Symmetric and asymmetric lithospheric extension: Relative effects of frictional-plastic and viscous strain softening

Authors


Abstract

[1] Strain-dependent rheologies may play a critical role in the deformation of the lithosphere and particularly in the development of focused shear zones. We investigate the effects of strain softening on lithospheric extension using plane strain thermomechanical finite element model experiments. Parametric softening is specified as a linear decrease of the effective internal angle of friction, the effective viscosity, or both in the model rheologies. The sensitivity of deformation to the choice of softening parameters is investigated in cases where the crust is either strongly or weakly coupled to the mantle lithosphere. Results are classified according to the symmetry (S) or asymmetry (A) of the deformation of the upper and lower lithosphere during rifting. Strain softening is required for rifting asymmetry but is not always sufficient. A range of model tectonic styles occurs including pure and simple shear modes with focused shear zones. Mode selection is mostly determined by the feedback between two primary controls, the “dominant” rheology and the parametric strain-softening mechanisms listed above. Softening of the dominant rheology promotes asymmetric extension of that part of the lithosphere controlled by the dominant rheology. Model results are consistent with the proposed primary controls and the factors that contribute to these controls. In particular, decreasing and increasing the rifting velocity can change the mode by changing the dominant rheology. Asymmetry is strongest in coupled models which include a decrease in the internal angle of friction and have low rifting velocities.

1. Introduction

[2] The overall features of continental rifts and passive margins have been interpreted both in terms of large-scale symmetric and asymmetric kinematic models of extension [Boillot et al., 1992; Brun and Beslier, 1996; Keen et al., 1989; Lister et al., 1986; Louden and Chian, 1999; Mutter et al., 1989; Sibuet, 1992; Wernicke and Burchfiel, 1982; White, 1990]. Though on a smaller scale, deformation on faults and shear zones is inherently asymmetric, the question of the role of shear zones on large-scale asymmetry of passive margins is still unanswered. This is mainly a result of the lack of good constraints on the structure of the mantle lithosphere and its evolution during continental lithosphere extension. Available data do point to asymmetries in regard to geometry of conjugate margins, large-scale low-angle detachments [Boillot et al., 1992; Louden and Chian, 1999; Sibuet, 1992], and features interpreted to be high-angle shear zones that extend into the upper mantle [Brun and Gutscher, 1992; Whitmarsh et al., 2001].

[3] Although a range of geometries can be described by kinematic lithosphere extension models, including pure shear [McKenzie, 1978], simple shear [Wernicke, 1985] and combinations of these geometries [Lister et al., 1986], these kinematic models provide little insight into the overall dynamical processes and, particularly, mechanisms favoring asymmetry. Most dynamical models consider only first-order extension of a layered lithosphere with viscoplastic or elastic-viscous-plastic rheologies that depend only weakly on strain [Bassi et al., 1993; Braun and Beaumont, 1987; Chéry et al., 1992]. These models are either intrinsically symmetric or predict symmetric extension. In contrast, large-scale numerical models of mantle convection apparently require some form of strain-dependent weakening for the self-consistent formation of first-order features of plate tectonics such as well-defined subduction, transform faulting and ridge formation. [Tackley, 1998; Zhong and Gurnis, 1996].

[4] Several mechanisms may contribute to strain or strain rate dependent weakening but their relative and absolute importance is poorly constrained. Cohesion loss [Buck, 1993], fluid pressure variations [Sibson, 1990], gouge formation and mineral transformations [Bos and Spiers, 2002] may reduce the brittle and frictional strength, whereas, in the viscous domain a transition from dislocation to grain size sensitive creep [Braun et al., 1999; Karato et al., 1986; Poirier, 1980] can reduce the effective viscosity. A combination of these mechanisms in frictional-viscous flow [Bos and Spiers, 2002] may also produce weakening. Strain rate dependent weakening has been used to represent the rate dependence of frictional strength, which is observed in lab experiments [Behn et al., 2002] and may characterize frictional behavior of faults on seismic timescales [Scholz, 1990]. In contrast, our modeling addresses fault weakness at geological timescales. We therefore focus on the strain dependence of material properties.

[5] The purpose here is to determine the effect of strain-dependent softening of frictional-plastic and viscous rheologies on the geometry of models of lithospheric extension. Specifically, forward thermomechanical modeling is used to investigate the consequences of reducing the effective internal angle of friction and/or reducing the scaling viscosity with increasing strain on strain localization and the mode of lithospheric extension. This approach extends recent model studies on the development of faults and ductile shear zones, which show that frictional-plastic and viscous strain softening during extension can lead to large offsets on localized normal faults [Buck, 1993; Frederiksen and Braun, 2001; Huismans and Beaumont, 2002; Lavier et al., 1999].

[6] In the following, we present: (1) the numerical model setup, choice of material parameters and boundary conditions; (2) the results of the numerical models; (3) a proposed rift style classification and model modes template; (4) the primary controls on mode selection; and (5) interpretation of the model results in terms of the controlling factors.

2. Model Description

[7] We use an arbitrary Lagrangian-Eulerian (ALE) finite element method for the solution of thermomechanically coupled, plane strain, incompressible viscous-plastic creeping flows [Fullsack, 1995; Willett, 1999]. The model is used to investigate extension of a layered lithosphere with frictional-plastic and thermally activated power law viscous rheologies (Figure 1). The finite element model solves the force balance equations of equilibrium for quasi-static incompressible creeping flows in two dimensions:

equation image

where summation is implied over repeated indices, p is the pressure, vi are the velocity components with associated strain rates, equation imageij, and stress, σij, given by equation imageij = equation image (∂vi/∂xj + ∂vj/∂xi), and σij = −pδij + 2ηequation imageij, δij = 1 when i = j, = 0 when ij, η is the viscosity, gj is the acceleration due to gravity, ρ is the density, and xj are the spatial coordinates. This system can be solved for the unknowns p and vj subject to mass conservation (incompressibility,) and the boundary conditions.

Figure 1.

(left) Model geometry showing crust and mantle lithosphere layer thicknesses, the weak seed, their corresponding properties, and the velocity boundary conditions, Vext, with Vb chosen to achieve a mass balance. Extension is driven by velocity boundary conditions and seeded by a small plastic weak region. Velocity boundary conditions are chosen to achieve a mass balance in the system. The model has a free top surface, and the other boundaries have zero tangential stress (free slip). Whether materials deform plastically or viscously depends on the ambient conditions. At yield, flow is plastic; below yield, deformation is viscous. Sedimentation and erosion are not accounted for in the model apart from surface smoothing resulting in small amounts of surface diffusion. Eulerian grid dimensions in horizontal and vertical dimension, nx, ny are 401 and 151, respectively, which amounts to double the model resolution of previously published results [Huismans and Beaumont, 2002]. The initial temperature field is laterally uniform and increases with depth from the surface, T0 = 0°C, to base of crust, Tm = 550°C, following a geotherm for uniform crustal heat production, A = 0.8 μW/m3 and a basal heat flux, qm = 20 mW/m2. The temperature increases linearly with depth in the mantle lithosphere and the sublithospheric mantle is isothermal at Ta = 1330°C. The boundary conditions are specified basal temperature, 1330°C, and insulated, q = 0, lateral boundaries. Thermal conductivity K = 2.25 W/m/°C and thermal diffusivity, κ = Kcp = 1 × 10−6 m2/s. Densities of crust and mantle at 0°C are ρ0 = ρc(T0) = 2800 kg/m3 and ρ0 = ρm(T0) = 3300 kg/m3, respectively, and depend on temperature with a thermal expansivity αT = 3.1 × 10−5 °C−1, ρ = ρ0[1 − αT (TT0)]. (right) Initial (solid lines) and strain softened (dashed lines) friction angle, ϕ, 7° → 1° representative strength envelopes of coupled and decoupled models when Vext = 0.3 cm/yr. Frictional-plastic strain softening behavior is shown at top.

[8] When the state of stress is below the frictional-plastic yield the flow is viscous. The constitutive rheologies for the viscous (ductile) flow are based on laboratory measurements on “wet” quartzite [Gleason and Tullis, 1995] and “dry” olivine [Karato and Wu, 1993], where the latter also includes the pressure dependence of viscosity. The effective viscosity in the model is of the general form

equation image

where ηeffv is the effective viscosity (see also Figure 2), equation image is the second invariant of the deviatoric strain rate tensor (equation image), n is the power law exponent, A is a scaling factor, Q is the activation energy, V is the activation volume, which makes the viscosity dependent on pressure, p, and R is the universal gas constant. A, n, Q, and V are derived from the laboratory experiments. For wet quartz, A = 1.1 × 10−28 Pa−n, n = 4.0, Q = 223 kJ/mol, and we use V = 0 because low pressure in the crust makes V unimportant [Gleason and Tullis, 1995] and for dry olivine, A = 2.4 × 10−16 Pa−n, n = 3.5, Q = 540 kJ/mol, V = 25 × 10−6 m3/mol [Karato and Wu, 1993].

Figure 2.

Flowchart of presented models. Models are discussed in the text. Velocity is total extension velocity applied. Initial and strain-softened friction angle define pressure dependence of Drucker-Prager pressure-dependent plasticity. Effective scaling viscosity η used in viscous strain softening defined as η = equation image 3(n + 1)/2/2. Subscripts η denote mineralogy used. Total strain softening interval ϵ gives interval over which weakening occurs. Model codes denote crust in models is either coupled (C) with mantle lithosphere (i.e., completely frictional) or decoupled (DC) with wet quartz rheology in lower crust. Models are nonstrain softening (NSS), frictional-plastic strain softening (FPSS), viscous strain softening (VSS), or combined strain softening (CSS) of both frictional-plastic and viscous rheologies.

[9] The reference parameter values for wet quartz listed above lead to a viscous lower crust in the models discussed here that can “decouple” from the mantle. Models of this type are described as decoupled models. In other models, ηeffv (wet quartz) is increased by a factor of 100 to achieve a crust that is totally in the frictional-plastic regime and is “coupled” to the mantle. Models of this type are termed coupled models. The viscosity scaling (Figure 1) represents a simple technique that creates either strong frictional lower crust or moderately weak viscous lower crust without recourse to additional flow laws, each with its own uncertainties. The scaling can either be interpreted as a measure of the uncertainty in the flow properties of rocks where flow is dominated by quartz (e.g., wet or dry), or taken to represent strong lower crust dominated by minerals that deform in the frictional regime for the conditions (temperature, strain rate) chosen for the model experiments.

[10] The plastic (frictional or brittle) deformation is modeled with a pressure-dependent Drucker-Prager yield criterion which, with suitable adjustment of constants, is equivalent to the Coulomb yield criterion for incompressible deformation in plane strain. Yielding occurs when

equation image

where J2′=$\frac{1}{2}$σij′σij′ is the second invariant of the deviatoric stress, c is the cohesion, ϕ is the internal angle of friction and α = α(ϕ) ∼ 1. With appropriate choices of c and ϕ this yield criterion can approximate frictional sliding in rocks.

[11] The incompressible plastic flow becomes equivalent to a viscous flow if an effective viscosity, ηeffv, for a plastic material is defined such that

equation image

[12] Setting the viscosity to ηeffv in regions that are on frictional-plastic yield satisfies the yield condition and allows the velocity field to be determined from the finite element solution of equation (1). The overall nonlinear solution is determined iteratively using η = ηeffp (for regions of plastic flow) and η = ηeffv (for regions of viscous flow).

[13] A value of ϕ ∼ 30° for sliding friction is observed in many rock types and is a good approximation to Byerlee's law. However, fluid pressure and other effects, some of which are listed in the introduction, may alter the state of stress in the frictional-plastic regime by altering material properties or effective strength through pore pressure effects [Connolly and Podladchikov, 2000; Ingebritsen and Manning, 1999; Rice, 1992; Ridley, 1993]. We do not calculate particular effects explicitly because there are large associated uncertainties. Instead, we determine the sensitivity of the model results to frictional-plasticity that is weaker than ϕ = 30° by defining an effective angle of friction, ϕeff. When Drucker-Prager yielding occurs, pore fluid pressure can be directly related to ϕeff, p sin ϕeff = (ppf). The sin ϕ (pf is the pore fluid pressure) which gives ϕeff ∼ 18° in crust in our models with hydrostatic pore fluid pressure. There may be additional transient fluid pressure effects owing to tectonic deformation, such that when all effects are considered the values of ϕeff may vary from dry/no weakening mechanism, ϕeff ∼ 30°, through the hydrostatic value to overpressured/materially weakened ϕeff ∼ 0°. By using reference initial values of ϕeff = 30°, 15°, and 7° we investigate the sensitivity of the results to a range of tectonic effective frictional-plastic strength regimes.

[14] Strain softening is introduced in the models as frictional-plastic strain softening (FPSS), and viscous strain softening (VSS) by respective linear decreases of the internal angle of friction and the scaling viscosity of the creep law with the second invariant of the deviatoric strain (Figures 1 and 2). Frictional-plastic faults and brittle shear zones may be weakened by high transient, or static, fluid pressures, or mechanical weakened by gouge, or mineral transformations [Bos and Spiers, 2002; Sibson, 1990; Streit, 1997]. For the viscous domain ductile shear zones may be weakened by a change from dislocation to diffusion creep caused by grain size reduction [Braun et al., 1999; Karato et al., 1986]. In most of our models, FPSS uses an initial friction angle ϕ = 7° and a softened friction angle ϕ = 1° and the transition occurs for 0.5 < (I2′)1/2 < 1.5. In other models ϕ, 30° → 4° and ϕ, 15° → 2° combinations are used (Figure 2). Equivalently, ηeffv is reduced by a factor 10 over the same range of strain in VSS. Both types of strain softening occur in combined strain softening (CSS).

[15] We have tested the dependence of the model results on mesh resolution and on the range of strain over which weakening occurs. Increasing the mesh resolution leads to narrower shear zones and to earlier localization because strain accumulates at a higher rate. This does not, however, alter the main character of any particular model result (e.g., compare models 1, 2, 3, and 9 to those of Huismans and Beaumont [2002], which where computed at half the resolution of the models presented here). Also the models do not show a strong sensitivity to the range of strain over which softening occur. The tests indicate that the major control on the model behavior are the threshold at which strain softening starts, the amount of softening that occurs, and the positive feedback between softening and strain accumulation. The mechanical model and initial conditions are shown in Figure 1.

[16] In addition to the mechanical system we also solve the thermal evolution in two dimensions (i = 1, 2) using the finite element method:

equation image

[17] Here T is the temperature, cp is the specific heat, K is the thermal conductivity, A is the heat production per unit volume and repeated indices imply summation. The mechanical and thermal systems are coupled though the temperature dependence of viscosity and density and are solved alternately during each time step. Initial conditions and other thermal properties are given in Figure 1.

3. Results

[18] A range of model experiments was undertaken (Figure 2). Most of our rift models could be classified as narrow rifts [e.g., Buck, 1991]. In section 3.1 we describe the coupled reference model with no strain softening (model 1), models with frictional-plastic strain softening (models 2 and 3), models with viscous strain softening (models 4 and 5), and models with both frictional and viscous strain softening (models 6 and 7). All of these models extend at the reference velocity, 0.3 cm/yr. In the second subsection these models are compared with others in which the initial and final strain-dependent property values are varied (models 2A, 2B, 3A, and 3B). In the final subsection the sensitivity of a subset of models to the extension velocity (0.06, 10.0 cm/yr) is investigated (models 8 and 9). Models are labeled C (coupled) or D (decoupled) (Figure 2) according to whether, or not, the creep viscosity for the wet quartz has been scaled up by a factor of 100.

3.1. Models With Reference Properties

3.1.1. Reference Model 1, Coupled, No Strain Softening (C-NSS)

[19] The reference model (Figures 2 and 3) is coupled and there is no strain softening (NSS). The frictional-plastic crust has an internal angle of friction of 7°. The tectonic style of the deformation is symmetric and evolves in two different phases. Early in phase one deformation is controlled by two cross-conjugate frictional-plastic shears, S1A/B (see Figure 3 for shear zone terminology), that emanate from the weak seed as plastic characteristics (Figure 3a, the early part of phase one is shown by the inset). These shears are relatively focused in the frictional-plastic layers and project at depth into the ductile mantle lithosphere as diffuse forced shears, T1A/B. The effect of the shears is to sever the frictional-plastic lithosphere and the regions external to these shears separate with little deformation except flexural upwarping as the crust above them is tectonically removed. During extension the frictional-plastic mantle is largely “cut out” by this shearing and the ductile mantle lithosphere is brought into close proximity to the lower crust. There is little extensional thinning of the ductile lower mantle lithosphere during phase one, but second-generation shears S2A/B are initiated in the crust (Figure 3a).

Figure 3.

Model 1, C-NSS, coupled model (ηeff (wet quartz × 100)), no strain softening, showing deformed Lagrangian mesh, velocity vectors, and sample isotherms after extension of (a) 84 and (b) 150 km for dashed area in Figure 1. Model layers from top down denote upper and lower crust, strong frictional upper mantle lithosphere, ductile lower lithosphere, and ductile sublithospheric mantle. Scaling of quartz viscosity makes all three upper layers frictional-plastic with the same ϕ = 7°. Note symmetric extension. The drawn overlay lines are interpretation that shows where shears are located on basis of shearing of the mesh and on strain rate plots that show the incipient shears. Inset provides interpretation of the deformation either at the same time as the main panel or shortly before and approximately identifies active and relic shears. The following labeling conventions are used for the shears: S, shears in the frictional-plastic crust and mantle; T, shears in the ductile mantle lithosphere, and D, subhorizontal shear zones in the viscous lower crust. Shears that reach or project to the surface on the left side of the model are labeled A; those projecting to or surfacing on the right side are labeled B. First, second, third, etc., generation shears are labeled 1, 2, 3, etc. Shears in the inset are shown bold when active and fine when inactive. Dashed lines surrounded by elliptic zones are forced diffuse shears and may be active (bold) or inactive (fine). This convention is used in all model result figures. (Animations of model evolutions can be found at http://adder.ocean.dal.ca/huismansetalJGR2003/animations.html).

[20] During phase two (Figure 3b), strain in the ductile mantle is focused as near pure shear necking beneath the overlying rift axis. In the crust, the S1A/B shears are abandoned and shears S3A/B develop as conjugates to S2A/B. T1A/B are abandoned as the ductile mantle necking progresses.

3.1.2. Model 2, Coupled, Frictional-Plastic Strain Softening (C-FPSS)

[21] Model 2 is the same as model 1 except that the internal angle of friction decreases linearly from 7° to 1° as (I2′)1/2 increases from 0.5 to 1.5. The tectonic style again has two phases but the deformation is asymmetric. The earliest parts of phase one are the same as model 1, but once the threshold strain is reached strain softening preferentially focuses strain onto one of the cross-conjugate frictional shears, in this instance S1B (Figure 4a). Asymmetry is caused by the positive feedback between increasing strain and strength reduction (decreased internal angle of friction). Any random perturbation of the symmetric strain field leads to the development of one shear at the expense of the other so that the polarity of the asymmetry is also random. By 22 Myr (Figure 4a), S1B has been rotated out of a favorable orientation for further deformation and is progressively abandoned in favor of S2A. S2A becomes strongly focused by strain softening and links with the forced shear T1B in the ductile mantle.

Figure 4.

Model 2, C-FPSS, coupled model (ηeff (wet quartz × 100)), with frictional-plastic strain softening (see Figures 1 and 2), showing same model properties as in the reference model after extension of (a) 66 and (b) 120 km. Scaling of quartz viscosity makes upper three layers frictional plastic with the same frictional-plastic parameters. Note asymmetric extension, particularly in Figure 4a.

[22] The phase one development is dominated by the frictional model properties, the ductile behavior having no asymmetric tendency. The asymmetry offsets the zones of crustal and mantle thinning by approximately 50 km (Figure 4a). Shearing on S2A and the mantle parts of S1B and T1B uplifts the lower plate and the midmantle lithosphere is juxtaposed first against the lower crust by shearing on S1B and then against the upper crust, when the lower crust is cut out at C by the development of S2A (Figure 4a).

[23] Later, during phase two (Figure 4b) continued exhumation of the lower plate, by shearing on S2A and T1B, upwarps and rotates S1B to a horizontal position and the crustal block above S1B is also rotated as S2B develops (note rotated lower crust, LC). By 40 Myr (Figure 4b) ductile pure shear extension and exhumation of the trailing lower plate, lower lithosphere is well advanced (note advection of P). S1B now has a reverse dip where it is nearly exposed at the surface in the footwall of S2A at P and would appear as a thrust sense shear. Extension is completed by necking of the lower lithosphere beneath P, aided by shear on S3B-T3B, as the two margins separate.

[24] During phase two, S2A, and its continuation at depth, T1B, are the key shears that determine the geometry of the rift and that facilitate extraction, exhumation and exposure of the lower plate. The two protomargins have pronounced differences in width, geometry and structural style. The lower plate margin is wide with mid/lower lithosphere structurally emplaced beneath extended crust. There is also an allochthonous rotated crustal block, LC, distal to the main extensional crustal basin. The upper plate margin is narrow. It contains the early phase one extensional basin formed by shearing on S1A and is bounded and truncated by S2A, the main shear of phase two which exposes mantle in its footwall.

3.1.3. Model 3, Decoupled, Frictional-Plastic Strain Softening (DC-FPSS)

[25] Model 3 is the same as model 2 except that the wet quartz viscosity has the nominal value and the crust can therefore decouple. The early part of phase one is characterized by distributed crustal extension as the crust decouples from the mantle on D1A/B (Figure 5a). Upper mantle extension is focused on S1B, which projects to depth diffusively as T1B. By 23 Myr (Figure 5a) asymmetric extension is controlled by S1B, which has localized in the crust, and is linked to the near-symmetric shears D1A/B. Unlike models 1 and 2, S1A is only weakly active during phase 1. The surface rift flank uplifts are also diminished owing to distributed crustal extension and thinning above D1A/B. The mantle lithosphere, however, retains its flank uplifts at the Moho level, as in model 2.

Figure 5.

Model 3, DC-FPSS, decoupled model (ηeff (wet quartz)), with frictional-plastic strain softening (see Figures 1 and 2), showing same model properties as in the reference model after extension of (a) 69 and (b) 120 km. No scaling of quartz viscosity makes lower crust viscously weak. Note weak early asymmetry followed by symmetric lithospheric mantle necking.

[26] At the end of phase one and during phase two the crust develops a series of localized frictional shears, first S1A, then S2A/B, and finally S3A/B (Figure 5b). There are also more minor conjugate shears on the flanks of the rift, labeled SF. The upper crustal style comprises a set of fault blocks separated by listric frictional shears that sole out into the ductile lower crustal shears D1A/B. These fault blocks are arrayed above, and juxtaposed against, mid/lower lithosphere that has much less asymmetry in its geometry than model 2. The mid/lower lithosphere undergoes mostly distributed necking during phase 2 (Figure 5b). Overall, the tectonic style has more distributed crustal extension than model 2 and, because S2A is less important, there is less dramatic exhumation and no exposure of mantle lithosphere.

3.1.4. Model 4, Coupled, Viscous Strain Softening (C-VSS)

[27] Model 4 is the same as model 2 except that frictional-plastic strain softening is replaced by viscous strain softening. The final results of the two models differ significantly. During the early part of phase one both models are similarly controlled by the S1A/B shears. The difference is that localization in model 4 occurs on the continuation of S1B, T1B, into the upper ductile mantle lithosphere (Figure 6a), whereas localization was in the frictional-plastic region in model 2. Later, in phase one, the S2A/B shears develop in the crust (Figure 6a), and unlike model 2 at this stage, S2B projects into the ductile mantle as T2B, where it is localized. At this stage S1A/B cease activity and the evolution takes the form of mildly asymmetric exhumation of a wedge shaped zone of mantle lithosphere and overlying crust bounded by S2A-T1B and S2B-T2B (Figure 6a). The crustal geometry during phase one is nearly symmetric.

Figure 6.

Model 4, C-VSS, coupled model (ηeff (wet quartz × 100)), with viscous strain softening (see Figure 1), showing same model properties as in the reference model after extension of (a) 75 and (b) 150 km. Scaling of quartz viscosity makes upper three layers frictional plastic. Note early symmetry followed by late asymmetry in lower lithosphere.

[28] During phase 2 (Figure 6b) the crust between S2A/B extends by pure shear and the lower plate ductile mantle lithosphere is extracted and stretched. The result is asymmetric mantle extension in which T1B and T2B are responsible for most of the exhumation, giving a similar asymmetry to that of model 2. The crustal geometry is, however, more symmetric.

3.1.5. Model 5, Decoupled, Viscous Strain Softening (DC-VSS)

[29] Model 5 is the decoupled equivalent of model 4. In the early part of phase one there are no focused upper crustal shears. Instead, two nearly symmetric localized ductile shears in the mantle lithosphere, initially T1A and then T2B, link to localized detachments, D1A/B, at the base of the crust. All of these shears are more focused than those of model 3 owing to the viscous localization, which occurs in the viscous parts of the crust and mantle. Later in phase one S1B, a weak conjugate, S2B, and the flank shears, SF, develop and control the crustal extension (Figure 7a). Ductile extension of the mantle lithosphere is asymmetric at this stage with most shear taken on T1B.

Figure 7.

Model 5, DC-VSS, decoupled model (ηeff (wet quartz)), with viscous strain softening (see Figure 1), showing same model properties as in the reference model after extension of (a) 66 and (b) 150 km. Scaling of quartz viscosity makes lower crust viscously weak. Note asymmetry in crust during both phases.

[30] In phase two a set of plastic characteristics propagates across the rift zone to form conjugate shears, S3-8, and SF in the detached crust. D1A/B also propagate outward, further decoupling the crust and mantle over a wide region (Figure 7b). The crustal geometry resembles a series of plugs with no preferred asymmetry. Extension of the mantle lithosphere is highly asymmetric and has a style similar to that of model 4. At different positions across the rift progressively lower regions of the lower plate lithosphere are structurally emplaced in contact with the base of the lower crust. The model differs significantly from most of those described above because crustal extension is distributed, whereas mantle extension remains more focused. This gives a tectonic style similar to model 3 in which ductile mid and lower mantle is exhumed to the base of the crust but is unlikely to be exposed at the surface.

3.1.6. Model 6, Coupled, Combined Frictional-Plastic and Viscous Strain Softening (C-CSS)

[31] Model 6 combines both frictional-plastic and viscous strain softening mechanisms and should be compared with models 2 and 4 (Figures 4, 6, and 8). Evolution during the early part of phase one is the same as that of model 2 except that the continuation of S1B into the ductile mantle, T1B, is more localized owing to the viscous strain softening (Figure 8a). The two softening mechanisms combine to make deformation highly asymmetric and largely focused on one shear, S1B-T1B. Later in phase one the S2B shear develops (Figure 8a), and the overall geometrical evolution is similar to model 2.

Figure 8.

Model 6, C-CSS, coupled model (ηeff (wet quartz × 100)), with combined frictional-plastic and viscous strain softening (see Figure 1), showing same model properties as in the reference model after extension of (a) 66 and (b) 150 km. Scaling of quartz viscosity makes upper three layers frictional plastic. Note strong asymmetry during both phases.

[32] The evolution during phase two (Figure 8b) is also similar to that of model 2 (Figure 4b). The trailing lower plate mantle is stretched and exhumed as S1B-T1B is rotated and uplifted in the footwall of S2A. Minor differences from model 2 occur because shearing continues on S1B-T1B and because deformation in the trailing ductile mantle is more complex. As in model 2, the two protomargins have pronounced differences in geometry. The lower plate margin is wide with mantle lithosphere structurally emplaced beneath extended upper crust. The narrow upper plate margin, is dominated by S1A and S2A, and has lower mantle lithosphere, and what was originally sublithospheric mantle, structurally emplaced in the footwall of S2A.

3.1.7. Model 7, Decoupled, Combined Frictional-Plastic and Viscous Strain Softening (DC-CSS)

[33] Model 7, which also combines frictional-plastic and viscous strain softening mechanisms, should be compared with the decoupled models 3 and 5 (Figures 5, 7, and 9). Model 7 evolution early in phase one is the same as that of model 5. This similarity is significant because the two models with both frictional-plastic and viscous strain softening (models 6 and 7, respectively) mimic opposite end-member models (5 and 2) with frictional-plastic and viscous strain softening, respectively. Evidently, coupled models favor the expression of frictional-plastic strain softening, whereas decoupled models express viscous strain softening more strongly. For example, the only significant difference between models 7 and 5 in phase one is that in model 7 S1B undergoes frictional strain softening and is therefore more localized (Figure 9a).

Figure 9.

Model 7, DC-CSS, decoupled model (ηeff (wet quartz)), with combined frictional-plastic and viscous strain softening (see Figure 1), showing same model properties as in the reference model after extension of (a) 63 and (b) 150 km. Scaling of quartz viscosity makes lower crust viscously weak.

[34] The evolution of the ductile mantle in phase two is nearly identical to that of model 5 with the same asymmetry and secondary shears, e.g., T2B. In the crust, however, the plugs do not propagate across the rift zone. Instead, the crust in the center of the rift remains intact and new localized shears, S3A and SF, equivalent to S1B and SF, nucleate on the opposite protomargin above the necking mantle lithosphere (Figure 9b). The overall result is a highly asymmetric zone of mantle extension overlain by paired graben structures with opposite polarity. In this case it is anticipated that the lithosphere will rupture beneath the second generation of crustal structures leaving the first generation as relicts on the lower plate margin.

3.2. Sensitivity of Model Results to Strain-Dependent Frictional Properties

[35] Although space precludes a complete sensitivity analysis of all of the reference model properties, the initial, and final, strain-softened frictional-plastic properties are central to the conclusions and it is therefore important to investigate the effect of variations in these properties. The reference ϕ, 7° → 1° frictional-plastic properties tend to represent the weak limit and in this section we investigate higher values of the initial internal angle of friction for coupled and decoupled models equivalent to models 2 and 3. We express the relative reduction of the strength of a frictional shear by its softening ratio, sin (ϕinitial)/sin(ϕfinal) because the shear strength scales as p sin (ϕ) (p is pressure). The softening ratio is approximately 7 for models 2 and 3 and we keep the same ratio for the sensitivity tests but set the initial friction angle to 30° (approximately the dry frictional value in Byerlee's law) and 15°, respectively.

[36] The coupled model (model 2A) results with ϕ, 15° → 2° are similar to those for model 2. The only important difference is that the S2 shears localize later in the evolution. The resulting tectonic style remains asymmetric but lacks the rotated crustal allochthon (Figure 4b), which is mainly the product of exhumation by shearing on S2A. The coupled model (model 2B), with ϕ, 30° → 4° is also asymmetric but lacks the S2 shears. The lower plate is extracted by continued shearing on S1B assisted by a conjugate shear that develops in the mantle of the conjugate margin. This has the effect of transferring most of the crustal allochthons to the upper plate and increases the potential for surface exposure of midlithosphere on the lower plate margin.

[37] Overall, it appears that increasing the reference frictional-plastic strength of the upper lithosphere in coupled models promotes continued deformation on existing primary shears, thereby causing a highly asymmetric tectonic style of crustal extension. Furthermore, the amplitude of the rift flank uplifts increases with increasing reference frictional-plastic strength in such a manner to suggest that models with the initial friction angle approaching 30° exceed natural examples. Except for the positioning of the localized shear beneath the conjugate margin the character of the lower lithosphere extension during phase two of models 2A and 2B remains similar to that of model 2.

[38] The results of the equivalent decoupled models (models 3A and 3B) are also similar to the reference model, model 3 (Figure 5). The significant difference is that the segmentation of the upper crust during phase two into distinct blocks bounded by localized shears or into boudins is more strongly developed than in the reference ϕ, 7° → 1° model. Necking of the viscous lower lithosphere during phase two remains similar to that of model 3.

3.3. Sensitivity of Model Results to Rifting Velocity

[39] In all of the previous models the lithosphere extends at a velocity of 0.3 cm/yr, a slow rifting velocity. We now investigate the sensitivity to extreme values of the extension velocity, 0.06 cm/yr and 10 cm/yr to demonstrate that rifting velocity is an important control of the model tectonic styles. As in section 3.2 the sensitivity analysis is limited, but the results have important implications.

3.3.1. Model 8, Coupled, Frictional-Plastic Strain Softening and Slow Extension (C-FPSS, LOW-V)

[40] Model 8 is the same as model 2 except that the boundary extension velocity is decreased by a factor of 5 to 0.06 cm/yr to demonstrate the sensitivity of lithospheric extension to strain rate and thermal evolution (Figure 10). The model represents an end-member in which the lithosphere remains close to conductive thermal equilibrium (thermal Peclet number much less than one). The thickness of the upper frictional-plastic region is therefore maintained during extension in contrast to the factor of two thinning in model 2. Both the maintenance of the frictional-plastic layer, and the reduced viscous strain rates and associated stresses, imply that model 8 will be more strongly controlled by frictional-plasticity than model 2, or for that matter any of the models described above.

Figure 10.

Model 8, C-FPSS LOW V, coupled model (ηeff (wet quartz × 100)), with strain softening (see Figure 1) but rifting velocity decreased to 0.06 cm/yr, showing same model properties as in the reference model after extension of (a) 60 and (b) 223 km. Scaling of quartz viscosity makes upper three layers frictional plastic. Note strong asymmetry during phase 1 and 2.

[41] In phase one the behavior is very similar to model 2 with the development of the conjugate seeded shears and the asymmetric extension focused on S1B-T1B owing to frictional-plastic strain softening (Figure 10a). This style persists in phase two (Figure 10b) and it is only much later that shears, S2A, S3A, equivalent to S2A/B nucleate and strain soften (Figure 10b).

[42] The overall geometry matches the concept of lithospheric-scale simple shear in which the lower plate has been progressively extracted and exhumed, in this case with little internal deformation, from beneath the upper plate on the throughgoing lithospheric shear S1B. The protomargins contrast strongly with crustal extensional fault blocks on the upper plate and mantle lithosphere structurally exposed at the surface on the lower plate margin. An interesting late stage feature is the localized frictional shear, S2B, that develops in the lower plate mantle (Figure 10b).

3.3.2. Model 9, Coupled, Frictional-Plastic Strain Softening and Fast Extension (C-FPSS, HIGH-V)

[43] Model 9 is the same as model 2 except that the boundary extension velocity is increased by a factor of 33.3 to 10 cm/yr to demonstrate the effects of extreme thermal advection (thermal Peclet number much greater than one), and high strain rates and associated high viscous stresses. It represents the opposite end-member to model 8 and should, therefore, be more strongly controlled by ductile flow, which in this model does not exhibit strain softening.

[44] The results (Figure 11) contrast strongly with models 2 and 8. The early part of phase one is similar to both models 1 and 2, but no strong preference is shown for localization on one of the frictional shears. Instead both S1A/B localize (Figure 11a). Extension is essentially symmetric. The same symmetry persists throughout phase one with the development of S2A/B, and also during phase two with the narrow pure shear necking of the underlying ductile mantle lithosphere (Figure 11b).

Figure 11.

Model 9, C-FPSS HIGH V, coupled model (ηeff (wet quartz × 100)), with strain softening (see Figure 1), but rifting velocity increased to 10 cm/yr, showing same model properties as in the reference model after extension of (a) 75 and (b) 105 km. Scaling of quartz viscosity makes upper three layers frictional plastic. Note return to symmetric rifting style caused by increased rifting velocity.

4. Proposed Rift Style Classification and Model Mode Template

[45] At this stage it is helpful to classify the model results according to a template of rifting styles as a first step toward the development of an understanding of the controls of these styles. The proposed rift classification and mode template is based on the evolving geometries of the upper and lower lithosphere during rifting phases one and two. These two regions potentially have fundamentally different influences on the deformation because they have contrasting frictional-plastic and viscous rheologies. Deformation of these two regions is to first order either symmetric (S) or asymmetric (A), which leads to four fundamental modes as proposed by Huismans and Beaumont [2002]: (1) SS, fully symmetric rifting, implying symmetry, or near symmetry in the extension geometry of both the upper (frictional-plastic) and lower (viscous) regions; (2) AA, fully asymmetric rifting, implying asymmetry of both upper and lower regions; (3) AS, asymmetric upper lithosphere rifting concomitant with symmetric lower lithosphere rifting; and (4) SA, symmetric upper lithosphere rifting concomitant with asymmetric lower lithosphere extension.

[46] Figure 12 shows how the phase one and two model rifting styles fit into the proposed mode template. We divide each model into two main phases, early and late rifting (corresponding to a/b Figures 311) because these phases tend to be respectively dominated by the frictional-plastic and viscous model properties and, therefore, they potentially have different modes. In some cases phase one consists of two different modes that are superimposed incrementally. In these cases additional information from the model results has been used to draw the diagrams and assign the modes.

Figure 12.

Rift style classification and model mode template. The diagrams show how the simplified model geometry evolves and are labeled by both the incremental and cumulative mode. For example, AA, +AA → SA, +SS → SS means that the initial AA mode is incremented by a further AA mode to give an SA geometry at the end of phase one, phase two has an incremental SS mode, and the final geometry is SS.

[47] The diagrams show how the simplified model geometry evolves and they are labeled by both the incremental and cumulative mode. For example, for model 3 AA, +AA → SA, +SS → SS means that the initial AA mode is incremented by a further AA mode to give a cumulative SA geometry at the end of phase one, phase two has an incremental SS mode and the final geometry is SS. It may appear counterintuitive that superposition of two AA modes gives SA but this is a consequence of the change in polarity of the asymmetric crustal extension leading to a cumulative symmetry. Although the cumulative geometries are important the incremental modes are more fundamental. All of the models fit the conceptual mode framework to a first approximation, including models 2A, 2B and 3A, 3B discussed, but not illustrated, in section 3.2.

[48] During phase one, nearly all of the models exhibit incremental SS and AA modes (Figure 12), an expression of the degree to which the FPSS, VSS, and CSS mechanisms are, or are not, able to trigger asymmetry. Strain softening is the fundamental cause of asymmetry in all models, and therefore NSS models are always SS. However, some models with strain softening are also symmetric. During phase two most models retain their phase one modes, but some do switch.

5. Primary Controls on Mode Selection-Dominant Rheology and Strain Softening

[49] The mode selected by a model during extension depends on the interaction of two main controls, the “dominant” rheology and strain softening. In this section we explain the concept of dominant and subordinate rheologies, briefly discuss a simple illustrative example, and list the three factors that determine which rheology is dominant and which is subordinate in the type of model considered here. We then list the additional two factors that arise from strain softening. In section 6 we interpret the model behaviors and mode selection (Figure 12) in terms of these two main controls and the five factors.

5.1. Dominant Rheology

[50] From a simple conceptual point of view the relative “strength” of the frictional-plastic and viscous rheologies can be seen to be one of two main controls of the rifting mode. When one of the rheologies becomes the main load bearing part of the system it can be considered to be the dominant rheology and may force the weaker, subordinate part of the system into a compliant style of deformation. This compliant style may be different from the intrinsic or natural mode that would have been selected in the absence of forcing by the dominant rheology.

[51] This concept is illustrated more quantitatively by considering the rate of energy dissipation in a simple plane strain, two-layer system with a uniform thickness upper frictional-plastic layer bonded to lower uniform thickness, constant viscosity layer with a free slip boundary at the base of the system. This example is a simplification of the coupled models. When the laminate extends under symmetric uniform velocity boundary conditions the deformation mode selected will be the one with minimum energy dissipation rate (in this case the rate of internal work, provided rate of gravitational work, etc. is small). The two main contributions to the energy dissipation rate are from the deformation in the frictional-plastic layer, deformation which may be distributed or focused onto one or more shears, and from the coupled viscous boundary layer that develops below the plastic layer when there is differential extension between the layers. Which mode is selected depends on the trade-off between reduced plastic dissipation, when the plastic deformation is focused, versus increased viscous dissipation when there is differential motion between the layers.

[52] Elsewhere, we report on simple numerical model experiments of this type and use analytical estimates that approximate their energy dissipation rate to make quantitative estimates of mode selection. Here, we note the result that modes are selected in the following order of decreasing domination by viscous dissipation; pure shear extension of the entire laminate, focused symmetric “plug” type mode of the frictional-plastic layer, and focused asymmetric simple shear mode of the frictional-plastic layer.

[53] In the simple example above the mode progression reflects the change from a state dominated by the viscous rheology to a state dominated frictional-plastic rheology. That is, when the viscous coupling is sufficiently weak the viscous dissipation does not influence the mode selected. Conversely, when the viscous dissipation is high, the viscous rheology dominates and the frictional-plastic layer is forced to extend by pure shear even though this is not the preferred mode for this layer. This result implies that the same system will behave differently at different extension rates because the viscous dissipation depends on the strain rate, a result we use in section 6.

[54] The concept of the dominant/subordinate rheologies can be applied to the more complex models (Figure 12) in a qualitative way if the factors that determine which rheology is dominant are classified. The important factors are the evolving model geometry, FEG; the strength distributions of the viscous and frictional model regions, FSD; and the effect of the thermal evolution in determining the rheologies and their strengths, FTH. We use the simplified concept of strength for this discussion but note that a complete quantitative analysis will be based on the energy dissipation rate because it has a firmly based minimization principle, whereas “strength” does not.

5.1.1. FEG, Evolving Geometry of the Frictional-Plastic and Viscous Model Regions

[55] The initial and evolving geometries of the frictional-plastic and viscous regions of the model contribute to the determination of the dominant rheology. For example, the coupled and decoupled models have different viscous and frictional-plastic regions. As models evolve their geometries also change. Typically, during phase one the frictional-plastic rheology dominates the rift zone, but this phase of rifting may sever and dismember the frictional-plastic part of the lithosphere leaving the viscous material in the rift axis to dominate the second rifting phase. Such a change in the dominant rheology, owing to evolving geometry, can produce a change in the rift mode. Alternatively, evolving geometry may reinforce a mode, an example is the necking boudinage instability.

5.1.2. FSD, Evolving Strengths of the Frictional-Plastic and Viscous Model Regions

[56] Relative strengths, as opposed to the geometries, of the frictional-plastic and viscous regions also contribute to the dominant rheology. Viscous stress increases with strain rate, higher stress at higher rifting velocities, whereas frictional strength is independent of strain rate. Models with exactly the same material geometries may therefore have different dominant rheologies and select different modes depending on the rift velocity. The dominant rheology may also change if the extension rate changes during rifting.

5.1.3. FTH, Evolving Thermal State of the Model

[57] The initial temperature distribution contributes to determine the dominant rheology through its effect on the geometry of the viscous and frictional-plastic model regions, and on the viscous strength through the temperature dependence of viscosity [Buck and Lavier, 2001; Lavier and Buck, 2002]. The thermal evolution also influences the dominant rheology. Slow extension, at small thermal Peclet number, allows the lithosphere to cool as it extends, thereby continuously renewing the frictional-plastic regions and promoting frictional-plasticity as the dominant rheology. Fast extension at high thermal Peclet number has the opposite effect.

5.2. Strain Softening

[58] Strain softening is the second of the two main factors that determines the rift mode. Strain softening has the intrinsic capability to induce asymmetry through the positive feedback between softening and increased strain rate, leading and to further softening and localization of strain onto shears. Strain softening also reduces the strength of the model region that softens. For models with fixed velocity boundary conditions reduced strength commonly implies reduced dissipation, that is, strain rates do not increase more than stresses are reduced. Whether the asymmetric tendencies that accompany softening are expressed in the models depends on two factors: strain softening of the dominant rheology, FSS, and; geometrical hardening of localized shears, FGH.

5.2.1. FSS, Strain Softening of the Dominant Rheology

[59] The importance of strain softening in selecting asymmetric modes is not solely determined by the intrinsic softening. Mode selection depends on the interaction of the softening mechanism with the dominant/subordinate rheologies. Strain softening of the dominant rheology promotes asymmetry because the subordinate rheology cannot inhibit the asymmetric mode. Conversely, strain softening of the subordinate rheology alone may not be expressed by asymmetry if the nonsoftening dominant rheology forces a compliant symmetric mode. In both cases the behavior of the dominant rheology determines the mode with the minimum rate of energy dissipation. The interactions may change if there is a switch in dominant rheology as the model evolves.

5.2.2. FGH, Geometrical Hardening of Localized Shears

[60] Even when strain softening has occurred, leading to the development of localized shears, further deformation on these shears may be opposed or reinforced by the evolving geometry [Buck, 1993; Hassani and Chéry, 1996; Lavier et al., 1999]. Geometrical hardening occurs when the model geometry evolves so that the active shears lock and are no longer viable. Even weak shears will lock and be abandoned and this may cause a change in the rift mode. In contrast, strain localization may enhance geometrical softening and could lead to mode switching if the dominant rheology changes as a consequence of softening. We regard geometrical effects to be most important in mode switching, that is in killing the currently active mode. The other factors, listed above are perhaps more important in selecting and promoting modes.

6. Interpretation of Model Results in Terms of the Factors that Control Mode Selection

[61] Figure 13 provides a summary of our qualitative interpretation of the modes selected by the models and shown in Figure 12. Each of the incremental modes (Figure 12) has a corresponding set of factors (shown in brackets, Figure 13) that contribute to the mode selection, arranged with those determining the dominant rheology in the first line. The contribution from strain softening, the second main control, is in the second line together with the type of strain softening, NSS, FPSS, VSS, or CSS. The incremental mode is given above the square bracket and the cumulative mode is indicated by the arrow. The overall mode is listed in the last column. Note that the cumulative mode may never have occurred as an incremental mode. For example, superimposed phases of the AA mode with opposite polarity can result in an overall SS geometry. It is therefore important to focus on the increments in the evolution and not just the final product.

Figure 13.

Summary of model results in terms of primary controls, dominant rheology, and strain-softening rheology and the factors that determine these controls discussed in section 5. Factors controlling dominant rheology are defined in text as FEG, evolving geometry of the frictional-plastic and viscous model regions; FSD, evolving strengths of the frictional-plastic and viscous model regions; and FTH, evolving thermal state of the model. Factors controlling strain softening are FSS, strain softening of the dominant rheology; and FGH, geometrical hardening of localized shears. For each model the factors that act to select the mode are given in brackets together with the dominant rheology. FSS (strain softening) is also labeled with the rheology that softens. The incremental mode progression is included above the brackets and the cumulative and final geometrical modes are indicated by arrows. For example, for model 2 during phase the incremental mode is +AA one because FEG, FSD, and FTH make the frictional-plastic rheology dominant. This combines with FSS, frictional-plastic strain softening, which allows positive feedback and the resulting rift mode for phase one is AA. During phase two the incremental rift mode is +AS because FEG and FSD make the viscous rheology dominant. This combines in the lower lithosphere with the strain-softening rheology being subordinate and a resulting S mode. The dominant viscous rheology, however, fails to force compliance on the overlying frictional plastic region, which continues to extend asymmetrically giving an incremental +AS mode. The final geometry is AA.

[62] No model selects an overall pure shear mode, which would be preferred at high viscous coupling and dissipation. The modes selected are therefore restricted to symmetric and asymmetric deformation of the upper and lower lithosphere.

[63] In the early stages of phase one the models have a relatively uniform layered structure; therefore the initial geometry, strength distributions and thermal state will be the factors that determine the dominant rheology. The interaction between the dominant rheology and strain softening will then determine the mode. This means that the guiding principles described in section 5.1 should apply. Later in the evolution the relative contributions from each factor become more complex because the evolving model properties may lead to a change in dominant rheology and the effect of geometrical hardening of strain-softened shears becomes important. We focus on the reasons for symmetry or asymmetry in phase one and then explain why the mode persists or is replaced in phase two.

[64] Model 1 has an incremental SS mode in both phases because there is no FSS (strain softening) and therefore no tendency for toward asymmetry. In phase one the frictional-plastic rheology is dominant giving the plug mode. During phase two, FEG (evolving geometry) and FSD (evolving strength distribution) transfer control to the viscous rheology and both model layers in the rift zone neck mainly by pure shear.

[65] Frictional-plasticity is the dominant rheology in phase one of model 2 for the same reasons as model 1. The AA mode is selected because FSS (FPSS) acts on the dominant rheology to create asymmetry and the underlying viscous region deforms in a compliant manner. In phase two the viscous rheology is less subordinate (Figure 12), owing to the effects of FEG (evolving geometry) and FSD (strength distribution). The nonstrain softening lower lithosphere therefore selects the S mode because it is less compliant. The overlying frictional-plastic region continues to extend asymmetrically giving an overall incremental AS mode. The final geometry has the AA mode.

[66] Variations on model 2 (models 2A and 2B) are even more strongly dominated by the frictional-plastic rheology owing to their higher internal angles of friction. FSS (FPSS) causes the AA mode to be selected in phase one of model 2A but model 2B remains more symmetric. This is probably a result of FGH (geometrical hardening) because there is less geometrical incompatibility in the plug mode than in the asymmetric mode. For example, in model 2 (Figure 4a) the S2A shear develops as a consequence of geometrical hardening as the shear S1B rotates. It will be more difficult to initiate S2A in the frictionally stronger crust of model 2B [Buck, 1993; Hassani and Chéry, 1996; Lavier et al., 1999]. Both models 2A and 2B are asymmetric in phase two but there is a tendency for the viscous lower lithosphere to achieve some symmetry because the frictional-plastic region is less dominant than it was in phase one.

[67] In model 3 the frictional-plastic rheology is less dominant than in model 2 because there is viscous decoupling in the lower crust, FEG and FSD. However, FSS (FPSS) still combines with the dominant rheology to select the AA mode early in phase one. Later, but still in phase one, deformation switches to the other side of the rift and a second phase of fundamentally AA mode extension is superimposed. FGH (geometrical hardening) is the probable cause of the polarity flip owing to increasing offset between the localized shear zones in the upper crust and in the mantle lithosphere. This change is equivalent to the development of the S2A shear in model 2. In phase two the frictional-plastic control is reduced owing to FEG and FSD and the viscous lower lithosphere extends symmetrically. Extension of the frictional-plastic region is also symmetric at the large scale, but asymmetric at the crustal block level. This tendency to distributed deformation may be controlled by the coupled viscous dissipation as explained in section 5.1. The incremental mode is SS and the final geometry is also SS.

[68] Variations on model 3 (models 3A and 3B) behave in the same way as model 3 and for the same reasons, with two incremental phases of AA mode deformation followed by SS deformation in phase two. It is noteworthy that the polarity flip and onset of the second incremental AA mode occurs earlier in the models with increased internal angles of friction, FSD (increased frictional-plastic strength), which supports FGH as the cause of the polarity flip.

[69] Model 4 can be contrasted with model 2 because FSS (VSS) now acts on the subordinate rheology as opposed to the dominant rheology. Although the initial mode is SA, showing some manifestation of the FSS feedback, the rest of phase one is SS, demonstrating that the nonsoftening frictional-plastic rheology is dominant. This behavior is a variation on that described in section 5.1. Here the dissipation in the coupled viscous boundary layer is minimized when both rheological layers extend in mirror symmetry plug modes reflected across their boundary. In phase two FEG and FSD pass control to the viscous rheology, which combines with FSS (VSS) acting on the new dominant rheology to select the SA mode.

[70] Model 5 is initially SS because frictional-plasticity is the dominant rheology and FSS (VSS) acts on the subordinate rheology creating an inverted plug mode. However, model 5 is decoupled and FEG and FSD allow the viscous rheology to dominate later in phase one. FSS (VSS) then acts to select the AA mode. The cumulative mode is AS at the end of phase one. In phase two the AA mode is retained until the latest stages when deformation propagates across the crust in the rift zone, most likely owing to FGH, to convert what has been an incremental AA mode to a final SA state.

[71] Model 6 is similar to model 2 but is even more strongly AA because FSS (CSS) acts on both the dominant frictional-plastic and subordinate viscous rheologies. This mode persists in phase two because FSS (CSS) continues even when control is transferred to the viscous rheology.

[72] Model 7 selects the same mode progression as model 5 for the same reasons. It becomes AA when FSS (CSS) acts on both rheologies. The AA mode persists through most of phase two. However, late in the evolution the polarity of crustal extension reverses and deformation flips to the opposite side of the rift, most likely the consequence of FGH.

[73] Model 8 is like model 2 in phase one except that the viscous dissipation is reduced owing to the reduced rifting velocity. The model is therefore strongly dominated by the frictional-plastic rheology. FSS (FPSS) strongly favors the AA mode. This mode persists in phase 2 because the low dissipation viscous layer remains compliant and FTH (thermal evolution) maintains a thick strong frictional-plastic dominant rheology, which when reinforced by FSS (FPSS) gives a dramatically asymmetric extension style.

[74] In model 9 the high rifting velocity produces the converse of model 8. The viscous dissipation is increased and the viscous rheology dominates. The SS mode is selected because FSS (FPSS) acts only on the subordinate rheology. The asymmetry is suppressed because asymmetric extension requires higher dissipation in the underlying viscous region. In phase two the viscous rheology continues to dominate as the dismembered frictional-plastic region geometrically weakens (FGH) and becomes more subordinate. The SS mode is maintained.

[75] Taken together, the different modes exhibited by models 8, 2, and 9 are a good demonstration of the role of the viscous rheology in suppressing the preferred asymmetry in the frictional-plastic rheology with strain softening. The increase in viscous dissipation with increasing rift velocity forces the mode to change from AA, for models 8 and 2, to SS, for model 9, because this is now the minimum dissipation mode. This transition is exactly the one described for the simple two-layer model in section 5.1.

7. Conclusions

[76] We have explored the effects of frictional-plastic and viscous strain softening on the asymmetry of lithospheric extension using plane strain thermomechanical model experiments. Parametric strain softening was specified as a linear decrease of the effective internal angle of friction, the effective viscosity, or both, over a range of strain.

[77] The sensitivity of the deformation to the choice of softening parameters was investigated in two types of models; coupled, where the crust is strongly coupled to the mantle lithosphere, and, decoupled, where the corresponding coupling is weak. Results are classified according to the symmetry or asymmetry of the respective deformation of the upper and lower lithosphere during early (phase one) and later (phase two) phases of rifting. We draw the following conclusions.

[78] 1. Predicted rift modes belong to four fundamental types: (1) fully symmetric in which the geometry of both upper and lower lithosphere is approximately symmetric; (2) fully asymmetric rifting; (3) asymmetric upper lithosphere rifting concomitant with symmetric lower lithosphere extension; and (4) symmetric upper lithosphere rifting concomitant with asymmetric lower lithosphere extension (Figure 12).

[79] 2. Mode selection in the early stages of rifting can be understood to be the result of interaction between strain softening and the “dominant-subordinate” rheologies. These interactions are summarized in Figure 14 and the guiding principles are explained in conclusions 3–5.

Figure 14.

Factors controlling extension and the corresponding cumulative modes. (a) Feedback relations between dominant rheology and strain softening mechanism for coupled models for phase one and two. (b) Changes to crustal modes as a result of decoupling in lower crust. Crustal modes are highlighted and mantle lithosphere modes are shown in light gray, as changes occur mostly to crustal rift modes and mantle lithosphere rift modes are the same as in coupled models, apart from phase one of the CSS model. Note increased symmetry in FPSS models and increased asymmetry in VSS models. (c) Sensitivity of mode selection of FPSS coupled models to extension velocity. Note very strong asymmetry at low extension velocity and symmetry with high extension velocity.

[80] 3. Strain softening breaks symmetry and promotes asymmetry through the positive feedback between softening and increased strain, leading to further softening which focuses deformation preferentially onto shears that soften. Models with no strain softening are inherently symmetric if no other mechanism breaks the symmetry. Models with strain softening are intrinsically asymmetric unless asymmetric deformation is inhibited.

[81] 4. Consideration of the rate of internal energy dissipation leads to our definition of dominant and subordinate model rheologies (section 5.1). The dominant rheology has the potential to force compliant deformation of the subordinate rheology if this results in the minimum energy dissipation rate.

[82] 5. Strain softening of the dominant rheology promotes asymmetric extension of the part of the lithosphere that is controlled by the dominant rheology. The subordinate rheology does not inhibit the asymmetry. Conversely, strain softening of the subordinate rheology will not lead to asymmetric extension if the dominant rheology does not strain soften and forces compliant deformation on the subordinate rheology.

[83] 6. Most of the models investigated have a two-phase development. Phase one is generally controlled by the rupture and extension of the upper frictional-plastic lithosphere and phase two is controlled by the viscous (ductile) lithosphere which upwells in the rift axis. When this occurs there is also a change in the dominant rheology between phases. Fully asymmetric rifting is observed in phase one when there is frictional-plastic strain softening or combined strain softening of both rheologies. Viscous strain softening, or combined strain softening promotes asymmetry in phase two. These results are consistent with conclusions 3–5.

[84] 7. The mode selected depends on the rifting velocity because higher strain rates increase the rate of viscous energy dissipation and the viscous rheology can become dominant. This explains the mode change between models 2 and 9 (Figure 14), deformation of the frictional-plastic rheology becomes symmetric when the nonsoftening viscous rheology is dominant. This result is also consistent with conclusions 3–5.

[85] 8. The predicted model modes are similar to the pure shear [McKenzie, 1978], simple shear [Wernicke, 1985], and mixed mode [Lister et al., 1986] kinematic models, respectively, of lithospheric extension. Our model results provide a physical basis for these geometrical models and guidance concerning which mode may be selected. Some general implications follow for natural systems if the same mechanisms operate and the lithosphere behaves as a simple frictional-plastic/viscous laminate with strain softening rheologies. Strain softening promotes asymmetry but the tendency for asymmetric deformation can be suppressed by the behavior of an overlying or underlying coupled layer that inhibits localization and the strain-softening feedback mechanism. Although mode selection may depend on rifting velocity, the intrinsic viscosity of viscous layers in the system is at least as important in determining mode selection. Extension of frictional upper crust is more likely to be asymmetric if it is underlain by a low-viscosity middle or lower crust.

[86] 9. The interaction of the dominant rheology with strain softening does not explain all of the complexities seen in the models. Certainly, some of these result from additional mechanisms including geometrical hardening, which forces the abandonment of active shears and the development of new ones, and the thermal evolution of the models.

[87] 10. The model results should be regarded as a first approximation. They can be improved using methods that include finite element mesh refinement and better, physical, strain-softening mechanisms. In addition, rifting is three, not two dimensional.

Acknowledgments

[88] This work was funded through an NSERC Research Grant to Beaumont. Chris Beaumont also acknowledges support through the Inco Fellowship of the Canadian Institute for Advanced Research and through the Canada Research Chair in Geodynamics. Numerical calculations used software developed by Philippe Fullsack. We acknowledge discussions with Philippe Fullsack, and Jean Braun, which improved the research. We thank the reviewers Roger Buck and Jean Chéry and the Associate Editor Jean Braun for their detailed and constructive comments.

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