# The role of frictional plasticity in the evolution of normal fault systems
**Authors:** David E. Dempsey, Susan M. Ellis, Julie V. Rowland, Rosalind A. Archer
**Published in:** Journal of Structural Geology (2012)
**DOI (version of record):** https://doi.org/10.1016/j.jsg.2012.03.001
**This copy:** Accepted manuscript (peer-reviewed, author's version). Text extracted from the PDF at https://ddempsey.github.io/papers/2012-dempsey-role-frictional-plasticity-evolution-normal/dempsey2012-role-frictional-plasticity.pdf; equations, tables and figure placement may be imperfect, so cite the DOI.
**Notice:** Accepted manuscript. © 2012. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/. The version of record is available at https://doi.org/10.1016/j.jsg.2012.03.001
**Licence:** CC BY-NC-ND 4.0 https://creativecommons.org/licenses/by-nc-nd/4.0/
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- 1 **The role of frictional plasticity in the evolution of normal fault systems**
- 2 David E. Dempseya* , Susan M. Ellisb , Julie V. Rowlandc , Rosalind A. Archera
- 3
- 4 a Department of Engineering Science, University of Auckland, Private Bag 92019,
- 5 Auckland 1142, New Zealand
- 6 b GNS Science, 1 Fairway Drive, Lower Hutt 5011, New Zealand
- 7 c School of Environment, University of Auckland, Private Bag 92019, Auckland 1142,
- 8 New Zealand
- 9
- 10 **Abstract**
- 11 Tectonic extension applied to the upper crust commonly results in the development of
- 12 sub-parallel arrays of faults and fault belts. A computational model of an extending
- 13 elastic-viscous-plastic crust is presented, which describes the evolution of stress, strain
- 14 and displacement in the vicinity of an active normal fault. Within the upper crust discrete
- 15 episodes of fault slip accommodate up to half the applied extension rate, with the balance
- 16 accommodated by bands of frictional plastic shearing representing young proto-faults.
- 17 Repeated coseismic shear stress reductions in the near-fault vicinity impose a stress-
- 18 failure shadow where the formation of new faults is prohibited. A near-surface
- 19 compressional stress regime in the footwall causes the failure shadow to extend
- 20 disproportionally into this fault block, thereby favoring the formation of new faults in the
- 21 hanging wall. A continuous shallowing of fault dip is also documented and supports a
> * Corresponding author: Ph: +64 9 373 7599 ext 87061, Fax: +64 9 373 7468, email: d.dempsey@auckland.ac.nz
- 22 scenario of migration of faulting into the hanging wall of existing faults, consistent with
- 23 descriptions of the structural evolution in the western Gulf of Corinth, Greece.
- 24
- 25 **Keywords**
- 26 Normal fault, frictional plasticity, fault rotation, numerical model, extension budget
- 27
- 28 **1. Introduction**
- 29 Extension of the lithosphere in rifting basins is accommodated through displacement on
- 30 normal dip-slip faults or by the emplacement of dikes, i.e., intrusive magmatism. Such
- 31 structures are observed at rifting at both convergent (Taymaz et al., 1991; Wilson et al.,
- 32 1995) and divergent plate margins (Ebinger, 1989), and intra-plate settings (Illies, 1972;
- 33 Puskas et al., 2007). Over time the character of normal faulting evolves, with individual
- 34 fault strands organizing into sub-parallel en-echelon arrays and, with continued linkage,
- 35 into large fault belts (Cowie et al., 2005). However, fault planes may also become
- 36 inactive (Jackson, 1999). Extension can cause fault planes to rotate about a horizontal
- 37 axis, shallowing the dip angle to a point where continued reactivation of the fault
- 38 becomes unlikely (Sibson, 1985). In addition, stresses generated by an existing fault have
- 39 been hypothesized (Goldsworthy and Jackson, 2001) to induce a migration of faulting in
- 40 a basinward direction (Dart et al., 1995; Horton and Schmitt, 1998; Goldsworthy and
- 41 Jackson, 2001; Sanchez et al., 2010).
- 42
- 43 During discrete episodes of dip-slip on normal faults, extension is accommodated by
- 44 translation of one fault block relative to the other. Displacement is primarily parallel to
- 45 the fault plane, and therefore horizontal movements are accompanied by vertical
- 46 displacements. This behavior translates to uplift of the footwall block and a
- 47 comparatively larger subsidence of the hanging wall (e.g., Barr, 1987). Conceptual
- 48 models employ lithospheric flexure (Weissel and Karner, 1989; Kusznir et al., 1991),
- 49 domino rotation (Wernicke and Burchfiel, 1982), or isostatic rebound (Barr, 1987;
- 50 Westaway, 1992) on either planar or listric fault surfaces to describe this deformation.
- 51 However, field studies indicate that fault slip aggregated across a rift does not always
- 52 account for the full geodetic extension rate (e.g., Villamor and Berryman, 2001; Wallace
- 53 et al., 2004). Additional deformation mechanisms or movements on undiscovered faults
- 54 must account for the shortfall.
- 55
- 56 Normal fault systems evolve and grow over many thousands of years in response to the
- 57 continuous application of far-field tectonic extension (e.g., Villamor and Berryman,
- 58 2001; Bell et al., 2009). Such evolution involves the initiation of new faults (Jackson and
- 59 McKenzie, 1983), which is itself an indication that extension is not completely
- 60 accommodated by existing structures. Potential causes of this shortfall include: a
- 61 migration of the locus of extension away from the faulted region; a suspension of activity
- 62 on pre-existing faults due to fault rotation (Jackson, 1999), restrengthening of the fault
- 63 (Li et al., 2003; Muhuri et al., 2003) or stress shadowing by other faults (Cowie 1998);
- 64 distributed frictional deformation (see section 2, below); or a suspension of alternative
- 65 mechanisms of extension accommodation, e.g., dike emplacement (Rowland et al., 2010).
- 66
- 67 The purpose of this work is to investigate specifically the role that frictional plasticity
- 68 plays in the evolution of a normal fault system. From a theoretical perspective, frictional
- 69 plasticity is the non-recoverable deformation of the crust through frictional sliding on
- 70 hypothetical fracture planes. Such planes are presumed to exist abundantly throughout the
- 71 crust (Ranalli and Murphy, 1987) at orientations that are optimal for reactivation within
- 72 the prevailing stress field. Within the crust, these structural discontinuities may represent
- 73 joints, dilational jogs, bedding planes or extensional shear fractures (Sibson, 1996; 2000)
- 74 across which frictional shear displacements might be expected to occur. Coalescence of a
- 75 multitude of these shear offset structures within a well constrained deformation band may
- 76 be a central process leading to the formation of a new fault (e.g., Pollard and Fletcher,
- 77 2005). Thus, an investigation of frictional plasticity, treated as a precursor to fault
- 78 formation, can address where subsequent faults should localize and provide insight into
- 79 fault zone evolution.
- 80
- 81 We build on earlier models of Ellis et al. (2006) and Nüchter and Ellis (2010) that
- 82 prescribe fault surfaces directly and model frictional sliding between fault blocks over
- 83 many seismic cycles. We model the evolution of normal fault systems shortly after the
- 84 onset of rifting. Modeled deformation includes: discrete episodes of slip between fault
- 85 blocks that displace independently along the plane of contact; frictional plastic yield
- 86 within the brittle upper crust; and viscous flow in the lower crust. The relative importance
- 87 of each mechanism in accommodating the regional extension rate indicates the likelihood
- 88 of development of additional fault structures and the positions of these relative to the
- 89 primary fault. We observe a gradual shallowing of the dip angle of the fault plane due to
- 90 rotation, which is a prerequisite for the development of cross-cutting faults (Profett, 1977;
- 91 Nur et al., 1986) as older structures rotate beyond the limit for reactivation (Sibson,
- 92 1985).
- 93
- 94 Results from the computational model are interpreted in the context of the western Gulf
- 95 of Corinth, an asymmetric graben in mainland Greece, comprised of several major north-
- 96 dipping faults accommodating extension of 12-14 mm yr-1 (e.g., Clarke et al., 1997;
- 97 Briole et al., 2000; Floyd et al., 2010). Geomorphological and stratigraphic evidence
- 98 indicate a migration of faulting in a basinward direction (Goldsworthy and Jackson,
- 99 2001; Flotté et al., 2005; Rohais et al., 2007) as new faults developed in the hanging
- 100 walls of older structures (Jackson, 1999). Flotté et al. (2005) propose a tectonic model
- 101 utilizing a low-angle detachment fault underlying steeper surface faults, although this
- 102 model is not universally accepted and is not applied by us (e.g., McNeill et al., 2005). In
- 103 this paper, we propose a model for basinward fault migration based on modeled
- 104 distributions of stress and frictional plasticity in the vicinity of existing faults.
- 105
- 106 **2. Previous numerical experiments**
- 107 Computational modeling of fault systems on short timescales has been used to examine
- 108 rupture propagation along a fault under rate-state friction constraints (Dieterich, 1979;
- 109 Ruina, 1983), rupture propagation in an asymmetric geometry (Oglesby et al., 2000), and
- 110 near-fault coseismic deformation of an elastic crust (e.g., Freund and Barnett, 1976;
- 111 Gupta and Scholz, 1998). Such studies constrain the dynamics of fault rupture and are
- 112 useful in modeling ground acceleration and quasi-static coseismic displacements;
- 113 however, they do not account for deformation in fault systems over many seismic cycles.
- 114
- 115 Over longer timescales, numerical models of fault architecture and rift evolution, such as
- 116 an extension to the propagator matrix method of Rundle (1980) by King et al. (1988) and
- 117 Stein et al. (1988), have explored topographical development of normal faults over
- 118 multiple earthquake cycles and in continual depositional and erosional processes. Their
- 119 model indicates that the degree of lithospheric flexure and the width of the deformation
- 120 region are controlled by the effective elastic thickness of the crust. Finite element models
- 121 have expanded on these findings, investigating the effects of finite elastic strength in the
- 122 upper crust, and tracking patterns of failure in the vicinity of a fault (e.g., Melosh and
- 123 Williams, 1989; Bott, 1997). Plastic deformation reduces the effective elastic thickness of
- 124 the crust, thereby controlling graben dimension by migrating the outer flexure hinge
- 125 closer to the fault scarp (Bott, 1997).
- 126
- 127 Melosh and Williams (1989) detailed a finite element model that accommodates slip
- 128 between 65° dipping conjugate elements and which tracks proximity to Mohr-Coulomb
- 129 failure in the vicinity of a dip-slip fault. Their analysis indicates a maximum propensity
- 130 to failure in the near surface of the hanging wall, and also at the fault tip where the
- 131 maximum gradient in dip-slip is encountered. On the basis of strain-energy minimization,
- 132 they argue that antithetic faults should form in the hanging wall in response to coseismic
- 133 stress changes caused by rupture of the main fault. This result, which is specific to
- 134 systems in which the main fault dips at 65°, does not account for coupling between
- 135 gravitational potential and strain energy (Dempsey et al., 2012).
- 136
- 137 Henk (2006) uses a thermo-mechanical model with a viscoelastic-plastic rheology to
- 138 investigate lithospheric extension and deformation about a reactivated thrust fault, for a
- 139 variety of strength parameters and strain rates. These models also demonstrate strain
- 140 localization in the near-surface of the hanging wall, interpreted as a crestal collapse
- 141 graben (McClay and Ellis, 1987). Others have used 3-dimensional thermo-mechanical
- 142 models (Braun et al., 2008; Popov and Sobolev, 2008) to model rifting over large spatial
- 143 and temporal scales (100's of km and millions of years), and have reproduced rift
- 144 bounding faults through bands of localized strain (Huismans et al., 2005).
- 145
- 146 **3. Numerical model**
- 147 The finite element method, implemented here using the software suite Abaqus/Standard
- 148 (http://www.simulia.com/products/abaqus_standard.html), is used to solve deformation
- 149 and stress equations for an extending crust. Far-field boundary conditions and a cyclical
- 150 fault-friction regime drive the evolution of stress and deformation in time. This approach
- 151 was previously used by Ellis et al. (2006) and Nüchter and Ellis (2010) to develop 2-D
- 152 models of fault systems in contractional and extensional tectonic settings.
- 153
- 154 We consider deformation parallel to the strike of a dip-slip fault as negligible, which
- 155 allows us to model a 2-dimensional slice of the crust under plane strain conditions. This
- 156 slice is oriented perpendicular to the axis of a generic rift system and measures 60 km
- 157 horizontally and 25 km vertically. Such dimensions are presumed to be sufficient to
- 158 represent the relatively shallow seismogenic zone in thermally weakened rifts
- 159 (approximately 8 km, e.g., Bibby et al. (1995)) and isolate fault behavior from the
- 160 influence of side and lower boundaries.
- 161
- 162 Deformation prior to anelastic yield is described by linear, isotropic elasticity,
- 163 parameterized by Young's Modulus, _E_ , and Poisson's ratio, ν. For a complete summary of
- 164 mathematical terms and values see Table 1.
- 165
- 166 When stress in the upper crust exceeds the threshold for frictional plastic failure,
- 167 subsequent deformation is non-recoverable. This yield criteria can be defined in terms of
- 168 the Von Mises equivalent stress, _q_ ,
- 170 where the stress deviator tensor is defined _sij_ = σ _ij_ – σ _kk_ δ _ij_ /3. Under plane strain
- 171 conditions, _q_ reduces to the difference between maximum and minimum principal
- 172 stresses and is thus equivalent to the differential stress and equal to twice the maximum
- 173 shear stress, i.e., _q_ = σ _1_ - σ _3_ = 2τ. Frictional plastic yield occurs when the shear stress
- 174 exceeds a Mohr-Coulomb failure criterion
- 175 _p_ = ( _p_ − _pf_ ).sin() + _C_ .cos() (2)
- 176 where _p_ = -σ _kk_ /3 is the mean stress, _pf_ is the pore fluid pressure, assumed to be
- 177 hydrostatic, _C_ is the cohesion and φ is the internal angle of friction of the crust. As _p_
- 178 increases with depth in the crust so too does the shear stress required to achieve frictional
- 179 yield. Frictional-plastic strain, εp, that accumulates at yield is predicted to localize along
- 180 conjugate shear bands whose dip depends on the resolution of heterogeneities, the use of
- 181 dynamic pressure in Eq. (2) (Buiter, 2012) and the friction angle (Kaus, 2010). In these
- 182 models, we use a modified internal angle of friction, sin(φ’)=sin(φ).(ρ- ρf)/ ρ, where ρ is
- 183 rock density and ρf is the saturating fluid density. This modification reflects weakening of
- 184 the crust by the hydrostatic pore pressure term in Eq. (2) (Townend and Zoback, 2000).
- 185 For a modified friction angle of 19°, Kaus (2010) predicts shear bands to localize along
- 186 the Roscoe angles, which for a non-dilational crust are at 45° to σ1. This dip angle is
- 187 shallower than would be expected for a dilational crust at greater friction angles. The
- 188 effects of strain softening are not considered in these simulations. Strain softening is not
- 189 necessary to cause localization of shear deformation into bands (Rudnicki and Rice,
- 190 1975), although it does introduce a finite length-scale independent of mesh dimension
- 191 (Needleman, 1988). Other workers, e.g., Huismans et al. (2005), have incorporated strain
- 192 softening for modeling the onset of faulting during lithospheric extension.
- 193
- 194 Anelastic deformation of the lower crust by dislocation creep is modeled using a viscous
- 195 power-law rheology. The viscous strain rate, _v_ , depends upon the local stress state via
- 196 the Von Mises equivalent stress
- 197 _v_ = _Aqn_ exp(− _Q_ / _RT_ ) (3)
- 198 where _Q_ is the activation energy, _A_ is a constant containing material and geometrical
- 199 factors, _n_ is the power-law exponent, _R_ is the gas constant and _T_ is the temperature in
- 200 Kelvin. In these models, we use viscous parameters from Paterson and Luan (1990) (see
- 201 Table 1). A linear crustal temperature gradient of 35°C km-1 (Figure 1(b)) is used, which
- 202 is consistent with the elevated heat flux of active rifts, approximately 0.7 W m-1
- 203 (Hochstein, 1995). At depths in the crust where viscous deformation is the dominant
- 204 mechanism accommodating extension, we can rearrange Eq. (3) for the Von Mises stress,
- 205 _qv_ , required to accommodate the imposed bulk strain rate
- 207 In this form, it is clear that at shallower depths, where the crust is cooler, a higher Von
- 208 Mises stress is required to accommodate extension by viscous flow. At any depth in the
- 209 crust, both _qv_ and _qp_ (=2τ _p_ ) can be calculated as a function of temperature and pressure. If
- 210 _qv_ < _qp_ and _qv_ < _q_ , then viscous flow occurs, and if _qp_ < _qv_ and _qp_ < _q_ , then deformation is
- 211 frictional plastic. When _qv_ = _qp_ , there is a transition between the two behaviors and the
- 212 depth at which this occurs is called the brittle-ductile transition (BDT).
- 213
- 214 Deformation of the crust by tectonic extension occurs due to velocity boundary
- 215 conditions imposed at the lateral boundaries. GPS investigations of the western Gulf of
- 216 Corinth indicate that extension is on the order of 12-14 mm yr-1 (e.g., Clarke et al., 1997;
- 217 Briole et al., 2000; Floyd et al., 2010). In the models detailed here, we apply a horizontal
- 218 velocity boundary condition of -6 and +6 mm yr-1 to the left and right boundaries,
- 219 respectively, to account for this forcing (see Figure 1(a)). This differs from previous
- 220 models (Bott, 1997; Ellis et al., 2006; Nüchter and Ellis, 2010) where one boundary is
- 221 constrained by a `no displacement' condition and the full tectonic forcing applied to the
- 222 other boundary. The lower boundary is fixed at zero vertical displacement and the upper
- 223 surface is unconstrained. Gravity is applied throughout the model as a body force. A
- 224 constant density of 2800 kg m-3 is applied throughout the crust.
- 225
- 226 A single normal fault is modeled as a contact surface that intersects the crust, dipping at
- 227 an angle θ=50° to a depth of 15 km. This geometry is shallower than previous numerical
- 228 experiments of faulting (e.g., Melosh and Williams, 1989), but is consistent with normal
- 229 fault dips used by Bell et al. (2009) to calculate fault slip rates, and by Briole et al. (2000)
- 230 to model normal fault earthquakes in the Gulf of Corinth. The fault is prescribed as either
- 231 `locked', where the surfaces are prevented from sliding against one another, or `slipping',
- 232 where frictional sliding according to Amonton's law is permitted on the interface
- 233 _d n_ (5)
- 234 where τ is the shear stress, σ _n_ is the normal stress and μ _d_ is the dynamic coefficient of
- 235 friction on the fault surface, taken here as a slip weakened value of 0.2 to account for
- 236 heating and fluid effects during dynamic rupture propagation (Scholz and Engelder,
- 237 1976). The fault is arbitrarily switched between locked and slipping modes and therefore
- 238 this model is not suitable for investigating earthquake rupture initiation. Instead we
- 239 investigate the stress and deformation of a faulted crust during the seismic cycle, i.e., the
- 240 static response to earthquakes.
- 241
- 242 The model domain is meshed using ~10 000 plane-strain quadrilateral elements, with
- 243 smaller elements used near the surface and on the fault contact surfaces. Element
- 244 dimensions range from a minimum of 250 m on the fault to 1000 m at the base of the
- 245 model (see Figure 1(a)).
- 246
- 247 An initial stress state for the model is evolved by applying the velocity boundary
- 248 conditions over geological time (30 kyr, see Figure 1(d)). This stress ‘spin-up’ (Hetland
- 249 and Hager, 2006; Ellis et al., 2006) phase produces a lithostatic vertical stress, σ _z_ , and a
- 250 horizontal stress, σ _x_ , such that the differential stress is equal to the lower of _qp_ or _qv_ , i.e., a
- 251 critically stressed crust at frictional plastic yield at shallow depths and viscous yield
- 252 below the BDT (Townend and Zoback, 2000). Deformation that occurs during the stress
- 253 spin-up period is discarded at the beginning of a new simulation, while the stress state is
- 254 retained as an initial condition. The initial stress state, characterized by the differential
- 255 stress (and equivalent to the Von Mises stress under plane strain conditions), is shown as
- 256 a function of depth in Figure 1(c).
- 257
- 258 The seismic cycle, in which long periods of fault dormancy are punctuated by short,
- 259 violent episodes of fault slip and followed by a readjustment period of afterslip and
- 260 aftershock activity, is approximated in the model by a two-step cycle. The interseismic
- 261 period refers to the interval between earthquakes in which extension applied at the model
- 262 boundaries alters the displacement and stress distribution within the model domain. Each
- 263 interseismic period is followed by a seismic period, in which the friction coefficient on
- 264 the fault surface is substantially lowered and displacements of the fault blocks are driven
- 265 by an imbalance between gravitational and elastic forces. The model reaches dynamic
- 266 equilibrium, where elastic changes accumulated during the interseismic period are offset
- 267 by those of the seismic period, after 10 seismic cycles. In this paper we analyze results
- 268 from the 12th cycle (Figure 1(d)). Displacements of the model boundaries caused by the
- 269 velocity boundary conditions are negligible during the 60 second seismic period. Any slip
- 270 that occurs during this period is due to stress imbalances accumulated during the
- 271 interseismic period. Fault slip that occurs during the seismic period includes all possible
- 272 afterslip.
- 273
- 274 **4. The seismic cycle**
- 275 Figure 2 summarizes salient aspects of the modeled seismic cycle including (a) coseismic
- 276 surface displacements, (b) fault parallel displacement that occurs during the 12th fault
- 277 rupture, (c) the distribution of the differential stress after 12 seismic cycles, (d) elastic
- 278 strain accumulated during the 12th fault rupture, and the total accumulated (e) viscous and
- 279 (f) frictional-plastic strain after 12 seismic cycles.
- 280
- 281 A sudden decrease in friction on the fault surface during the seismic period permits
- 282 displacement of the fault blocks relative to one another. As the problem is modeled under
- 283 plane-strain conditions, along-strike displacement is not considered and the resulting
- 284 motions are approximately parallel to the dip of the fault. Figure 2(b) plots the fault-
- 285 parallel displacement field of the crust during the seismic period, in which “up-dip” is
- 286 taken as the positive direction. Maximum displacements are observed at the fault scarp
- 287 and decay with distance, consistent with geodetic evidence of large normal fault ruptures
- 288 at Hegben Lake, Montana and Borah Peak, Idaho (Barrientos et al., 1987). Along the
- 289 fault plane the footwall exhibits 0.6 m of upward-directed fault-parallel displacement
- 290 from the surface to 5 km depth (Figure 2(b) inset). Similarly the hanging wall is displaced
- 291 a constant 0.9 m in a down-dip direction for the same range of depths. The total single-
- 292 event offset is therefore 1.5 m, which consistent with the order of magnitude of historical
- 293 ruptures in the Gulf of Corinth (Koukouvelas and Doutsos, 1996; Roberts, 1996). In
- 294 addition to material (Gudmundsson, 2004) and fault geometry (Section 4) considerations,
- 295 the length of the loading period preceding an earthquake, i.e., the interseismic period,
- 296 exercises an important control on the size of the subsequent fault rupture. In these
- 297 simulations we have selected an interseismic period of 220 years that produces fault
- 298 ruptures of an appropriate size for the region under consideration. Fault slip decays from
- 299 its maximum value to zero over a depth interval of 5.0-8.5 km, below which viscous
- 300 deformation maintains a stress state that inhibits slip. The true depth of the modeled fault
- 301 is therefore determined by material parameters and the applied strain rate and is not a
- 302 prescribed feature of the geometry.
- 303
- 304 The distribution of the differential stress (Figure 2(c)) reflects the brittle and viscous
- 305 rheologies prevailing in the upper and lower crust, respectively (see Figure 1(c) and Eq.
- 306 (2) and (4)). The maximum differential stress, marking the transition from brittle to
- 307 ductile behavior, occurs at a depth of 6 km. Perturbations to the profile in Figure 1(c) are
- 308 observed close to the fault surface and are caused by coseismic movements of the fault
- 309 blocks and gradients in slip at the fault tip. In the hanging wall, a broad maximum in _q_ is
- 310 centered on the BDT, while in the adjacent footwall there is a corresponding reduction
- 311 compared to background levels for that depth.
- 312
- 313 Viscous deformation is the dominant contribution to deformation at the high temperatures
- 314 and pressures found in the lower crust (>9 km depth). Frictional plastic deformation or
- 315 slip on the fault surface are absent from this region of the crust so that, during the
- 316 interseismic period, 2.64 m (12 mm yr-1 × 220 yr) of tectonic extension must be
- 317 accommodated by viscous strain. As the strain rate, _v_ , is proportional to _q4_ (see Table 1
- 318 and Eq. (3) with _n_ = 4), the additional accumulation of εv in Figure 2(e) along the fault
- 319 root at depths >6 km points to an increase in the stress above the background level for
- 320 this depth. This increase, evident in Figure 2(c), is attributed to the upper crustal
- 321 displacements of the fault blocks and the accommodation of these motions within the
- 322 mid-crust. The extra viscous straining directly beneath the fault alters post-seismic
- 323 deformation at the surface (Ellis et al., 2006) and helps restore the stress state on the fault
- 324 to its equilibrium state (Nüchter and Ellis, 2010).
- 325
- 326 Frictional-plastic strain accumulates exclusively in the upper crust at depths shallower
- 327 than 6 km and localizes in shear bands that dip at approximately 45° from the horizontal,
- 328 as discussed in section 2. Qualitatively, frictional-plastic strain occurs preferentially in a
- 329 triangular wedge of the hanging wall block and is conspicuously absent from much of the
- 330 footwall. Regions of increased plastic strain are also observed at the lateral boundaries of
- 331 the model; however, these regions are outside the influence of the fault and represent
- 332 background strain behavior. The presence of multiple shear bands, as opposed to one or
- 333 two bands of high strain, is consistent with the “distributed” mode of fracturing described
- 334 by (Schueller et al., 2010). They showed that if the applied extension rate cannot be
- 335 accommodated by displacements on existing frictional plastic shear zones, which in turn
- 336 are limited by the slip velocity in both brittle and ductile media, then new shear zones
- 337 form to accommodate the shortfall. However, with further deformation such that these
- 338 shear bands experience strain softening due to accumulated plastic strain, we expect that
- 339 displacement would localize onto fewer structures.
- 340
- 341 **5. Extension budget**
- 342 The extension rate of the modeled rift, which is applied as a velocity boundary condition
- 343 at the lateral boundaries, must be accommodated within the crust by some combination of
- 344 the available means of deformation, i.e., by fault slip, or the accumulation of elastic,
- 345 frictional plastic, or viscous strain. The relative contribution of each to the total extension
- 346 of the crust during a single seismic cycle comprises the extension budget. For instance, if
- 347 during a single seismic cycle tectonic extension of 2.64 m is comprised of 0.66 m of
- 348 horizontal elastic strain, 0.66 m of horizontal frictional-plastic strain, and 1.32 m
- 349 horizontal component of fault slip, then the relative contribution of each component is
- 350 0.25, 0.25 and 0.5, respectively. Depending on the means of deformation available, the
- 351 extension budget will vary with both depth (e.g., viscous strain does not accumulate in
- 352 the upper crust) and time (e.g., fault slip occurs only during the seismic period). Figure 3
- 353 shows a representative extension budget with depth for the (a) interseismic and (b)
- 354 seismic periods, and (c) a full seismic cycle, where each component is normalized against
- 355 the total extension (2.64 m) during the full cycle.
- 356
- 357 Figure 3(a) indicates that viscous strain is the sole contributor to deformation at depths
- 358 greater than 9 km and, due to its slow accumulation rate, occurs principally during the
- 359 interseismic period. Small changes in elastic strain are also observed at these depths,
- 360 although, summed over the interseismic and seismic periods, their net effect is nearly
- 361 zero, which is characteristic of the dynamic equilibrium of the system.
- 362
- 363 Figure 3(c-d) indicates that the presence of a rupturing fault permits extension of the
- 364 upper crust to be distributed between the two brittle failure mechanisms of fault slip and
- 365 frictional plastic deformation. However, the accumulation of each component is mutually
- 366 exclusive, i.e., fault slip occurs rapidly during the seismic period (Figure 3(b)) whereas
- 367 frictional-plastic strain accumulates slowly during the interseismic period (Figure 3(a)).
- 368 Coseismic frictional-plastic strain accumulation is significant for steeper dipping faults
- 369 (see Section 6). A significant component of transient elastic strain accommodates
- 370 extension during the interseismic period, but is replaced by fault slip in the following
- 371 seismic period. It also causes a reduction of the maximum shear stress, (σ1 – σ3)/2,
- 372 throughout the crust wherever the least principal compressive stress is horizontal, i.e.,
- 373 σ3=σ _xx_ . Accumulation of frictional-plastic strain ceases for a brief post-seismic period
- 374 until extension restores the preseismic stress state to frictional yield (Dempsey et al., 375 2012).
- 376
- 377 Figure 3(a) and (c) show that the transition from purely brittle deformation, including
- 378 fault slip and frictional-plastic straining, to purely viscous deformation occurs over a 5.0-
- 379 9.0 km depth interval. Frictional plastic deformation terminates over a shallower 5.0-6.5
- 380 km depth interval, with a small overlap between the brittle and viscous rheologies
- 381 indicative of the exponential decrease in viscous strain rate with decreasing temperatures
- 382 (Equation (3)). Fault slip penetrates further into the brittle-ductile transition zone,
- 383 terminating over depths of 6.5-9.0 km. This bimodal truncation of the two frictional
- 384 behaviors is due to the relative frictional weakness of the fault (μ=0.2 compared with the
- 385 intact crust μ=tan(φ’)≈0.3) and indicates a mixed brittle-viscous region between a purely
- 386 brittle upper crust and purely viscous lower crust. In this region weakened, embrittled
- 387 faults are able to penetrate the mostly viscous lower crust (Scholz, 1988, 1998), while
- 388 frictional plasticity is limited to shallower depths.
- 389
- 390 Figure 4(a) considers the composition of the full cycle extension budget within the upper
- 391 crust (< 5.0 km depth) as a function of the loading period that precedes an earthquake.
- 392 For this analysis each simulation was “spun-up” to the same dynamic stress equilibrium,
- 393 using a nominal interseismic period of 220 years, and followed by a single seismic cycle
- 394 at the modified length. Thus, Figure 4(a) considers the effect of seismic cycle length
- 395 independent of the previous rupture history. It also illustrates extension accommodation
- 396 in the upper crust at different times during the interseismic period. As in Figure 3,
- 397 components were normalized against the total extension applied over the full cycle. For
- 398 longer seismic cycles, distributed frictional-plastic strain accounts for a proportionally
- 399 larger share, e.g., approaching 50% of the total extension for an interseismic period 220
- 400 years in length. During the early stages of the interseismic quiescence following fault
- 401 rupture, tectonic extension is mostly taken up by the accumulation of elastic strain
- 402 throughout the upper crust, which is later exchanged for fault slip. Extension reduces
- 403 horizontal stress, σ _xx_ , which causes the differential stress to approach the limit of
- 404 frictional plastic failure, _qp_ , and switches large areas of the crust from elastic to frictional
- 405 plastic behavior. Elastic strain is temporarily saturated in these regions, and subsequent
- 406 accumulated strain is frictional plastic. Each successive increment of extension during the
- 407 interseismic period then has two compounding effects: (i) to effect stress changes that
- 408 switch ever larger regions of the crust from elastic to frictional plastic deformation and
- 409 (ii) to accumulate frictional-plastic strain within an increasing volume of yielding crust
- 410 and elastic strain within a decreasing volume of elastic crust. Horizontal stress is
- 411 increased in the subsequent earthquake step due to coseismic displacements of the fault
- 412 blocks.
- 413
- 414 Fault dip also exerts control on the composition of the upper crust extension budget (see
- 415 Figure 4(b)). For a system in dynamic equilibrium with a seismic cycle 220 years in
- 416 length, a shallow fault, θ = 40-50°, accommodates more than half of the applied
- 417 extension through fault slip, with the shortfall accommodated by frictional plasticity.
- 418 However, steeper faults accommodate a decreasing proportion of the total extension
- 419 through displacement of the fault blocks. This decrease is because the production of a
- 420 given amount of horizontal extension requires an increasing component of vertical
- 421 movement, and thus a correspondingly larger earthquake, at steeper angles of fault dip. A
- 422 reduction in the fault-slip component of the extension budget must be compensated by an
- 423 increase in other components, which for an upper crust at dynamic equilibrium must be
- 424 frictional-plastic strain. This result is largely independent of the elastic properties of the
- 425 crust, and the partitioning of upper crustal extension between the accumulation of
- 426 frictional-plastic strain and episodic fault slip appears to be controlled principally by the
- 427 geometry of the system.
- 428
- 429 **6. Distribution of frictional plasticity**
- 430 By definition, frictional-plastic strain accumulates in those regions where the differential
- 431 (plane-strain Von Mises) stress exceeds the threshold for failure, i.e., _q>qp_ . The
- 432 accumulation of frictional-plastic strain then, to some extent, reflects the stress state of
- 433 the crust. Figure 5(a-b) plots the proximity of the differential stress to the limit of
- 434 frictional plastic yield, i.e., _qp-q_ , to a depth of 8 km, prior to and following fault rupture.
- 435 The frictional plastic yield limit is pressure dependent and must therefore be recalculated
- 436 following fault rupture to account for changes in mean stress. The sharp transition from
- 437 brittle to ductile failure is evident at a depth of 6 km, consistent with the thickness of the
- 438 frictional plastic layer in the extension budget (see section 5). Below 6 km, viscous
- 439 deformation maintains low stresses and ensures that _q|σ _zz_ |, and
- 464 the stress regime is compressional (see hatched regions in Figure 5(a-b)). The shallow
- 465 extent of these zones is principally a reflection of the low lithostatic vertical stress at
- 466 shallow depths, which is exceeded by only a modest horizontal compression.
- 467
- 468 The presence of a slipping fault perturbs the differential stress in its vicinity, imposing a
- 469 failure shadow on a crust that is otherwise at frictional yield. This failure shadow
- 470 broadens near the surface, particularly within the footwall block, and tapers to a narrow
- 471 width near the BDT. Figure 5(c) shows that frictional-plastic strain does not accumulate
- 472 within the shadow, but rather the greatest accumulation is outside the shadow in the
- 473 hanging wall block. In this region, we observe frictional plastic shear-bands dipping at
- 474 45°, synthetic to the main fault, and intersecting the surface ( _z_ =0) outside the edge of the
- 475 failure shadow. Low levels of frictional plasticity are also observed in the footwall block
- 476 at depths of 3-7 km where upward-directed fault slip and compression have not
- 477 significantly altered the stress state. Except for a small region at the root of the fault, the
- 478 majority of the fault plane is free from plastic deformation, although we cannot preclude
- 479 failure from dynamic loading during rupture propagation, which we have not modeled.
- 480
- 481 Previous numerical experiments by Melosh and Williams (1989) indicated that formation
- 482 of antithetic faults in the hanging wall of a main fault was energetically favorable over
- 483 synthetic faults of similar a similar dip. This result was derived assuming immediate
- 484 secondary fault formation in response to failure stresses induced by slip on the main fault,
- 485 i.e., coseismic antithetic faults. A similar result is replicated in our model by considering
- 486 frictional-plastic strain accumulated during the coseismic period in response to slip on a
- 487 65° dipping master fault (Figure 6(b)). For the model containing a 50° dipping fault,
- 488 frictional-plastic strain during the seismic period is negligible as local bending at the
- 489 surface of the hanging wall is less pronounced that for steeper faults. This result suggests
- 490 a preference for coseismic antithetic faulting may be limited to steeper dip angles.
- 491
- 492 In contrast, in Figure 5(c) and Figure 6(b) we demonstrate a preference for synthetic fault
- 493 formation, caused by a steady accumulation of frictional-plastic strain in shear bands
- 494 during the comparatively longer interseismic period. It should be emphasized this result
- 495 is not inconsistent with Melosh and Williams (1989), but rather indicates that the
- 496 configuration of secondary faults is sensitive to the fault dip relative to optimally oriented
- 497 fault dip, and to the stage of the seismic cycle during which they form.
- 498
- 499 **7. Fault plane rotation**
- 500 Coseismic displacement of the fault blocks parallel to the fault plane, which
- 501 accommodates extension and reduces near-fault shear stresses, is perhaps the most
- 502 conspicuous aspect of deformation associated with faulting (e.g., Barrientos et al., 1987;
- 503 Segal, 2010). However, observations of normal faults dipping at shallow angles (<30°)
- 504 that preclude reactivation within an Andersonian stress regime (Wernicke, 1995), or a
- 505 regime of cross-cutting faults (Profett, 1977; Nur et al., 1986), suggest a secondary
- 506 process of rotation via a bookshelf or domino-style mechanism. Such motion, resolved on
- 507 the fault plane, is equivalent to a displacement of the fault block perpendicular to the fault
- 508 plane. Further, for a rotation as opposed to a rigid body translation, displacement must
- 509 increase linearly with distance from some pivot, i.e., a point of no displacement.
- 510
- 511 Figure 7 plots fault perpendicular displacement along the fault and against depth during
- 512 (a) the interseismic and (b) seismic periods, with a regression line fit to the data in the
- 513 brittle layer (<8 km depth). During both periods, displacement profiles in the upper crust
- 514 are approximately linear and have gradients of 59 and 66 mm km-1 (i.e., millimeters of
- 515 fault perpendicular displacement per kilometer of depth) for the interseismic and seismic
- 516 periods, respectively.
- 517
- 518 The magnitude of fault plane rotation can be calculated from the gradients of the
- 519 regression lines in Figure 7. A linear fault-normal displacement profile, _u_ , is expressed as
- 520 a function of depth, _u_ = _m_ ( _z-z0_ ), where _m_ is the gradient of regression line through the
- 521 displacement data, and _z_ 0 is the depth of the fixed horizontal axis, or pivot, about which
- 522 rotation occurs. The depth of the pivot corresponds to the intersection of the regression
- 523 line in Figure 7(a-b) with the vertical _u =_ 0 axis. During the interseismic period, the pivot
- 524 ranges in depth from 18 to 18.7 km, with a mean of 18.4 km. For the seismic period,
- 525 rotation of the fault plane occurs about a pivot at a depth of 4.7 km. A displacement, _u_ ,
- 526 some vertical distance _z_ - _z_ 0 above the pivot induces a rotation, _d_ θ, of the fault plane
- 527 (Figure 7(c)), given by
- 528 _d_ = tan− 1 ( _u_ sin/( _z_ − _z_ 0 )) = tan− 1 ( _m_ .sin) (6)
- 529 Using m = 59 and 66 mm km-1 , we calculate a shallowing of dip of 2.4 and 2.7
- 530 millidegrees during the interseismic and seismic periods, respectively. This corresponds
- 531 to a long term rotation rate of 23.5° every million years (for a seismic cycle that repeats
- 532 every 220 years), although a reduction in this rate would presumably be observed as the
- 533 fault rotates out of the range for reactivation.
- 534
- 535 The total rotation during a seismic cycle is distributed approximately equally between the
- 536 interseismic and seismic periods. The amount of coseismic rotation is proportional to the
- 537 length of the interseismic period that precedes it, and is thus also proportional to
- 538 earthquake magnitude. Further, coseismic rotation about an axis at a depth of 4.7 km, 14
- 539 km shallower than that for the interseismic period, suggests that a different forcing
- 540 operates each rotation. For instance, during the interseismic period, far-field tectonic
- 541 extension deforms the entire crust, causing vertical contraction in addition to horizontal
- 542 spreading, effectively flattening the crust and the fault that is hosted within. The effect of
- 543 this deformation on fault dip is evident in Figure 8(a), which presents the exaggerated
- 544 change in position of the fault surface and topography during the interseismic period.
- 545
- 546 In contrast, displacements at the lateral boundaries during the short seismic period are too
- 547 small to account for the observed rotation. However, large and rapid stress changes in the
- 548 mid-crust, which cause uplift of the footwall and accommodate collapse of the hanging
- 549 wall, may be responsible. Footwall uplift is caused by a decrease in stress and rapid
- 550 decompression of the crust between 6 and 9 km depth of the footwall block. Figure 8(b)
- 551 shows the decrease in pressure associated with this uplift, superimposed on the deformed
- 552 topography and fault plane. A drop in pressure corresponds to an increase in volume in
- 553 that region, which, while primarily directed up-dip along the fault plane, also includes a
- 554 fault normal component. This expansion applies a force to the hanging wall and a torque
- 555 about the pivot at 4.7 km depth. This expansion also accounts for a component of the
- 556 pressure increase in the adjacent hanging wall. The expanding footwall does work on the
- 557 hanging wall, which, in addition to the compression caused by dip-slip of the fault block
- 558 above, yields a reduction in volume and an increase in stress. As this rotation occurs due
- 559 to stress changes at the fault tip, the depth of the pivot about which rotation occurs is
- 560 primarily controlled by the depth of faulting.
- 561
- 562 **8. Evolution of normal fault systems**
- 563 Where pre-existing faults do not adequately accommodate the geodetic extension rate,
- 564 parts of the crust are subjected to a horizontal, positive strain. In regions of the crust
- 565 where the shear stress is below the Mohr-Coulomb failure limit, extension is
- 566 accommodated by elastic strain. However, continued elastic stretching generates shear
- 567 stresses that eventually induce failure on any pre-existing fracture planes and material
- 568 discontinuities. Shear stresses are maintained at near critical values (Townend and
- 569 Zoback, 2000) by incremental frictional failure on these minor structures in response to
- 570 each subsequent increment of extension. These processes constrain the maximum stress
- 571 state attainable by the crust and therefore impose an upper limit on the amount of
- 572 extension that is temporarily accommodated as elastic strain.
- 573
- 574 However, no such constraint exists for accommodation of extension through frictional-
- 575 plastic strain, which accumulates following the saturation of elastic strain. In our model,
- 576 frictional plastic deformation localizes in shear bands that form at 45° to the maximum
- 577 principal compressional stress, which is vertical in an extensional tectonic regime. That
- 578 these shear bands represent the development of new faults is speculative, although they
- 579 do satisfy two criteria associated with fault structures: (i) the shear bands accommodate
- 580 dip-slip displacement and (ii) are localized, rather than distributed (Schueller et al.,
- 581 2010), zones of deformation, in this case the width of several elements (approximately 1
- 582 km). In these models, shear zone width is sensitive to mesh dimension, as we have not
- 583 incorporated a description of strain softening behavior (Needleman, 1988). Physically, a
- 584 localization of frictional plasticity represents a continuous reactivation within a particular
- 585 region of those small discontinuities, e.g., joints, dilational jogs, fractures, etc.,
- 586 accommodating shear displacements. Providing each subsequent reactivation increases
- 587 the length of the activated structure, a region of frictional plastic shear may be regarded
- 588 as a sort of proto-fault. We note that the evolution of a proto-fault-style shear zone into a
- 589 true fault that perturbs stresses in its vicinity is not modeled here. Instead, we are
- 590 considering the early onset of proto-fault development.
- 591
- 592 Basinward migration of normal fault structures has been inferred from stratigraphic
- 593 studies in Turkey (Dart et al., 1995), the central Basin and Range province, USA (Horton
- 594 and Schmitt, 1998), the Gulf of Corinth, Greece (Goldsworthy and Jackson, 2001), and
- 595 the strike-slip Gar Basin in western Tibet (Sanchez et al., 2010). Structurally, new faults
- 596 develop in the hanging wall block of older faults producing a series of sub-parallel faults
- 597 spaced every 2-5 km. Goldsworthy and Jackson (2001) proposed several mechanisms for
- 598 the cessation of rupture on one fault and concurrent development of an adjacent structure
- 599 including: topographic stresses in a domino-style topography modulated by the strength
- 600 of the active fault; density contrast between sedimentation of the hanging wall basin and
- 601 erosion of the uplifted footwall; and rolling-hinge evolution along a migrating lower
- 602 crustal flow front (Buck, 1988; McKenzie et al., 2000). However, they note that, with the
- 603 exception of the rolling hinge, which itself requires a migrating magma source, these
- 604 mechanisms do not account for the preferred basinward direction of migration.
- 605 Alternatively, inward migration within a rift may result from brittle necking of strong
- 606 crustal layers, where thinner brittle layers are easier to fault cf. thicker brittle layers
- 607 outside the rift (Buiter et al., 2006).
- 608
- 609 The migration of fault activity first requires that new faults form, which occurs when
- 610 existing faults do not accommodate the total geodetic extension rate. For instance, in
- 611 Figure 3 we show that for a 220 year interseismic period, a single fault dipping at 50°
- 612 accounts for approximately half the extension applied to the upper crust, with the balance
- 613 accommodated by off-fault frictional-plastic strain. As fault slip and frictional-plastic
- 614 strain accumulate during the seismic and interseismic periods respectively, we consider
- 615 the competing accumulation of elastic versus frictional-plastic strain during the
- 616 interseismic period, with the caveat that the elastic component is replaced by fault slip
- 617 during an earthquake. Elastic strain accumulates only in those regions of the upper crust
- 618 maintained below the limit of frictional plastic failure, which is largely within the near-
- 619 fault stress-failure shadow (Figure 5(a)). Elsewhere, tectonic extension applied for long
- 620 time periods ensures the crust is close to failure and thus strain is accommodated by
- 621 frictional plasticity.
- 622
- 623 For a system containing a single fault, frictional-plastic strain accommodates a significant
- 624 component of the extension budget (Figure 4(b)). In terms of rift basin evolution, our
- 625 model describes an area soon after the onset of extension in which only a single mature
- 626 fault developed (see Figure 9(a)). The full extension rate cannot be accommodated by this
- 627 structure, and thus frictional-plastic strain accommodates the shortfall. Frictional-plastic
- 628 strain localizes into shear zones in the hanging wall. In our model these shear zones are
- 629 synthetic to the main fault, and are therefore consistent with a scenario of migration of
- 630 fault activity into the hanging walls of existing faults. However, the formation of
- 631 secondary faults antithetic to the main fault, i.e., a graben geometry, is observed in many
- 632 locations (e.g., Jackson and McKenzie, 1983) and also in numerical experiments (Melosh
- 633 and Williams, 1989). Preference of one over the other has been considered from an
- 634 energy minimization perspective (Melosh and Williams, 1989); however, Figures 5 and 6
- 635 indicate that the geometry of the primary fault, and the precise phase of the seismic cycle
- 636 during which secondary fault formation occurs, may also play a role. Other factors that
- 637 may control the geometry of secondary structures include the initial localization of shear
- 638 bands onto pre-existing planes of weakness or crustal heterogeneities, and the effects of
- 639 gravitational spreading and basal detachment shear during rifting (e.g., Brun et al., 1994).
- 640
- 641 Localization of frictional plastic deformation into proto-faults occurs outside the stress-
- 642 failure shadow of the main fault in the hanging wall block (Figure 9(b)). The extent of the
- 643 failure shadow therefore exercise a degree of control on the location of these proto-faults
- 644 and perhaps also the spacing of a sub-parallel array of faults. If the -1 MPa contour is
- 645 used to estimate the dimension of the failure shadow, Figure 5(a) would suggest a
- 646 minimum spacing of 10 km for footwall faults and 2 km for hanging wall faults. The
- 647 latter is slightly smaller than the average fault spacing in the western Gulf of Corinth of
- 648 5-8 km (Rohais et al., 2007); however, as the dimensions of the stress-failure shadow
- 649 depend on the thickness of the brittle upper crust and fault dip, direct comparison
- 650 between model and field studies is not possible. For example, crustal thickness in the
- 651 Gulf of Corinth of 10-12 km (Briole et al., 2000) is somewhat thicker than the 6-8 km in
- 652 our model. For this crust, we would expect larger failure shadows than those modeled and
- 653 consequently larger minimum fault spacings for both fault blocks. In addition, fault dips
- 654 in the western Gulf of Corinth are not well constrained, although the dip of our modeled
- 655 fault planes are consistent with those used by Briole et al. (2000) in modeling normal
- 656 fault earthquakes in the Corinth Rift.
- 657
- 658 Coalescence of frictional plastic deformation into a single shear zone, culminating in a
- 659 new discrete fault structure has important implications for the extension budget. The
- 660 introduction of a second stress-failure shadow (Figure 9(c)) increases the area available
- 661 for interseismic elastic-strain accumulation and reduces that available for frictional-
- 662 plastic straining. Proto-faults previously developing adjacent to the new fault would
- 663 become inactive as deformation within its stress-failure shadow is switched to
- 664 elastic/fault slip (Figure 9). Fault slip will then account for a greater component of the
- 665 extension budget. Frictional-plastic straining will initiate in the hanging wall of the new
- 666 fault, although at a reduced rate reflecting the extension accommodated by the second
- 667 fault.
- 668
- 669 Rotation of faults to shallower dip angles, either through interseismic stretching or
- 670 coseismic stress changes in the mid-crust (see section 7), has the opposite effect on the
- 671 upper crustal extension budget. The eventual removal of a fault that was accommodating
- 672 extension (Figure 9(d)), and the subsequent recovery of stresses within its failure shadow
- 673 to Mohr Coulomb yield (Figure 9(e)), increases the extension shortfall, which is then
- 674 accommodated by frictional plasticity. The formation of proto-faults in the hanging wall
- 675 of the second fault then accelerates and the cycle continues. Alternatively, the recovery of
- 676 the stress-failure shadow of the inactive fault would permit the development of new,
- 677 cross-cutting faults in this region (Nur et al., 1986).
- 678
- 679 The net effect of the processes discussed above is a preference for the migration of
- 680 faulting into the hanging walls of existing faults. However, an appropriate timescale to
- 681 attach to this evolutionary scenario is unclear, largely due to ambiguity in the rate of
- 682 formation of new faults. In our models, the frictional plastic shear strain rate in the
- 683 hanging wall where proto-faults form is approximately 7×10-16 s-1 , which for faulting to
- 684 seismogenic depths (6-8 km) represents a slip rate on the order of 0.15 mm yr-1 . At this
- 685 rate, a proto-fault should exhibit several 10's of meters of offset after several hundred kyr.
- 686 Goldsworthy and Jackson (2001) described uplift of coastal faults in the western Gulf of
- 687 Corinth and suggested that the morphology might be generated in less than a million
- 688 years. One might further expect acceleration of the slip rate with increasing rift maturity
- 689 due to along-strike linkage of fault segments (Cowie and Roberts, 2001; Bell et al.,
- 690 2009).
- 691
- 692 **9. Conclusion**
- 693 The onset of extension and subsequent evolution of normal faulting within the brittle
- 694 crust was investigated via a computational model for an extending elastic-viscous-plastic
- 695 crust. Extension applied at the boundaries of the model is accommodated in the upper
- 696 crust through dip-slip on a fault contact surface or the accumulation of frictional-plastic
- 697 strain, and in the lower crust through viscous deformation. Between these two layers, a
- 698 mixed brittle-viscous region develops, which exhibits a rapid decrease in frictional-
- 699 plastic straining, a slower decay in fault slip, and the onset of viscous deformation.
- 700
- 701 Frictional-plastic strain localizes preferentially within the hanging wall due to the
- 702 distribution of shear stress near the fault. Coseismic displacements of the fault blocks
- 703 create a stress-failure shadow that extends disproportionally into the footwall thereby
- 704 favoring the formation of new faults in the adjacent hanging wall block. The dimension
- 705 of the failure shadow at the surface is broadly consistent with the observed fault spacing
- 706 in normal fault arrays of several kms.
- 707
- 708 Rotation of the fault plane about a horizontal axis at depth is observed to occur during
- 709 both phases of the seismic cycle. A long term rotation rate of 23.5o per Myr is divided
- 710 into almost equal components of far-field tectonic rotation during the interseismic period
- 711 and rapid rotation induced by mid-crustal stress changes during the seismic period. The
- 712 subsequent shallowing of the fault plane over many seismic cycles provides one avenue
- 713 by which faults may be become inactive, accelerating the frictional-plastic straining and
- 714 proto-fault formation accommodates the extension shortfall.
- 715
- 716 The juxtaposition of these competing processes within an extending crust is consistent
- 717 with evolution of subsequent faults into existing hanging walls. However, a basinward
- 718 migration of faulting cannot occur indefinitely without a corresponding migration of the
- 719 locus of extension. Where the position of extension is static in time, recovery of the
- 720 stress-failure shadow around inactive faults may permit the eventual return of faulting to
- 721 the older footwalls. Cross-cutting of old, shallow structures by younger, steeper faults is
- 722 permitted, providing that they form outside the stress-failure shadow that extends
- 723 disproportionally into the footwall block.
- 724
- 725 As a final note, this simplified model for normal fault evolution requires careful
- 726 interpretation in the context of a richly complex crust. Material properties, stratigraphy
- 727 and tectonic forcings that vary along-strike and with depth also exert important controls
- 728 on the evolution of deformation and stress around faults, and by extension the
- 729 development of new structures. The mechanisms and scenarios described here do not
- 730 constitute a complete or prescriptive theory, but rather serve as an illustration of the
- 731 evolutionary processes that may prevail in specific locales, e.g., the western Gulf of
- 732 Corinth.
- 733
- 734 **Acknowledgements**
- 735 This research was supported by scholarships from the Tertiary Education Commission,
- 736 New Zealand and Energy Education Trust of New Zealand. A license for
- 737 Abaqus/Standard was supplied by the Faculty of Engineering, University of Auckland,
- 738 New Zealand. S. E.’s contribution to this research is supported by GNS Science’s Direct
- 739 Crown Funded Programme ‘Geological Hazards’. The authors thank P. Resor and R.
- 740 Davies for reviews that improved the quality of the manuscript.
- 741
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- 1031 **Table 1:**
- 1032 Symbols used in the text. Unless otherwise stated rheological parameters are set at the
- 1033 bracketed values.
- 1034
- 1035 **Figure 1**
- 1036 (a) A typical model geometry comprising a 60×25 km crust that hosts a single fault
- 1037 dipping 50° at the surface to a depth of 15 km and intersecting the surface 20 km from
- 1038 the left vertical boundary. The lower boundary is constrained to zero displacement in the
- 1039 _z_ direction and the left and right boundaries move at a constant -6 and 6 mm yr-1 in the _x_
- 1040 direction, respectively. The upper boundary is unconstrained. Model mesh (~10 000
- 1041 elements) is shown superimposed on the domain. (b) Linear vertical temperature gradient
- 1042 of 35 °C km-1 imposed through the crust. (c) Vertical differential stress, _q_ , profile as
- 1043 prescribed by Eq. (2) and (4). (d) Flow diagram detailing time periods, fault properties
- 1044 and boundary conditions associated with various stages of the model.
- 1045
# 1046 **Figure 2**
- 1047 Changes to displacement, stress and strain distributions due to fault slip observed during
- 1048 a seismic cycle of loading and slip. (a) Coseismic vertical surface displacements (adapted
- 1049 from Dempsey et al. (2012)) produced by the model (solid line) and at Borah Peak, Idaho
- 1050 (1983, M=6.9), and Hegben Lake, Montana (1959, M=7.3). Error bars for historical data
- 1051 are smaller than the marker dimension. (b) Change in fault parallel displacement during
- 1052 the 12th seismic period, i.e., fault slip. Note that down-dip displacements are taken to be
- 1053 negative. The inset profiles show fault parallel displacement (m) at the fault plane for the
- 1054 footwall (dashed) and hanging wall block (solid). The vertical scale is the same as for the
- 1055 contour plot. (c) Distribution of the differential stress, _q_ , at the end of the 12th seismic
- 1056 cycle. (d) Change in fault parallel elastic strain during the 12th seismic period. Note that a
- 1057 non-linear contour interval has been used. (e) Total accumulated viscous strain after 12
- 1058 seismic cycles. (f) Total accumulated frictional-plastic strain after 12 seismic cycles. The
- 1059 position of the fault is marked by the black and white dashed line.
- 1060
- 1061 **Figure 3**
- 1062 (a) Interseismic and (b) seismic components of the (c) full seismic cycle extension
- 1063 budget. The contribution of fault slip (no marker), elastic (square), frictional plastic
- 1064 (circle) and viscous (triangle) deformation mechanisms at each depth add to give the unit
- 1065 normal extension profile. This extension budget has been constructed for deformation
- 1066 between 10 < _x_ < 40 km. (d) Extension budget for a crust at a critical stress state in the
- 1067 absence of faulting.
- 1068
- 1069 **Figure 4**
- 1070 Elastic (grey), frictional plastic (black) and fault slip (black hatched) normalized
- 1071 components of the extension budget averaged over the upper crust (0 < _z_ < 5 km). (a)
- 1072 Extension budget components against the length of the seismic cycle. (b) Extension
- 1073 budget components against the dip angle of the fault, for a seismic cycle length of 220
- 1074 years.
- 1075
- 1076 **Figure 5**
- 1077 Proximity of the differential stress to Mohr-Coulomb failure, _qp-q_ , (a) prior to and (b)
- 1078 following an earthquake for a 50° dipping fault and a crust at dynamic equilibrium. The
- 1079 BDT is marked by a sharp decrease in proximity to failure and black hatching indicates
- 1080 regions where the crust is in tectonic compression, i.e., |σ _xx_ |>|σ _zz_ |. (c) Distribution of
- 1081 frictional-plastic strain accumulated during the 12th interseismic period (Figure 1(d)). The
- 1082 position of the fault is marked by the black and white dashed line.
- 1083
- 1084 **Figure 6**
- 1085 Frictional-plastic strain accumulated during (a) the 12th interseismic period, and (b) the
- 1086 12th seismic period, for a model containing a 65° dipping fault. The position of the fault is
- 1087 indicated by the black and white dashed line.
- 1088
- 1089 **Figure 7**
- 1090 Fault perpendicular displacement along the fault surface during (a) the interseismic and
- 1091 (b) seismic period. A linear regression profile has been fit to the data above a depth of 8
- 1092 km. (c) Geometric arguments used to derive fault rotation from fault perpendicular
- 1093 displacement profiles. Note that negative displacements in (a) and (b) correspond to the
- 1094 direction of the vector **u** (positive up, right) in (c). The position of the axis of rotation is
- 1095 indicated in (b) and (c) by a black circle.
- 1096
- 1097 **Figure 8**
- 1098 Deformation of the entire model domain and fault plane, exaggerated 2000 times, during
- 1099 (a) the interseismic and seismic periods. Boundary extension of 1.3 m at both lateral
- 1100 boundaries, along with a vertical contraction of 1 m, is marked. (b) Coseismic
- 1101 deformation in finer detail for the grey box in (a). Pivots about which rotation occurs are
- 1102 shown by open circles. Contours indicate coseismic mean stress change: light grey,
- 1103 unfilled = -0.25 MPa, light grey, filled = -0.5 MPa, dark grey, unfilled = +0.25 MPa, dark
- 1104 grey, filled = +0.5 MPa.
- 1105
- 1106 **Figure 9**
- 1107 Conceptual evolution of a normal fault system over time. (a) Shortly after the onset of
- 1108 rifting the crust hosts a single fault that imposes a stress-failure shadow as in Figure 5
- 1109 (grey shading). (b) Frictional plastic proto-faults, and (c) eventually new faults, develop
- 1110 in the hanging wall. (d) Continual rotation of faults out of the angle for reorientation
- 1111 reduces the component of extension accommodated by fault slip and promotes the
- 1112 development of new proto-faults. (e) Recovery of the failure shadow around inactive
- 1113 faults allows cross-cutting proto-faults to develop.
|Symbol|Explanation|
|---|---|
||_Rheological parameters_
|
|ρ|density [2800 kg m-3]|
|_E_, ν|Young’s Modulus [10 GPa], Poisson’s ratio [0.25]|
|_C_, φ|cohesion [1 MPa], material angle of friction [30°]|
|μd
_v_
|dynamic fault friction coefficient [0.2]
viscous strain rate [6.3×10-15s-1]|
|_A_,_Q_,_n_,_R_|pre-exponential constant [4×10-34Pa-4s-1], activation energy [135 kJ
mol-1], power-law exponent [4], gas constant [8.314]|
|_T_|temperature|
||_Stress terms, components and conventions_|
|σ_ij_,_sij_|stress and deviatoric stress tensors|
|σ1, σ3|maximum, minimum principal compressive stress|
|_p_,_pf_|mean stress, pore fluid pressure|
|_q_,_qp_,_qv_|differential stress (plane strain Von Mises), differential stress at
frictional plastic yield, differential stress maintaining viscous strain rate|
|τ, σ_n_|shear, normal stress on fault surface|
|sign|negative values of stress indicate compression|
||_Displacement and strain terms_|
|_x_,_z_|Cartesian coordinates,_z_is negative when directed downward|
|_ux_,_uz_,_upara_|horizontal, vertical and fault parallel displacement|
**Table 1:** Symbols used in the text. Unless otherwise stated rheological parameters are set at the bracketed values.
Figure 1. (a) A typical model geometry comprising a 60×25 km crust that hosts a single fault dipping 50° at the surface to a depth of 15 km and intersecting the surface 20 km from the left vertical boundary. The lower boundary is constrained to zero displacement in the z direction and the left and right boundaries move at a constant 6 and 6 mm yr1 in the x direction, respectively. The upper boundary is unconstrained. Model mesh (~10 000 elements) is shown superimposed on the domain. (b) Linear vertical temperature gradient of 35 °C km1 imposed through the crust. (c) Vertical differential stress, q, profile as prescribed by Eq. (2) and (4). (d) Flow diagram detailing time periods, fault properties and boundary conditions associated with various stages of the model.
Figure 2. Changes to displacement, stress and strain distributions due to fault slip observed during a seismic cycle of loading and slip. (a) Coseismic vertical surface displacements (adapted from Dempsey et al. (2012)) produced by the model (solid line) and at Borah Peak, Idaho (1983, M=6.9), and Hegben Lake, Montana (1959, M=7.3). Error bars for historical data are smaller than the marker dimension. (b) Change in fault parallel displacement during the 12th seismic period, i.e., fault slip. Note that downdip displacements are taken to be negative. The inset profiles show fault parallel displacement (m) at the fault plane for the footwall (dashed) and hanging wall block (solid). The vertical scale is the same as for the contour plot. (c) Distribution of the differential stress, q, at the end of the 12th seismic cycle. (d) Change in fault parallel elastic strain during the 12th seismic period. Note that a nonlinear contour interval has been used. (e) Total accumulated viscous strain after 12 seismic cycles. (f) Total accumulated frictionalplastic strain after 12 seismic cycles. The position of the fault is marked by the black and white dashed line.
Figure 3. (a) Interseismic and (b) seismic components of the (c) full seismic cycle extension budget. The contribution of fault slip (no marker), elastic (square), frictional plastic (circle) and viscous (triangle) deformation mechanisms at each depth add to give the unit normal extension profile. This extension budget has been constructed for deformation between 10 < x < 40 km. (d) Extension budget for a crust at a critical stress state in the absence of faulting.
Figure 4. Elastic (grey), frictional plastic (black) and fault slip (black hatched) normalized components of the extension budget averaged over the upper crust (0 < z < 5 km). (a) Extension budget components against the length of the seismic cycle. (b) Extension budget components against the dip angle of the fault, for a seismic cycle length of 220 years.
Figure 5. Proximity of the differential stress to MohrCoulomb failure, qpq, (a) prior to and (b) following an earthquake for a 50° dipping fault and a crust at dynamic equilibrium. The BDT is marked by a sharp decrease in proximity to failure and black hatching indicates regions where the crust is in tectonic compression, i.e., |σxx|>|σzz|. (c) Distribution of frictionalplastic strain accumulated during the 12th interseismic period (Figure 1(d)). The position of the fault is marked by the black and white dashed line.
Figure 6. Frictionalplastic strain accumulated during (a) the 12th interseismic period, and (b) the 12th seismic period, for a model containing a 65° dipping fault. The position of the fault is indicated by the black and white dashed line.
Figure 7. Fault perpendicular displacement along the fault surface during (a) the interseismic and (b) seismic period. A linear regression profile has been fit to the data above a depth of 8 km. (c) Geometric arguments used to derive fault rotation from fault perpendicular displacement profiles. Note that negative displacements in (a) and (b) correspond to the direction of the vector u (positive up, right) in (c). The position of the axis of rotation is indicated in (b) and (c) by a black circle.
Figure 8. Deformation of the entire model domain and fault plane, exaggerated 2000 times, during (a) the interseismic and seismic periods. Boundary extension of 1.3 m at both lateral boundaries, along with a vertical contraction of 1 m, is marked. (b) Coseismic deformation in finer detail for the grey box in (a). Pivots about which rotation occurs are shown by open circles. Contours indicate coseismic mean stress change: light grey, unfilled = 0.25 MPa, light grey, filled = 0.5 MPa, dark grey, unfilled = +0.25 MPa, dark grey, filled = +0.5 MPa.
Figure 9. Conceptual evolution of a normal fault system over time. (a) Shortly after the onset of rifting the crust hosts a single fault that imposes a stressfailure shadow as in Figure 5 (grey shading). (b) Frictional plastic protofaults, and (c) eventually new faults, develop in the hanging wall. (d) Continual rotation of faults out of the angle for reorientation reduces the component of extension accommodated by fault slip and promotes the development of new protofaults. (e) Recovery of the failure shadow around inactive faults allows crosscutting protofaults to develop.