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plastic model used. For the Drucker-Prager model, plastic yield depends on both shear

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stress and the mean normal stress. In the gap of the releasing stepover, the mean normal stress is tensile, which promotes plastic yield; but it is compressive in the restraining stepover, thus impeding plastic yield. The two belts of high plastic strain rates result from a combination of relatively low mean normal stress and high shear stress there. More details on how the mean normal and shear stresses affect plastic yield around a restraining bend are given by Li et al. (2009).

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4. Effects of Model Parameters

We have shown the basic mechanics of crustal deformation and strain localization around stepovers: extension and subsidence around a releasing stepover,

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compression and uplift around a restraining stepover, and localized plastic strain within the stepovers that may eventually connect the stepovers into bends. In this section we explore the effects of key model parameters on these results.

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4.1. Effects of Fault Geometry

Stepovers are geometric irregularities of strike-slip faults, thus fault geometry

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should have important effects on the stress and strain fields, as suggested by previous studies (Connolly and Cosgrove, 1999; Dooley and Schreurs, 2012; Garfunkel, 1981;

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Segall and Pollard, 1980; Tikoff and Teyssier, 1994; Willemse et al., 1996). We

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conducted numerical experiments to explore the impacts of fault geometry.

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4.1.1. The stepover width and fault depth

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Previous theoretic analysis and physical models show that the fault gap between the stepover influence the stress and strain interaction between the fault segments, and so does the depth of these faults (Dooley and Schreurs, 2012; Segall and Pollard, 1980). Here we examine the effects of both the stepover width and fault depth in terms of the aspect ratio (width/depth). Figure 6 (case 3) shows the effects of increased aspect ratios (wider fault separation) on a releasing stepover with the same boundary conditions as in Figure 2. Although the results are similar in both cases, 15

ACCEPTED MANUSCRIPT increasing the aspect ratio reduces the rates of subsidence, which occurs over a broader area. More importantly, when the step is wider, off-fault strain is more diffuse (Figure 6b). When the aspect ratio is 6:1, the stepover prefers to develop into two subparallel faults rather than being connected into a releasing bend (Figure 6d).

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The effects of the aspect ratio are stronger for a restraining stepover (case 4) (Figure 7). As the aspect ratio increases, less uplift is predicted in a smaller area (cf. Figure 4b), and off-fault plastic strain localization may develop into subparallel faults

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rather than forming a bend.

Figure 6. Effect of the aspect ratio (step width: fault depth) on a releasing stepover (case 3). The upper panels are results for a model with a 4:1 aspect ratio. The stepover width is 80 km; the fault depth is 20 km. (a) Vertical velocity at the surface, negative for subsidence. Arrows show horizontal velocities. (b) Plastic strain rates outside the 16

ACCEPTED MANUSCRIPT faults. The lower panels show the vertical velocity at the surface (c) and plastic strain rates (d) for a model with a 6:1 aspect ratio. The stepover width is 120 km; the fault

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depth is 20 km.

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Figure 7. Effect of the aspect ratio (step width: fault depth) on a restraining stepover

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(case 4). The upper panels are results for a model with an aspect ratio of 4:1. (a) Vertical velocity at the surface, positive for uplift. Arrows show horizontal velocities.

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(b) Plastic strain rates outside the faults. The lower panels show the vertical velocity at the surface (c), and plastic strain rates (d) for a model with a 6:1 aspect ratio.

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ACCEPTED MANUSCRIPT 4.1.2. Fault obliquity

Most natural faults are not exactly parallel to the direction of relative crustal motion across the fault. The obliquity between fault strike and the direction of relative crustal motion could have important effects on strain partitioning across strike-slip

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fault systems (Garfunkel, 1981; Tikoff and Teyssier, 1994). Here we show two models with oblique faults, one for a releasing stepover (case 5) and one for a restraining stepover (case 6) (Figure 8). These models have the same stepover width

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and fault depth as the reference models, except that the faults are oblique to the direction of block motion by 5°, a value based on that of a natural fault (Garfunkel,

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1981) and used in a previous analogue model (Wu et al., 2009).

Figure 8. Effects of fault obliquity for a releasing stepover (case 5, upper panels) and a restraining stepover (case 6, lower panels). 18

(a, d) Vertical velocity at the surface,

ACCEPTED MANUSCRIPT negative for subsidence. Arrows show horizontal velocities. (b, e) The maximum shear stress (color contours) and the three-dimensional stress states (“beach balls”) as

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explained in Figure 3. (c, f) Plastic strain rates outside the faults.

Comparing the results with those in Figures 2-5, one can see that fault obliquity causes broader and more diffuse vertical deformation (Figures 8a and 8d) for both releasing and restraining stepovers. Similar results have been found in analogue

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models (Dooley and Schreurs, 2012; Wu et al., 2009). Fault obliquity also causes the stress state to be transtensional around the releasing stepover (Figure 8b) and

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transpressive around the restraining stepover (Figure 8e). The resulting plastic strain

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models (Figures 2-5).

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outside the fault zones are more diffuse (Figures 8c and 8f) than that in the reference

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4.1.3. Overlapping and underlapping faults

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Another geometric factor influencing the development of stepovers is the degree of over- and underlapping of the master faults (Connolly and Cosgrove, 1999; Dooley and McClay, 1997; Willemse et al., 1996). Underlapping occurs when the offset fault segments approach each other such that their strain fields interact. Overlapping is when the fault tips have passed each other. In our models, underlapping and

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ACCEPTED MANUSCRIPT overlapping are indicated by the offset angle θ (Figure 1). The fault segments underlap when θ is less than 90°, and overlap when θ is greater than 90°.

Figure 9 shows two cases (case 7 and case 8, Table 1) for a releasing stepover. In the case of underlapping master faults, the strain fields of the separated fault tips

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interact with each other, as seen in the connected area of subsidence (Figure 9a). The stress field is dominantly strike-slip everywhere except near the fault tips, where it is extensional (Figure 9b). The localized strain between the fault tips (Figure 9c) is

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higher than that for the reference model (Figure 3). This high strain is needed to compensate for the strain in the stepover that cannot be accommodated by the

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underlapping faults. When the fault tips are closer to each other, the off-fault strain is weaker because the fault can accommodate more relative block motion. Once the fault

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segments overlap, the strain partitioning and stress perturbation around the stepover

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changed significantly. The subsidence is concentrated within the stepover (Figure 9d), and the maximal subsidence rate is a little higher than that in the underlapping

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stepover. The overlapped faults also cause slightly higher shear stress than that in the

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underlapping case (Figure 9e), and the plastic strain within the stepover is weaker (Figure 9f), because the relative motion is more effectively accommodated when the faults overlap.

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Figure 9. Predicted results of overlapping (case 7, upper panels) and underlapping (case 8, lower panels) releasing stepovers. (a, d) Vertical motion, negative for

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subsidence. Arrows show horizontal velocities. (b, e) The maximum shear stress

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(color contours) and the three-dimensional stress states (“beach balls”). (c, f) Plastic

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strain rate outside the faults.

These results are consistent with previous studies (Corti and Dooley, 2015; Hu et al., 2016; Mann et al., 1983). Numerical models of fault interaction show that closely spaced overlapping faults act more efficiently in accommodating fault slips (Crider and Pollard, 1998; Willemse et al., 1996). Analogue models have shown that underlapping releasing sidestepping faults (offset angles of 30–75°) typically form elongate rhomboidal grabens; releasing offset with ~90° sidesteps produce shorter, 21

ACCEPTED MANUSCRIPT squat, rhomboid pull-apart basins; and overlapping releasing sidesteps (115–150°) produce box-like grabens with highly linked basin sidewalls (Corti and Dooley, 2015;

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Dooley and McClay, 1997).

4.2. Effects of Material Property

We also explored the effects of material properties on the model results. In our viscoelasto-plastic model, the key material properties include viscosity and plastic

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yield (determined by frictional coefficient and cohesion). We show here only results

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of releasing stepovers; the effects on restraining stepovers are similar.

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4.2.1. Viscosity

In the reference model (case 1) the viscosity for the lower crust and upper mantle

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is set to be 2.0×1020 Pa s. Lowering the viscosity to 4.0×1019 Pa s (case 9) promotes

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strain localization in the stepover (Figure 10a-c). The area of subsidence is smaller, and the rates higher, than those in the reference model (Figures 2), whereas the stress

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patterns (Figure 10b) are similar. The plastic strain is more localized in the stepover when the viscosity is lower (Figure 10c). Conversely, a higher viscosity (1.0×1021 Pa s, case 10) leads to more diffuse and higher shear stress around the stepover (Figure 10e), with more plastic strain distributed away from the stepover, mostly along two belts of localized strain that extend from the fault tips (Figure 10f). 22

ACCEPTED MANUSCRIPT In another case, we examined the effects of contrasting viscosity across the strike-slip fault (case 11) (Figure 10g-i). Here the viscosity is 2.0×1019 Pa s for the right block and 2.0×1021 Pa s for the left block. The asymmetric rheology produces broader subsidence and higher shear stress in the weaker right block. The distribution

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of off-fault plastic strain is also slightly broader in the right block (Figure 10i).

Figure 10. Effect of rheology contrast on a releasing stepover. The upper panels are results of lowering the viscosity relative to the reference model in Figure 2. (a) Vertical velocity at the surface. Arrows show horizontal velocities. (b) The maximum shear stress (color contours) and stress states (“beach balls”). (c) Plastic strain rates 23

ACCEPTED MANUSCRIPT outside the faults. The middle panels (d-f) show results of increasing the viscosity. The lower panels (g-i) are results of a case of contrasting viscosity: the viscosity is 10

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times higher in the right side of the fault than in the left side. See text for details.

Similar results are reported in analogue studies (Eisenstadt and Sims, 2005; Withjack et al., 2007). For example, dry sand with low ductility has been shown to

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cause wider deformation zone outside the major fault (Withjack et al., 2007).

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4.2.2. Fault strength

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In our models, plastic yielding occurs in both the fault zones and the surrounding crust, with different plastic yield strength (in terms of cohesion and internal friction).

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Figure 11 shows the effects of doubling the fault strength (case 12). The higher fault

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strength causes more diffuse deformation around the stepover, because less slip is accommodated on the fault plane. Relative to the reference model (case 1, Figures 2

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and 3), the subsidence is broader and rates lower (Figure 11a); the maximum shear stress is higher around the stepover (Figure 11b), and more plastic strain is distributed away from the stepover (Figure 11c).

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Figure 11. Model results of a releasing stepover with higher fault strength than that in the reference model (case 1, Figure 2). (a) Vertical motion, negative for subsidence. Arrows show horizontal. (b) The maximum shear stress (color contours) and stress

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states (“beach balls”). (c) Plastic strain rate outside faults.

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5. The mechanics of fault bends

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We have shown that, when the fault segments are sufficiently close across a

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stepover, strain would localize within the stepover that may form connecting faults, turning the stepover into a bend. Here we investigate how stepovers may evolve into

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bends.

Figure 12 shows the results of three cases of different releasing bends (cases 13 -15). The first case features a long, gentle bend similar to the case of an underlapping stepover (Figure 9a-c). In contrast to the stepover, the bend reduces the plastic strain outside the faults because, with the fault segments being connected, the bend is more efficient in accommodating the relative block motion. The same can be said for the

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ACCEPTED MANUSCRIPT second case (Figure 12d-f), when compared with the stepover in Figures 2 and 3. When the overlapping stepover is connected into a bend (Figure 12g-i), it is more efficient in accommodating the relative block motion than the stepover (cf. Figure 9d-f), hence the lower shear stress and off-fault plastic strain.

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Similarly, when restraining steopovers are connected to become restraining bends (cases 16-18), the relative block motion is more efficiently accommodated by the fault system, so off-fault plastic strain is reduced (Figure 13). Similar to

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restraining stepovers, restraining bends cause localization of plastic strain in two belts that extend from the bends, indicating the fault system’s tendency to either straighten

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the master fault or initiate new faults to more efficiently accommodate the relative

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block motion.

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Figure 12. Model results of releasing bends with different bend angles (cases 13-15).

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The upper panels show a case of a gentle bend formed by connecting the

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underlapping stepover shown in Figure 7a-c. The middle panels show a case with the bend formed by connecting the stepover shown in Figure 2. The lower panels show results of an overlapping releasing bend. The left panels (a, d, g) show vertical motion. Arrows show horizontal velocities. The middle panels (b, e, h) show the maximum shear stress (color contours) and stress state (“beach balls”). The right panels (c, f, i) show plastic strain rate outside the faults.

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Figure 13. Model results of restraining bends with different bend angles (cases 16-18).

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The upper panels show the results of a gentle restraining bend; the middle panels are

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results of the restraining bend formed by connecting the restraining stepover shown in Figure 4, and the lower panels show the results of an overlapping restraining bend. (a, d, g) uplift (background color) and horizontal velocities (arrows); (b, e, h) the maximum shear stress (color contours) and stress states (“beach balls”); (c, f, i) plastic strain rate outside the faults.

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6. Applications to natural strike-slip faults

We have kept our models simple and generic to illustrate the basic mechanics controlling strain partitioning and stress perturbation around stepovers and bends. In

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this section we apply these models to some natural stepovers and bends. The purpose here is not to investigate the detailed structures around these features, which are certainly much more complicated than those represented in our models. Rather, we show that some of the major structures in nature can be understood from the basic

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mechanics explored in the simple models.

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6.1. Pull-apart basins: the Dead Sea Trough and the Death Valley

Here we compare two pull-apart basins associated with releasing stepovers: the

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Dead Sea Trough and the Death Valley in California. The Dead Sea Trough formed

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around a releasing stepover on the Dead Sea fault system (Sobolev et al., 2005; Ten

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Brink et al., 1999; Ten Brink and Zvi, 1989), which is the left-lateral transform boundary between the African Plate to the west and the Arabian Plate to the east

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(Gomez et al., 2006). It is an overlapping stepover; the width of the fault step is less than 10 km, and the length to width ratio is ~7 (Gurbuz, 2010). The orientation of Dead Sea Trough is parallel along the main Dead Sea Fault system, and its geometry is elongated and nearly rectangular. Death Valley, on the other hand, is an underlapping steopover located in the eastern California shear zone (Burchfiel and Stewart, 1966; Jones, 1987). The spacing between strike-slip faults over the stepover is almost 40 km. The Death Valley pull-apart basin is highly oblique to the transform 29

ACCEPTED MANUSCRIPT direction. Given their similar crustal thickness (35-40 km) (Fliedner et al., 1996; Group et al., 2004), the ratios of the strike-slip fault spacing over crustal thickness of these two stepovers represent end-member cases for stepover geometry (Smit et al., 2008).

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We adapted our model for the Dead Sea and the Death Valley stepovers with their first-order geometry (Figure 14). The lateral loading is 6 mm/yr for the Dead Sea stepover model (Garfunkel and Ben-Avraham, 1996; Petrunin and Sobolev, 2006) and

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12 mm/yr for the Death Valley stepover model (Shen et al., 2011). The fault strength (10 MPa for cohesion) is chosen to fit the predicted surface velocity with that based

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on GPS measurements in both cases.

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The models predict patterns of crustal deformation comparable to those developed in the Dead Sea and the Death Valley pull-apart basins (Figures 14b and

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14e). As predicted in our simple models of different stepover widths and under- or

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over-lapping (cases 7-8), the narrow and overlapping Dead Sea stepover produces an elongated, nearly rectangular pull-apart basin between the two strike-slip fault

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segments (Figure 14b). The predicted maximum subsidence rate is ~1.3 mm/yr, which is in the same order of magnitude of long-term tectonic subsidence rate in the Dead Sea (Brink and Flores, 2012). The predicted rates of vertical motion need to be taken with caution, because some important processes, including isostasy and erosion, are not included in the model.

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ACCEPTED MANUSCRIPT In contrast, our model for the underlapping Death Valley stepover predicts diffuse subsidence around the step, with its orientation highly oblique to the master faults. The predicted maximum subsidence rate is about 2.5 mm/yr. These results are broadly consistent with field studies (Burchfiel et al., 1987; Burchfiel and Stewart,

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1966; Jones, 1987).

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Figure 14. Model results of strain patterns for the Dead Sea Trough (upper panels) and the Death Valley (lower panels) releasing stepovers. (a, d) Topographic relief and major faults (lines). (b, e) Vertical motion (color contour, negative for subsidence) and horizontal velocities (arrows). Heavy lines are the fault segments included in the numerical model. (c, f). Plastic strain rate outside the modeled faults. The numbers are predicted fault slip rates. DSES: Dead Sea eastern strand; DSWS, Dead Sea western

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ACCEPTED MANUSCRIPT strand; NDVF, Northern Death Valley fault zone; SDVF, Southern Death Valley fault zone.

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The different patterns of subsidence and faulting between the Dead Sea Trough and the Death Valley pull-apart basins can be largely explained by their fault geometry. On the Dead Sea faults, the master faults accommodate most (>90%) of the lateral loading. The pull-apart basin is shallow and developed largely between the

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master strike-slip faults. It indicates that the narrow and overlapping Dead Sea faults are efficient in accommodating the relative block motion. In contrast, the wide gap

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between the strike-slip fault segments across the Death Valley stepover

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accommodates <70% of relative motion along the faults. The rest is accommodated by localized plastic strain and subsidence in a broad region between the master faults.

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Thus the stepover geometry is a key factor for the broad pull-apart basin in the Death

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Valley, although the complex and variable tectonic history in this area is also

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important (Smit et al., 2008).

6.2. Push-up ranges: restraining bends on the San Andres Fault

We have shown that restraining stepovers and bends impede relative motion along strike-slip faults, causing crustal compression. Here we apply our model to two restraining bends along the San Andreas Fault: the Big Bend in southern California

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ACCEPTED MANUSCRIPT (Figure 15) and the restraining bend in the Santa Cruz mountains, northern California

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(Figure 16).

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Figure 15. (a) Topographic relief and major faults (lines) around the Big Bend in southern California. Numbers show geological slip rates (data from Calif. Geol. 2002,

http://www.consrv.ca.gov/CGS/rghm/psha/index.htm).

(b)

The

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Survey,

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predicted plastic strain outside the modeled faults (thick lines). The numbers are predicted slip rates on the faults. Circles show earthquakes. SAF: the San Andreas

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Fault; SJF: the San Jacinto Fault; EF: the Elsinore fault; ECSZ: the eastern California

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shear zone; PV &CB: Palos Verdes and Coronado Bank Fault zone; GF: the Garlock Fault; WTF: the western Transvers Ranges; SN: Sierra Nevada block.

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ACCEPTED MANUSCRIPT Figure 16. (a) Topographic relief and major faults (lines) of the Santa Cruz bend of the San Andreas Fault (SAF). The beach ball shows the focal mechanism of the 1989 Loma Prieta earthquake. Blue lines show regional active faults. (b) Predicted plastic strain rates outside the modeled faults (thick line); The numbers are predicted slip

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rates on the faults. The beach ball shows the predicted three-dimensional stress state at epicenter of the 1989 Loma Prieta earthquake. (c) Predicted vertical motion,

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positive for uplift.

The Big Bend is a major restraining bend along the San Andres Fault, formed ~5

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Ma ago as a result of the opening of the Gulf of California and involving complicated

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evolution (Atwater and Stock, 1998). Previous models have shown that the Big Bend causes high stress over a broad region, which explains the diffuse seismicity in

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southern California (Li and Liu, 2006). The Big Bend also localizes strain along the

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eastern California shear zone (Liu et al., 2010). Our models of simple restraining stepovers (case 2, Figure 5) and bends (case 16-18, Figure 13) reproduce the basic

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features of broadly distributed shear stresses and localized plastic strain. In a model adapted for the Big Bend, we included three dextral strike-slip faults subparallel to the southern SAF: the San Jacinto Fault, the Elsinore Fault, and the Palos Verdes and Coronado Bank fault zone (Figure 15).

Adding these faults reduced the off-shore strain localization that was not observed but predicted in the previous models with only the SAF proper included (Li 34

ACCEPTED MANUSCRIPT et al., 2009; Liu et al., 2010). The most significant result is the localized plastic strain in the western Transverse Ranges – a gap between the northern end of the Big Bend and these subparallel dextral faults (Figure 15b). Strain localizes here to make up for the inefficiency of fault displacements because of the fault gap. The results are similar

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to strain concentration within underlapping restraining stepovers (case 7, Figure 9a-c). Similar results are also found in a more specific model tailored for southern California plate boundary zone (Ye and Liu, 2017). The predicted strain localization in western Transverse Ranges is consistent with geological evidences of active deformation there

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(Hubbard et al., 2014; Seeber and Armbruster, 1995), although the surface strain rates

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are difficult to measure because of many blind thrust faults (Namson and Davis, 1988; Rodgers, 1979).

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The restraining bend across the Santa Cruz Mountains in northern California, on

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the other hand, is a relatively small and gentle bend along the San Andreas Fault where the 1989 Loma Prieta earthquake (Mw 6.9) occurred (Anderson, 1990;

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Schwartz et al., 1990) (Figure 16). Here the surface trace of the SAF changes trend

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from 320° to 310°. Adapting this fault configuration, our model elevated deviatoric stresses around it, but because of the small bend angle (θ), the predicted strike-slip on the bend is only ~2 mm/yr (about 10%) less than that on the northern and the southern segments. The plastic strain is weakly localized around the bend, without producing zones of strain localization away from the bend as predicted for sharper bends (Figure 16b). Similar results are found in analogue experiments that show strike-slip fault running through a bend without producing new faults, when the bend angle is <15° 35

ACCEPTED MANUSCRIPT (Cooke et al., 2013). The model predicts compressive stress state in the epicentral region of the 1989 Loma Prieta earthquake (Figure 16b), although the details differ because of model simplifications. The model also shows a narrow uplift zone in the

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Santa Cruz Mountains with <1.0 mm/yr uplift (Figure 16c).

7. Discussion

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Crustal deformation around stepovers and bends of strike-slip faults has been extensively studied, mostly via analogue modeling. These analogue models are

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usually designed for specific pull-apart basins or push-up ranges. While providing

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detailed strain evolution that can be constrained by local geology, analogue models offer no direct and quantitative information for why crust deforms in different ways

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around different stepovers and bends. In this study we investigated the basic

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mechanics for crustal deformation around stepovers and bends by numerically simulating the quasi-steady state displacement, stress, and strain fields around

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stepovers and bends along strike-slip faults with various fault configurations. In particular, by simulating plastic strain localization outside the faults, these models predict where new faults may initiate or old faults reactivate. Although the models presented here are simple and mostly generic, they produce the general strain patterns comparable with those from analogue models and field observations for similar fault configuration of stepvers and bends, thus the modeled stress and strain fields can be

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ACCEPTED MANUSCRIPT helpful for understanding the cause of the variable crustal deformation around different stepovers and bends.

7.1. Effects of 3D fault geometry on fault evolution

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Our results suggest that the geometry of stepovers is a primary factor determining how plastic strain would be localized to form connecting faults or subparallel faults (Figures 2-7). These results provide a mechanical understanding to similar deformation reported in previous analogue models and field observations.

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Analogue experiments showed that strain is partitioned between fault slip and

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off-fault deformation (Cooke et al., 2013), and deformation is often concentrated in the pull-apart basin produced by releasing stepover (Dooley and McClay, 1997).

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Analogue models also show the angle of offset between the master fault segments is an important parameter controlling the architecture of pull-apart basins (e.g. Corti and

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Dooley, 2015). Wider restraining stepover have lower mechanical efficiency than

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narrow stepovers (Cooke et al., 2013), and multiple new faults may propagate within

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high-angle restraining bends (Hatem et al., 2015). Our models explain these results in terms of off-fault plastic strain distribution arising from stress changes around the stepovers.

Some previous numerical and physical models have shown the width of stepovers as a controlling factor for deformation around stepovers, whereas others emphasized fault depth or the thickness of the brittle lithosphere (Crider and Pollard, 1998; Petrunin and Sobolev, 2006; Petrunin and Sobolev, 2008; Vaghri and Hearn, 37

ACCEPTED MANUSCRIPT 2012; Willemse et al., 1996). We have shown that the combined effects of the stepover width and fault depth can be represented by the aspect ratio (stepover width: fault depth), because both parameters indicate the extent to which the stress fields of the master faults interact. Stepovers with bigger aspect ratios (wide steps and shallow

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faults) have weaker stress interaction, hence less strain localization in the steps. This is consistent with previous results of numerical (Petrunin and Sobolev, 2006) and analogue models (Corti and Dooley, 2015) that suggest that a thinner brittle

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lithosphere, or shallower faults, produce a shallower pull-apart basins.

Whereas in most cases only vertical faults are used in our model, other studies

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have shown that the fault dip also affects the shape of subsidence or uplift around the stepovers or bend (Choi et al., 2011). In particular, the presence of subhorizontal

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detachments can have significant impact on bend/stepover evolution (Smit et al.,

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2008).

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7.2. Model limitations

To illustrate the basic mechanics, we kept the models simple. In most cases, we used homogeneous lithospheric viscosity and vertical faults. The lithosphere in nature is certainly heterogeneous. In case 11 we modeled a simple viscosity contrast across the fault (Figure 10g-i), the results are consistent with previous studies that show concentrated and higher strain rate in the weaker regions and more defuse strain in the stronger regions (Dayem et al., 2009; Le Pichon et al., 2005; Sobolev et al., 2005). 38

ACCEPTED MANUSCRIPT The rheologically induced strain localization may cause abandonment of old faults and propagation of new faults in the more active side of stepovers and bends to more efficiently accommodate deformation (Wakabayashi et al., 2004).

Different boundary conditions would affect modeling results. In our models we

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used the relative motion across the strike-slip fault as the boundary conditions and the driving force. This is justifiable, because our main goal here is to show how the relative block motion is accommodated by the evolving strike-slip fault. Other driving

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forces, such as the gravitational potential energy and basal shear, are not explicitly explored in our model but are implied in the model’s requirement of force balance. In

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doing so our results cannot directly show the effects of laterally variable gravitational potential energy, or basal shear, which are shown to affect the shape of the pull-apart

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basins (Brink et al., 1996; Katzman et al., 1995) and the orientation of the principal

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stress axes (Lynch and Richards, 2001). In some regions basal shear may be an important driving force (Al-zoubi and Brink, 2002; Brink and Flores, 2012; Platt et al.,

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2008; Smit et al., 2008; Teyssier and Tikoff, 1998). However, basal shear is difficult

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to constrain, so we choose to use side loading, in terms of imposed horizontal velocity, as the loading mechanism and boundary condition, which can be directly constrained by geodetic or geological observations in most regions.

7.3. Implications for strike-slip fault evolution and seismicity Our model results can be understood in the light of the strike-slip faults’ tendency to maximize their mechanical efficiency (Cooke and Madden, 2014). In our models, 39

ACCEPTED MANUSCRIPT the mechanical efficiency of the fault system is indicated by the off-fault plastic strain – less efficient fault system produces more off-fault strain. Stepovers and bends are irregularities of strike-slip faults that impede the relative block motion across the fault. As a result, plastic strain localizes in the surrounding crust to activate old faults or

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initiate news faults, all reflecting the fault system’s self-adjustment to accommodate the relative block motion. Similar conclusions were reached by Cooke et al. (2013) from analogue modeling. Thus our models show that, when the stepover is narrow, strain localizes across the stepover to form a cross-stepover fault that effectively

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straightens the fault zone. But when the stepover is too wide, subparallel faults may

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develop. When the bends are too sharp to efficiently accommodate the relative block motion, strain localizes in belts subparallel to the master strike-slip fault segments to

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initiate new and more optimally oriented faults, as also found in analogue models (Hatem et al., 2015). In this process some old fault strands are abandoned, while new

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faults initiated. These results are helpful for understanding the evolution history of

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many natural strike-slip fault systems.

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Our models are quasi-steady state, meant to simulate long-term geological evolution of faults. Nonetheless, the predicted stress perturbation and strain partitioning around stepovers and bends can be useful for understanding seismicity. For a major strike-slip fault, deviatoric stresses are elevated around stepovers and bends, especially for restraining ones. Li and Liu (2006) showed that the broad distribution of high shear stress around the Big Bend help explain the broadly distributed seismicity in southern California, and Liu et al. (2010) suggested that the 40

ACCEPTED MANUSCRIPT localized strain along the eastern California shear zone, resulting from the Big Bend, may be associated with the recent damaging earthquakes, including the 1992 Lander (Mw 7.3) and 1999 Hector Mine (Mw 7.1) earthquakes, across the Mojave Desert. The predicted stress variations around stepovers and bends could also provide the

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background stress field needed to better modeling dynamic ruptures. Whether or not fault ruptures can propagate through bends and jump over stepovers determines the maximum size of potential earthquakes (Liu and Stein, 2016; Wesnousky, 2006). The initial stress field along stepovers and bends is one of the key parameters for

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modeling dynamic ruptures along these fault irregularities (Duan and Oglesby, 2005;

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Harris et al., 2009). Most rupture models used homogeneous or some assumed initial stress for lack of constraints, the steady-state stress fields around various stepovers

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and bends presented here could be used for improving the modeling of dynamic

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ruptures.

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8. Conclusions

Using a three-dimensional viscoelasto-plastic finite element model, we have systematically investigated the stress perturbation, strain partitioning, and localization of plastic strain around stepovers and bends, and their implications for fault evolution and crustal deformation. We found that fault geometry plays a key role in controlling how the fault system evolves. Stepovers and bends along a strike-slip fault impede the relative block (crustal or lithospheric) motion, elevating deviatoric stress in the 41

ACCEPTED MANUSCRIPT surrounding region, and localize plastic strain outside of the active faults that may reactivate old faults or initiate new faults to improve the mechanical efficiency of the fault systems. A stepover may develop into a bend when the fault segments are deep and close. Otherwise it tends to develop into subparallel strike-slip faults.

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A bend forms when a stepover is linked by faults initiated or reactivated by strain localization between the stepover. It is mechanically more efficient in accommodating the relative block motion than the stepover. A gentle bend may

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elevate the deviatoric stress and cause transpression or transtension around it, but is unlikely to produce new faults. A sharp bend, however, tends to produce two belts of

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localized strain that may lead to formation of new and more optimally oriented faults. These model results illustrate the strike-slip faults’ self-adjustment to minimize the

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results from other workers.

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mechanical work in accommodating the relative block motion, in agreement with the

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Adapting the models for the Dead Sea and the Death Valley, we have shown that the different pull-apart basins developed there result primarily from their fault

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geometry: the narrow and overlapping Dead Sea stepover produces the narrow pull-apart basin largely bounded by the strike-slip faults, whereas the wide gap over the Death Valley stepover causes the broad pull-apart basin there. The restraining bends of the Big Bend and that across the Santa Cruz Mountains both elevate deviatoric stresses around them. The gentle bend in the Santa Cruz Mountains allow the San Andreas Fault to remain as the dominant path for relative block motion, without the need to produce new faults around the bend. In contrast, the abrupt Big 42

ACCEPTED MANUSCRIPT Bend causes strong strain localization across the Mojave Desert and in the western Transverse Ranges, where numerous damaging earthquakes occurred in recent years.

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Acknowledgments. We thank Kelin Wang, Tim Dooley, and an anonymous reviewer for their constructive comments, which improved the manuscript. This work was supported by Basic Research Project of Institute of Earthquake Science, CEA (Grant 2014IES0102), and National Natural Science Foundation of China (Grants 41774111, 41604062). M. L. acknowledges support from NSF (grants 0730154 and 1519980), and National Natural Science Foundation of China (Grant 41374104).

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