Rheological contrast at the continental Moho: Effects of composition, temperature, deformation mechanism, and tectonic regime

Tectonophysics 609 (2013) 480–490

Contents lists available at ScienceDirect

Tectonophysics journal homepage: www.elsevier.com/locate/tecto

Rheological contrast at the continental Moho: Effects of composition, temperature, deformation mechanism, and tectonic regime Giorgio Ranalli a,⁎, Mareike Adams b a b

Department of Earth Sciences and Ottawa-Carleton Geoscience Centre, Carleton University, Ottawa, Canada, K1S 5B6 Department of Mathematics and Statistics, McGill University, Montréal, Canada, H3A 2K6

a r t i c l e

i n f o

Article history: Received 14 February 2012 Received in revised form 10 October 2012 Accepted 30 October 2012 Available online 12 November 2012 Keywords: Moho discontinuity Rheological contrast at the Moho Rheology of lithosphere Deformation mechanisms

a b s t r a c t The rheological contrast at the Moho is an important factor in continental tectonics. This paper explores systematically the effects of composition, temperature, deformation mechanism, and tectonic regime on the strength contrast, considering four compositions for the lower crust (felsic granulite, mafic granulite, wet diabase, dry diabase) and two for the lithospheric mantle (dry and wet peridotite). The strength contrast of the resulting eight compositional layerings is estimated as a function of Moho temperature which is varied from 600 to 1500 K. The Moho temperature can be converted to surface heat flow if the thickness and composition of the crust are known. Besides a “standard” case (crustal thickness 35 km), the cases of a thick (50 km) and thin (20 km) crust are also considered (with wet quartzite crustal rheology in the latter case). Results show a great variety of strength contrasts according to different conditions. Excluding the case of very low Moho temperature (TM ≤ 600 K), when the behaviour of both lower crust and upper mantle is frictional brittle and therefore the strength contrast vanishes, the strength contrast (at a given strain rate) is a strong function of composition, temperature, and tectonic conditions. Weak compositional stratification (e.g., soft lower crust/soft lithospheric mantle or hard lower crust/hard lithospheric mantle) results in lower contrasts than strong compositional stratification. For any given compositional combination, the absolute value of the strength contrast is higher in compressional as compared to extensional tectonic environments, and tends to decrease with increasing temperature from a maximum of hundreds of MPa at low-to-intermediate Moho temperatures (600 b TM b 1000 K) to values less than a few MPa at higher temperatures (TM > 1200 K). Therefore, rheological layering is favoured by strong intrinsic (i.e., compositional) strength contrasts between lower crust and upper mantle and relatively low Moho temperatures. © 2012 Elsevier B.V. All rights reserved.

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Lithospheric structure and composition . . . . . . . . . . . . 2.2. Lithospheric geotherm . . . . . . . . . . . . . . . . . . . . 2.3. Rheological behaviour . . . . . . . . . . . . . . . . . . . . 3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Strength contrast as a function of composition and temperature 3.2. Effects of variations of strain rate and pore fluid pressure . . . 3.3. Effects of crustal thickness variations . . . . . . . . . . . . . 3.4. Possibility of high-pressure failure . . . . . . . . . . . . . . 4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

⁎ Corresponding author. E-mail address: [email protected] (G. Ranalli). 0040-1951/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.tecto.2012.10.037

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

481 480 481 481 481 482 483 484 486 487 488 488 489 489

G. Ranalli, M. Adams / Tectonophysics 609 (2013) 480–490

1. Introduction The continental lithosphere is a compositionally layered body. For the purpose of large-scale rheological modelling, it is usually assumed to consist of three layers: upper crust, lower crust and lithospheric mantle, whose brittle or ductile strengths are primarily dependent on composition and temperature. A large literature exists on the variations of rheological properties of the lithosphere as a function of depth, and the construction of “strength envelopes” (also termed “rheological profiles”) is a well-established procedure (cf. e.g. Afonso and Ranalli, 2004; Burov, 2010; Cloetingh and Burov, 1996; Kirby and Kronenberg, 1987; Kohlstedt et al., 1995; Ranalli, 1995, 1997). Despite being subject to considerable uncertainties, they provide a first-order estimate of the rheology of the lithosphere. Both modelling and observation show that the spectrum of rheological behaviours of the lithosphere varies between two end members, which have been termed “jelly sandwich” and “crème brulée” (Burov, 2010; Maggi et al., 2000). In the former, a frictionally brittle and relatively strong upper crust is separated from a strong (brittle or ductile) uppermost mantle by a soft ductile lower crust, and consequently the Moho is a major rheological discontinuity. In the latter, the only strong layer is the upper crust, which overlies a ductile lower crust and lithospheric mantle, and the Moho is a minor or negligible rheological discontinuity. However, when the rheological behaviour is modelled strictly on the basis of the constitutive equations of lithospheric materials, it becomes clear that the difference between the two models is not a dichotomy, but reflects the end members of a continuous spectrum of behaviours, depending on composition and temperature (Afonso and Ranalli, 2004). This paper focuses on the rheological contrast at the continental Moho (a discussion of the oceanic lithosphere as a function of age and composition can be found in Mahatsente et al., 2012). The Moho discontinuity is taken in the compositional sense (petrological Moho; for a discussion of the discrepancies between petrological and seismological Moho cf. O'Reilly and Griffin, 2013–this volume). The main purpose is to provide a baseline for predicting the strength variations at the Moho under different conditions of composition, crustal thickness, temperature, tectonic regime, and strain rate, which can be used as a basis for more detailed regional studies. Rheological predictions based on experimental results on rock deformation are subject to intrinsic uncertainties which are mainly a consequence of the necessary simplifications on the composition of the relevant rocks and the reproducibility of laboratory data. These are not easily quantifiable (cf. e.g. Karato, 2008; Kirby and Kronenberg, 1987; Korenaga and Karato, 2008; Ranalli, 1995 for discussions), but from the present viewpoint results of rheological modelling can be probably considered valid within an order of magnitude. Potentially more important limiting factors are the assumption of constant strain rate and the choice of predominant deformation mechanisms. The constant strain rate assumption can be overcome by incorporating the time dependence of stresses, strains, and strain rates and the effects of the applied force on bulk lithosphere deformation (cf. e.g., Ershov and Stephenson, 2006). The possibility of additional deformation mechanisms, besides the usual low-temperature frictional brittle fracture and high-temperature power law creep, is considered in this paper (see Section 2.3). The plan of the paper is as follows. Section 2 describes the main features of the model: choice of crustal thickness and composition (Section 2.1); estimation of crustal geotherms and corresponding surface heat flow for a given Moho temperature, crustal thickness and composition (Section 2.2); and rheological behaviour of the model materials (Section 2.3). Section 3 presents the estimated strength contrast at the Moho as a function of composition and temperature for the “standard case” (crustal thickness 35 km; Section 3.1), followed by a discussion of the possible effects of variations of strain rate and pore fluid pressure (Section 3.2), crustal thickness (Section 3.3), and the possibility of high-pressure failure (Section 3.4). Section 4 summarises the results

481

and discusses their implications and limitations for geodynamic modelling. 2. Model 2.1. Lithospheric structure and composition The “standard” model has a crustal thickness h = 35 km. The crust consists of two layers: upper crust (h1 = 20 km) and lower crust (h2 = 15 km), of densities ρ1 and ρ2 respectively, overlying the lithospheric mantle of density ρm (parameter values are shown in Table 1). For the purposes of comparison, we have considered also a thick crust (h = 50 km, h1 = 20 km, h2 = 30 km) and a thin crust (h = 20 km, consisting of one layer only). The thickness of the lithospheric mantle need not be specified, as the focus is on the change of rheological properties at the Moho, but the lithosphere/asthenosphere boundary can be assumed to be defined in the usual thermal sense as the intersection of the geotherm with the 1570 K mantle adiabat (Artemieva, 2011; Artemieva and Mooney, 2001). The behaviour of the upper crust is assumed to be controlled by the rheology of wet quartzite, but this assumption does not enter the estimation of strength contrast except in the case of thin crust where the entire crustal layer is assumed to have this composition. The behaviour of the lower crust is estimated for four different compositions, two of which can be considered relatively soft (felsic granulite, wet diabase) and two relatively hard (mafic granulite, dry diabase). The behaviour of the upper mantle is assumed to vary between relatively soft (wet peridotite) and relatively hard (dry peridotite). Therefore (except in the case of thin crust) there are eight possible compositional contrasts across the Moho, which we examine as functions of Moho temperature varying between 600 and 1500 K. 2.2. Lithospheric geotherm The thermal state of the continental lithosphere depends on many factors (cf. the extensive discussions in Artemieva, 2011; Jaupart and Mareschal, 2010). In the present context, we need to establish a correspondence between Moho temperature TM and surface heat flow Qo, so that the estimated strength contrast can be expressed as a function of both. For this purpose, we estimate type geotherms by solving the steady-state conductive heat transfer equation. The assumption of steady-state is strictly valid only for areas where the age of the last tectonothermal event is of the order of tens of Ma or more. However, since for a given crustal thickness and composition the heat flow across the Moho is a function of TM (see below), the transient response to relatively recent tectonothermal events is taken into account by increased mantle heat flow. Assuming continuity of both temperature and gradient at interfaces within the lithosphere, the temperature at the Moho depth z2 is given by (Afonso and Ranalli, 2004) T ðz2 Þ ¼ T o −


   A2 2 A Q A z 2 A1 A2 Z 2 þ 2 Z 1 þ M þ 2 ðZ 2 −Z 1 Þ Z 2 þ 1 − 2K 2 K2 K1 K1 2 K1 K2 ð1Þ

where To = 285 K is surface temperature; A and K are volumetric heat generation rate and thermal conductivity, respectively; z is depth to the bottom of the relevant layer (subscripts 1 and 2 refer to upper and lower crust); and QM is the Moho heat flow (heat flowing from the mantle across the Moho). Note that “Moho heat flow” as used in this paper is different from “basal heat flow” (see e.g. Artemieva and Mooney, 2001) which is the contribution to surface heat flow coming from below the enriched upper crust, i.e. including also the lower crustal heat production. The Moho heat flow can be written as Q M ¼ Q o −Q c ¼ Q o −ðA1 h1 þ A2 h2 Þ

ð2Þ

482

G. Ranalli, M. Adams / Tectonophysics 609 (2013) 480–490

Table 1 Thermal and creep parameters used in Eqs. (1)–(6). Density and thermal parameters from Afonso and Ranalli (2004); references for creep parameters given in the last column and footnote. Material

ρ (kg m−3)

A (μW m−3)

K (W m−1 K−1)

Ac (MPa−n s−1)

n

E (kJ mol−1)

Ref.

Quartzite (wet) Felsic granulite Mafic granulite Diabase (dry) Diabase (wet) Peridotite (dry) Peridotite (wet) LTP Peridotite (dry)

2640 2750 2880 2850 2850 3320 3320 Ap (MPa−2 s−1) 1.4 × 10−7

1.40 0.45 0.33 0.40 0.40 (1) (1)

2.5 2.1 2.1 2.1 2.1 (1) (1) Ek (kJ mol−1) 320

3.2 × 10−4 8.0 × 10−3 1.4 × 104 8.0 2.0 × 10−4 2.5 × 104 2.0 × 103

2.3 3.1 4.2 4.7 3.4 3.5 4.0 σp (MPa) 5.9 × 103

154 243 445 485 260 532 471

[1] [2] [2] [3] [4] [5] [5,6] [7]

(1)

Parameters not required (see Eqs. (1)–(3)). References: [1] Kirby and Kronenberg, 1987; [2] Wilks and Carter, 1990; [3] Mackwell et al., 1998; [4] Shelton and Tullis, 1981; [5] Chopra and Paterson, 1984; [6] Chopra and Paterson, 1981; [7] Mei et al., 2010.

where Qo is surface heat flow and Qc the heat flow contributed by crustal sources. Combining Eqs. (1) and (2) we obtain surface heat flow in terms of Moho temperature and crustal heat generation. Q o ¼ Q c þ ðT M −T o Þ

 z2 K1 K z þ A2 ðz1 −z2 Þ þ A2 1 2 −Z 1 þ 1 z2 K2 2 2z2

  K1 A2 −A1 K2

ð3Þ The values of the thermal parameters are listed in Table 1. They have been taken as constant in each layer, and fall within the usual estimates for the lithosphere (Artemieva, 2011; Hasterok and Chapman, 2011; Jaupart and Mareschal, 2010; Vilà et al., 2010). The case of a one-layer crust can be obtained in a similar way. Since the relation between QM and Qo is linear and Qc is uniquely determined by crustal thickness and composition, any change in surface heat flow is reflected in a change in Moho heat flow. This is a consequence of the adopted 1D model of the lithosphere, which does not take into consideration lateral variations of heat production in the crust. The resulting values of Moho heat flow for 700 b TM b 1300 K (that is, excluding extreme Moho temperatures) are in the range ~ 30–60% of Qo for crustal thickness 35 km and the assumed compositions, and ~ 20–50% of Qo for crustal thickness 50 km. These values are within the usual estimates (Artemieva and Mooney, 2001; Čermák, 1989; Hasterok and Chapman, 2011). The values for the case of a 20 km thick crust are much higher (70–80% of Qo), reflecting the fact that in this case most of the contribution comes from heat advection in the mantle. The results on strength contrast at the Moho (Section 3) are given in terms of both TM and Qo. 2.3. Rheological behaviour The rheological behaviour of lithospheric materials can be subdivided into a low-temperature brittle field and a high-temperature ductile field (cf. e.g. Ranalli, 1995). In the brittle field, failure usually occurs by frictional shear fracture (Coulomb–Byerlee criterion), which depends on pressure but is roughly independent of rock type. The shear failure criterion can be written as (Sibson, 1974) σ ¼ βo ρgzð1−λÞ

ð4Þ

where σ is the critical stress difference; g is gravity; ρ is the average rock density above depth z; and λ is the pore fluid factor (ratio of pore fluid pressure to lithostatic pressure). We assume hydrostatic pore fluid pressure, i.e. λ = 0.4. The numerical factor βo is a function of material parameters (cohesion and friction coefficient) and orientation of the stress field. Assuming negligible cohesion, a friction coefficient μ = 0.75, and an Andersonian stress field (one principal stress direction vertical), βo varies from 0.75 to 3.0 for extensional (normal

faulting) and compressional (reverse faulting) tectonics, respectively (cf. e.g. Ranalli, 1995). A vast literature is available on the ductile rheology of lithospheric materials (cf., among many others, Chopra and Paterson, 1981, 1984; Drury, 2005; Faul and Jackson, 2007; Faul et al., 2011; Hansen et al., 2011; Keefner et al., 2011; Kirby and Kronenberg, 1987; Kohlstedt et al., 1995; Mackwell et al., 1998; Rutter and Brodie, 1992; Shelton and Tullis, 1981; Wilks and Carter, 1990; and the discussions in Karato, 2008; Ranalli, 1995). The predominant creep mechanism at high temperature is usually power-law creep (PLC), where the ratecontrolling factor is some form of diffusion-controlled dislocation climb. Although grain-size-dependent grain boundary sliding and oxygen fugacity have been experimentally found to play a role in olivine, the variability of these parameters in the lithosphere is such that for the present purposes the PLC constitutive equation can be written as   E n ε_ ¼ Ac σ exp − RT

ð5Þ

where σ and ε_ are deviatoric stress and strain rate; Ac, n, and E are creep parameters; and R and T are gas constant and absolute temperature. The activation energy E characterises the temperature dependence of creep; the pressure dependence is negligible at lithospheric pressure, and in any case is not known except for olivine. As mentioned in Section 1, experimental uncertainties on creep parameters are rather wide, and moreover a rigorous statistical treatment of uncertainties has been applied only to olivine rheology (Korenaga and Karato, 2008). Since statistical analyses of uncertainties for other compositions are lacking, and in order to facilitate comparison with rheological models available in the literature, we use the most commonly accepted values in this paper, which are shown in Table 1. If the grain size is very small (b10 μm as an order of magnitude) diffusion creep, where the rate-controlling process is the migration of vacancies, becomes predominant in silicates (cf. e.g. Karato, 2008; Ranalli, 1995). This mechanism may be relevant in dynamically recrystallized shear zones, but it is unlikely to be the predominant creep mechanism in the bulk of the lithosphere, where average grain size is larger. Estimates of grain size in the shallow upper mantle, consistent with shear wave velocity and attenuation, are of the order of mm if the mantle is dry (Faul and Jackson, 2005) or of the order of cm in the presence of water (Behn et al., 2009). Thus, while the rheology, at least in the upper mantle, is likely to be composite, i.e., a combination of dislocation and diffusion creep mechanisms, the evidence favours the predominance of PLC. Consequently, following standard procedure in estimates of lithosphere rheology (Burov, 2010; Cloetingh and Burov, 1996; Ranalli, 1995), we use Eq. (5) to model the high-temperature ductile behaviour of lithospheric materials. Another flow law, which applies when creep is controlled by dislocation glide, has been observed in dry olivine and includes a

G. Ranalli, M. Adams / Tectonophysics 609 (2013) 480–490

483

stress-dependent activation energy (Faul et al., 2011; Goetze, 1978; Mei et al., 2010). It can be written as " 2 ε_ ¼ Ap σ exp −

Ek RT

sffiffiffiffiffiffi!# σ 1− σp

ð6Þ

where Ap is a creep parameter, σP is the Peierls stress, and Ek is a zero-stress activation energy. This flow law (low-temperature plasticity, LTP), is predominant at high stress and relatively low temperature and is potentially relevant to the ductile deformation of the lithospheric mantle (cf. the discussion in Mei et al., 2010 for olivine under anhydrous conditions). We have therefore included LTP in estimates of dry upper mantle rheology (parameter values are shown in Table 1). However, an estimate of the role of LTP in wet upper mantle rheology and for other lithosphere compositions is precluded by the lack of knowledge of the relevant creep parameters. The brittle strength predicted by Eq. (4) becomes unrealistically large at high confining pressure, unless the pore fluid factor approaches unity. There is experimental evidence, however, that the linear dependence between strength and pressure (i.e., depth) predicted by the Coulomb–Byerlee criterion breaks down at high pressure. Another type of brittle fracture, depending on temperature and strain rate in addition to pressure, becomes predominant at high pressure (P ≥ 1 GPa; Ord and Hobbs, 1989; Renshaw and Schulson, 2007; Shimada, 1993; Zang et al., 2007). Available results can be summarised by the relation (Zang et al., 2007)
 m 
  
  ε_ P T β σ ¼ Bo 1 þ k 1 þ α log 1 þ γ log ε_ r Bo Tr

ð7Þ

where P is pressure (MPa); T is temperature (degrees Kelvin); ε_ is strain rate; Tr =298 K and ε_ r = 10−5 s−1 are normalising factors; and Bo, k, m, α, β, and γ are empirical fracture parameters. Their values are approximately known only for three broad rock types (Zang et al., 2007; see Table 2), which makes it difficult to use Eq. (7) directly in this paper (cf. Section 3.4). A previous analysis has shown that high-pressure fracture is potentially relevant only in areas with low geotherm, especially if the ductile rheology is hard (Pauselli et al., 2010). Fig. 1 illustrates the relative differences in creep behaviour of crustal materials. Lower crustal strain rates are shown in Fig. 1A as a function of temperature at a reference stress of 10 MPa. Over the whole range of Moho temperatures, the hardest rheology is that of dry diabase (note that the parameters adopted here – from Mackwell et al., 1998 – result in a harder rheology than previous estimates; e.g. Carter and Tsenn, 1987), followed in order of increasing softness by mafic granulite, wet diabase and felsic granulite. Differences decrease with increasing temperature. The wet quartzite upper crustal rheology is shown for comparison in Fig. 1B. Under the same stress conditions, it is ~ 5 o.o.m. softer than felsic granulite for TM = 600 K, and the difference gradually decreases to b 1 o.o.m. for TM = 1500 K. Apart from the very soft upper crust, lower crustal ductile behaviour can be classified as relatively soft if the composition is felsic granulitic or wet mafic (these two rheologies are close to “intermediate,” i.e. plagioclase-controlled, rheology; cf. e.g. Ranalli, 1995), or relatively hard if the composition is mafic and dry. Mantle ductile behaviour is shown in Fig. 2 as a function of stress at the reference temperature T = 1250 K (Fig. 2A), and as a function of

Fig. 1. Ductile rheology of crustal materials, presented as strain rate as a function of temperature under constant stress: (A) Lower crust; (B) Upper crust.

temperature at the reference stress σ = 10 MPa (Fig. 2B), respectively. Both PLC and LTP are included, but a comparison between the two holds only in the case of dry rheology. At the chosen temperature, LTP becomes predominant over PLC in dry peridotite at σ ≥ 100 MPa; but even at the relatively low stress σ = 10 MPa, LTP is predominant in the low temperature range (T ≤1000 K). This mechanism, therefore, is relevant to flow in relatively cold dry lithospheric mantle, but its relevance to wet conditions cannot be assessed at present. 3. Results

Table 2 Parameters for high-pressure fracture (Eq. (7); Zang et al., 2007). Material

B0 (MPa)

k

m

α

β

γ

Granite Gabbro Peridotite

34.1 36.1 28.3

4.57 3.18 3.35

0.52 0.55 0.68

−1.128 −2.536 −1.875

1.732 2.340 1.310

0.035 0.035 0.035

For each combination of lower crust/upper mantle composition, the critical stress for shear fracture was estimated from Eq. (4), and the creep strength (stress as a function of temperature required to produce a given steady-state strain rate, here taken as 10 −14 s −1) was estimated for the relevant material from Eqs. (5) and (6) (the latter for dry peridotitic composition only). The strengths immediately above and below the Moho are those pertaining to the deformation

484

G. Ranalli, M. Adams / Tectonophysics 609 (2013) 480–490

mantle and in the lower crust (that is, those requiring the least critical stress in each case) are denoted by the following markers Marker

Rheology of upper mantle

Rheology of lower crust

X ♦ ● ■ + ▲

Frictional brittle Frictional brittle Low-T plasticity Low-T plasticity Power-law creep Power-law creep

Frictional brittle Power-law creep Frictional brittle Power-law creep Frictional brittle Power-law creep

We first present the results for the standard case; then discuss how different factors introduce uncertainties; finally we briefly compare results for different crustal thickness, and discuss the possibility of high-pressure fracture. 3.1. Strength contrast as a function of composition and temperature

Fig. 2. Ductile rheology of the lithospheric mantle: (A) Stress–strain rate relations at a reference temperature; (B) Strain rates as a function of temperature under constant stress.

mechanisms requiring the lowest stress. If these minimum stresses for lower crust and upper lithospheric mantle are denoted as σlc and σlm, respectively, the strength contrast at the Moho is defined as Δσ = σlm − σlc. The assumption of constant strain rate is an approximation which restricts the use of the rheological contrast at the Moho only as a measure of local relative strength. A discussion of its general applicability to dynamic models of lithospheric deformation is not within the scope of this paper, but some brief remarks on this problem are given in the final part of Section 4. Figs. 3–6 show the results for the standard model (h =35 km). The rheological contrast Δσ is given as a function of Moho temperature and surface heat flow (obtained from Eq. 3). Extensional tectonic settings (βo =0.75 in Eq. (4)) are denoted by a blue line and compressional settings (βo = 3) by a red line (when the two curves coincide the colour is simply red; note also that the contrast scale varies from figure to figure). The two predominant deformation mechanisms in the upper

A soft lower crust (rheology of felsic granulite or wet diabase) in an extensional tectonic environment fails by frictional shear fracture at T ≤ 600–700 K (given the various uncertainties, we give temperature to the nearest hundred), and flows by PLC at higher temperature. In a compressional tectonic environment, the transition temperature is ~ 100 K less. This is a consequence of the more rapid increase of brittle strength with depth (Eq. (4)) in compressive environments, which causes the intersection between brittle and creep strength curves to occur at a shallower depth. For a hard lower crust (rheology of mafic granulite or dry diabase), the transition temperature to ductile behaviour is slightly higher (700–900 K), with the highest values being associated with dry diabase in extension. The transition temperature between frictional shear failure and PLC for a soft upper lithospheric mantle (wet peridotite rheology) is ~ 800–900 K, with the higher values relevant to extension. If the rheology is dry, the lower-temperature part of the ductile field (comprised between frictional brittle behaviour and power-law creep) is occupied by LTP in a relatively narrow temperature range (900 b T b 1100 K) if the tectonic regime is extensional. In a compressional environment, however, LTP extends to lower temperatures. In both cases, PLC is predominant at T > 1100 K. The relevance of LTP for wet rheology cannot be evaluated since the constitutive law is reliably determined for olivine under anhydrous conditions only. The strength contrast Δσ is a strong function of composition and temperature (or the corresponding surface heat flow). Given the uncertainties in the values of creep parameters, we discuss only four types of lithological layering, grouping together felsic granulite and wet diabase compositions into a “soft lower crust,” and mafic granulite and dry diabase compositions into a “hard lower crust.” Combined with either wet or dry mantle, these four cases can be used as first-order estimates on which to base more accurate regional estimates. • Soft upper mantle/soft lower crust (wet peridotite/felsic granulite or wet diabase): The trends for the two lower crustal compositions are very similar (Fig. 3). For very low Moho temperatures (TM ≤ 600 K), materials above and below the Moho are both frictional brittle, and therefore Δσ = 0. As temperature increases, Δσ reaches a maximum (the lithospheric mantle being harder than the lower crust) at TM = 800 K, and then decreases with increasing temperature (where both deformation mechanisms are governed by PLC) to become negligible (Δσ b 10 MPa) for TM ≥ 1200 K. The high values of Δσ occur when the upper mantle is brittle and the lower crust ductile. As the brittle strength is governed by Eq. (4), peak values for extensional tectonic environments (~300–400 MPa) are four times less than peak values for compressional environments. • Hard upper mantle/soft lower crust (dry peridotite/felsic granulite or wet diabase): The trends for the two lower crustal compositions are again very similar, and reproduce the main characteristics of the

G. Ranalli, M. Adams / Tectonophysics 609 (2013) 480–490

485

Fig. 3. Strength contrast at the Moho as a function of Moho temperature and surface heat flow for the case soft upper mantle/soft lower crust: (A) Wet peridotite/felsic granulite; (B) Wet peridotite/wet diabase. Blue curve: extension; red curve: compression. Symbols for creep mechanisms listed in text, Section 3. (Same symbols used in Figs 3–6.)

previous case (Fig. 4). Both lower crust and upper mantle are brittle up to TM = 600–700 K. The strength contrast Δσ reaches a maximum (Δσ= 500–600 MPa in extension, ~800–1000 MPa in compression) in the 700–1000 K temperature range. The high values in extension occur when the upper mantle is brittle and the lower crust ductile; those in compression are associated with the occurrence of LTP in the upper mantle. • Hard upper mantle/hard lower crust (dry peridotite/mafic granulite or dry diabase): LTP in the upper mantle plays a role in this case (Fig. 5). For both lower crustal compositions, both lower crust and upper mantle are brittle (Δσ = 0) for TM ≤ 800–900 K in an extensional tectonic regime. The strength contrast reaches a maximum (Δσ= 200–400 MPa) for TM in the 900–1100 K range (these values are associated with brittle or LTP behaviour of the upper mantle and power-law creep in the lower crust). It then decreases to negligible values for TM ≥ 1200 K, where both lower crust and upper mantle deform by PLC. The trend in compression is similar, except that Δσ assumes large (up to 600–800 MPa) negative values in the lower

temperature range (TM ≤700–800 K). This result shows that the general assumption in geodynamic modelling that the strength of the upper mantle is always larger than, or equal to, that of the lower crust is not necessarily always valid. Negative values may occur when the lower crust is brittle and the upper mantle deforms by LTP. They are a consequence of two factors: a hard lower crust (so that strength for PLC is higher than frictional brittle strength at low temperature); and the large reduction in creep strength in the upper mantle brought about by the predominance of LTP. The experiments forming the basis of Eq. (6) (Mei et al., 2010) were performed in the temperature interval ~700–1300 K, and extrapolation to geological strain rates does not indicate a clear low-temperature cutoff for its validity. Consequently, the result that under some circumstances the upper mantle is softer than the lower crust must be tentatively accepted, although it must be subjected to further experimental confirmation. • Soft upper mantle/hard lower crust (wet peridotite/mafic granulite or dry diabase): The results in this case (Fig. 6) depend critically on the

486

G. Ranalli, M. Adams / Tectonophysics 609 (2013) 480–490

Fig. 4. Strength contrast for hard upper mantle/soft lower crust: (A) Dry peridotite/felsic granulite; (B) Dry peridotite/wet diabase.

composition of the lower crust. In the ductile field (PLC), which sets in above TM > 800 K for all three lithologies involved, the rheology of dry diabase is the hardest, followed in turn by wet peridotite and mafic granulite. Therefore, while the variations of Δσ with TM follow the usual trend in the case of lower crust of mafic granulite composition (the high value at TM = 800 K is a consequence of the fact that the upper mantle is still brittle in compression), the negative values of Δσ when the lower crust has the rheology of dry diabase are a consequence of the high creep strength of dry diabase, which causes the lower crust to be brittle in extension and ductile (but harder than the upper mantle) in compression. As we already stated in Section 2.3, the parameter values for PLC in dry diabase adopted here result in a very hard rheology and probably represent a limiting case. In terms of surface heat flow, estimated according to the procedure outlined in Section 2.2, high strength contrasts at the Moho (both positive and negative) correspond to surface heat flow less than ~ 70–80 mW m −2. This holds for all lithologies. However, in

the lower temperature range (TM b 800 K, i.e. Qo b 60 mW m −2), the contrast in some cases becomes nil or negligible (this trend is more pronounced in extensional tectonic regimes, especially for hard lower crust). As TM and Qo increase, the strength contrast monotonically decreases, and becomes negligible in all cases for TM ≥ 1100 K, corresponding to Qo ≥ 80 mW m −2 for the assumed crustal thickness and composition.

3.2. Effects of variations of strain rate and pore fluid pressure A change in strain rate affects ductile behaviour. An increase or decrease of one order of magnitude (thus covering the range 10 −13-10 −15 s −1) increases or decreases the PLC strength by a factor ~ 2. As long as both materials are in the PLC field, the strength contrast does not change. However, if one is in the brittle field and the other is ductile, an increase in strain rate decreases the strength contrast by an amount which depends on the reference strength contrast and therefore the Moho temperature. The corresponding increase/

G. Ranalli, M. Adams / Tectonophysics 609 (2013) 480–490

487

Fig. 5. Strength contrast for hard upper mantle/hard lower crust: (A) Dry peridotite/mafic granulite; (B) Dry peridotite/dry diabase.

decrease for LTP is ~ 3 to the first order, and consequently the changes in strength contrast associated with changes in strain rate follow a similar trend. Pore fluid pressure in the frictional brittle field has a potentially important effect, which however has to be evaluated on the basis of information in specific situations. In general, a hydrostatic pore pressure, as assumed in this paper, is a valid first-order assumption for the upper crust as long as the porosity is connected to the surface. At larger depths, the pore fluid factor could differ from the hydrostatic. The effects of pore fluid pressure depend on the interplay of the predominant deformation mechanisms. Completely dry conditions (λ = 0), especially when combined with low geotherms, would result in unrealistically high frictional strengths (cf. e.g. the discussion in Lamontagne and Ranalli, 1996), and therefore increase the strength contrast when the lower crust is ductile and the mantle is brittle. On the other hand, higher than hydrostatic pore fluid pressures (due to the occurrence of trapped fluids of sedimentary or metamorphic origin) could significantly reduce the strength contrast for the same rheological combination.

3.3. Effects of crustal thickness variations Increasing the crustal thickness to 50 km increases the frictional strength at the Moho (and therefore indirectly favours the prevalence of ductile deformation mechanisms) and changes the geotherm (the surface heat flow corresponding to a given Moho temperature decreases). These factors do not change the main trends of the strength contrast for the various lithological combinations, but tend to increase Δσ by a factor ≤ 2. In terms of surface heat flow, the high absolute values of Δσ correspond to Qo ≤ 60–70 mW m −2; the lower temperature range where Δσ tends to zero (especially in extension) corresponds to Qo ≤ 50 mW m −2. In the case of thinned soft crust (20 km thick, with the rheology of wet quartzite) high values of strength contrast (max Δσ= 200 and 900 MPa in extension and compression, respectively) occur in the low temperature range (TM b 1000 K) when the deformation mechanism of the lithospheric mantle is brittle or LTP (the lower crust is ductile in the whole temperature range considered). The corresponding surface

488

G. Ranalli, M. Adams / Tectonophysics 609 (2013) 480–490

Fig. 6. Strength contrast for soft upper mantle/hard lower crust: (A) Wet peridotite/mafic granulite; (B) Wet peridotite/dry diabase.

heat flow is Qo b 120–140 mW m −2, with 1/2 to 2/3 of this value coming from below the Moho, a characteristic of pronounced extension and consequently non-steady state environment. 3.4. Possibility of high-pressure failure The parameters for this type of failure (see Table 2) have been estimated only for three broad categories of rocks (granite, gabbro, and peridotite), and the pore fluid pressure dependence is uncertain. In an attempt to evaluate the potential effect of high-pressure brittle failure, we have compared the strength predicted by Eq. (7) with frictional, PLC, and LTP critical stresses for a mafic lower crust overlying the lithospheric mantle. For the standard model, high-pressure fracture is predominant in the lowermost crust for TM b 700 and 900 K (wet and dry diabase, respectively), and in the uppermost mantle for TM b 900 and 1200 K (wet and dry peridotite, respectively). In the range in which it predominates, its effect is to reduce the absolute value of the strength contrast to values always Δσ ≤ 200 MPa. This is the consequence of the temperature dependence of strength in the

empirical equation, and the relative decrease of the strength increase with depth when compared with frictional failure. However, we consider these results tentative and have not included high-pressure fracture in the results shown in Figs 3–6. An analysis of the possible effects on the total strength of the lithosphere can be found in Zang et al. (2007) and Pauselli et al. (2010).

4. Conclusions It should be emphasised that the aim of the present paper is to provide an overview of “type situations” affecting the possible rheological contrast at the continental Moho, and not a solution for specific regions and tectonic environments. For this reason, we have chosen rheologies for both crust and mantle that probably bracket the range of realistic conditions. Furthermore, the uncertainties in creep parameters and the limited applicability of the LTP mechanism are additional sources of uncertainty affecting the results presented as a function of Moho temperature and/or corresponding surface heat flow. Despite these

G. Ranalli, M. Adams / Tectonophysics 609 (2013) 480–490

limitations, some conclusions can be drawn which will be useful in future geodynamic modelling: • In addition to brittle frictional failure and PLC, at least one – and possibly more – deformation mechanisms can play a role: LTP for the lithospheric mantle, and high-pressure fracture in both crust and mantle. These two mechanisms have the common consequence of decreasing the critical strength maxima within the lithosphere, and consequently of reducing the total strength of the lithosphere in a given tectonic environment (Jiménez-Díaz et al., 2012; Pauselli et al., 2010; Zang et al., 2007). On the other hand, the occurrence of linear (diffusion) creep within the lithosphere is limited to areas of very small grain size (shear zones) and is not likely to affect the strength contrast at the Moho (though it can considerably decrease the total strength of the lithosphere under particular conditions). • The implicit assumption in most geodynamic models that the strength contrast at the Moho is ≥0 (i.e., the uppermost mantle has strength higher than, or equal to, that of the lowermost crust), while valid in a wide range of cases, has no general applicability. We have shown that, when the lower crust has hard (mafic) rheology, the reverse can be true, especially in compressive tectonic regimes. The geodynamic consequences of this rheological layering remain to be explored. For instance, the occurrence of lithospheric delamination (where only the lower part of the lithosphere is subducted after being detached from the upper part) has been hypothesised to account for coupled extension–compression belts as the Apennines (cf. e.g. Channel and Mareschal, 1989, and the discussion in Pauselli et al., 2010). It is commonly assumed that the detachment horizon is located in the soft lower crust. If the lower crust is harder than the upper mantle, however, delamination can be envisaged to occur below, and not above, the Moho. • In all cases of lower crust softer than the mantle, the largest strength differences (up to the order of hundreds on MPa) occur in the low-to-intermediate range of Moho temperature and surface heat flow (roughly, TM ≤1000 K and Qo ≤ 70 mW m −2). Consequently, the occurrence of geodynamic processes requiring a soft lower crustal layer sandwiched between harder layers is subject to compositional and thermal constraints. Processes related to lower crustal flow such as two-level plate tectonics (Lobkovsky and Kerchman, 1991), relaxation of Moho topography (Ranalli, 1997), and channel flow in collisional orogens (Gervais and Brown, 2011; Grujic, 2006) can be envisaged to take place only within specified compositional and thermal limits. • The rheological stratification of the lithosphere affects a whole range of processes, from the possible location of delamination and necking levels to flexural thickness and total strength (cf., among many others, Cloetingh and Burov, 1996; Fernández and Ranalli, 1997; Mikhailov et al., 2010; Ranalli and Murphy, 1987; Watts and Burov, 2003). Considerations on the depth distribution of seismicity have lead Maggi et al. (2000) to postulate what they term the “crème brulée” model, in which there is no strong uppermost mantle layer. The opposed “jelly sandwich” model assumes a soft lower crust between harder upper crustal and uppermost mantle layers, and its validity has been strongly argued by Burov (2010) on the basis of rheological and geodynamic considerations. However, these two rheological models are not alternative but represent the limits of a possible range of lithosphere rheologies, as shown by Afonso and Ranalli (2004) on the basis of an estimation of relative total strengths of lower crust and lithospheric mantle. For soft lower crust (felsic or wet mafic) and low surface heat flow (low Moho temperature), the lithosphere has a “jelly sandwich” rheological structure; for hard lower crust and high surface heat flow it has a “crème brulée” structure. The continental lithosphere can therefore be expected to show large lateral variations in rheological stratification and total strength, as confirmed by observation (Tesauro et al., 2009a, b). The rheological contrast at the Moho parallels these trends, and is therefore a quick and simple

489

(because it does not require computation of strength envelopes and of total strength of the various layers) indicator of first-order rheological structure. As a final remark, it should be emphasised that the estimated rheological contrast at the Moho is an indicator of intrinsic rheological properties under fixed conditions (in the present case, strain rate). As such, and as any other parameterization of rheological properties, it is a necessary but not sufficient input for models of actively deforming lithosphere, where dynamic boundary conditions leading to stress redistribution must be considered (cf., among others, Ershov and Stephenson, 2006; Fernández and Ranalli, 1997; Mikhailov et al., 2010; Regenauer-Lieb et al., 2006; Schmalholz et al., 2009). The results presented in this paper therefore provide a valid semi-quantitative estimate of rheological contrast at the Moho only in steady-state, 1D situations (in other words, within the framework of classical strength envelopes). They illustrate an important aspect of Moho properties (in the long-term, zero frequency range), and thus complement seismic and attenuation properties (short-term, high frequency range), but cannot be construed to remain unchanged during the time evolution of geodynamic processes. Acknowledgments The work reported in this paper has been financially supported by a research grant from NSERC (Natural Sciences and Engineering Research Council of Canada) to G. R. We thank Manel Fernández and an anonymous reviewer for constructive suggestions that have helped in the preparation of the final version. References Afonso, J.C., Ranalli, G., 2004. Crustal and mantle strengths in continental lithosphere: is the jelly sandwich model obsolete? Tectonophysics 394, 221–232. Artemieva, I.M., 2011. The Lithosphere—An Interdisciplinary Approach. Cambridge University Press . (773 pp.). Artemieva, I.M., Mooney, W.D., 2001. Thermal thickness and evolution of Precambrian lithosphere: a global study. Journal of Geophysical Research 106, 16387–16414. Behn, M.D., Hirth, G., Elsenbeck II, J.R., 2009. Implications of grain size evolution on the seismic structure of the oceanic upper mantle. Earth and Planetary Science Letters 282, 178–189. Burov, E.B., 2010. The equivalent elastic thickness (Te), seismicity and the long-term rheology of continental lithosphere: time to burn-out “crème brulée”? Insights from large-scale geodynamic modeling. Tectonophysics 484, 4–26. Carter, N.L., Tsenn, M.C., 1987. Flow properties of continental lithosphere. Tectonophysics 136, 27–63. Čermák, V., 1989. Crustal heat production and mantle heat flow in Central and Western Europe. Tectonophysics 159, 195–215. Channel, J.T.H., Mareschal, J.-C., 1989. Delamination and asymmetric lithospheric thickening in the development of the Tyrrhenian Rift. In: Coward, M.P., Dietrich, M.P.D., Park, R.G. (Eds.), Alpine Tectonics: Geol. Soc. Lond. Sp. Publ., 45, pp. 285–302. Chopra, P.N., Paterson, M.S., 1981. The experimental deformation of dunite. Tectonophysics 78, 453–473. Chopra, P.N., Paterson, M.S., 1984. The role of water in the deformation of dunite. Journal of Geophysical Research 89, 7861–7876. Cloetingh, S., Burov, E.B., 1996. Thermomechanical structure of European continental lithosphere: constraints from rheological profiles and EET estimates. Geophysical Journal International 124, 695–723. Drury, M.R., 2005. Dynamic recrystallization and strain softening of olivine aggregates in the laboratory and in the lithosphere. In: Gapais, D., Brun, J.-P., Cobbold, P.R. (Eds.), Deformation Mechanisms, Rheology and Tectonics: From Minerals to the Lithosphere: Geol. Soc. Lond. Sp. Publ., 243, pp. 143–158. Ershov, A.V., Stephenson, R.A., 2006. Implications of a visco-elastic model of the lithosphere for calculating yield strength envelopes. Journal of Geodynamics 42, 12–27. Faul, U.H., Jackson, I., 2005. The seismological signature of temperature and grain size variations in the upper mantle. Earth and Planetary Science Letters 234, 119–134. Faul, U.H., Jackson, I., 2007. Diffusion creep of dry, melt-free olivine. Journal of Geophysical Research 112, B04204 http://dx.doi.org/10.1029/2006JB004586. Faul, U.H., Fitz Gerald, J.D., Farla, R.J.M., Ahlefeldt, R., Jackson, I., 2011. Dislocation creep of fine-grained olivine. Journal of Geophysical Research 116, B01203. http://dx.doi.org/ 10.1029/2009JB007174. Fernández, M., Ranalli, G., 1997. The role of rheology in extensional basin formation modelling. Tectonophysics 282, 129–145. Gervais, F., Brown, R.L., 2011. Testing modes of exhumation in collisional orogens: synconvergent channel flow in the southeastern Canadian cordillera. Lithosphere 3, 55–75.

490

G. Ranalli, M. Adams / Tectonophysics 609 (2013) 480–490

Goetze, C., 1978. The mechanisms of creep in olivine. Philosophical transactions of the Royal Society of London. Series A: mathematical and physical sciences 288, 99–119. Grujic, D., 2006. Channel flow and continental collision tectonics: a review. In: Law, R.D., Searle, M.P., Godin, L. (Eds.), Channel Flow, Ductile Extrusion and Exhumation in Continental Collision Zones: Geol. Soc. Lond. Sp. Publ., 268, pp. 25–37. Hansen, L.N., Zimmerman, M.E., Kohlstedt, D.L., 2011. Grain boundary sliding in San Carlos olivine: flow law parameters and crystallographic-preferred orientation. Journal of Geophysical Research 116, B08201. http://dx.doi.org/10.1029/2011JB008220. Hasterok, D., Chapman, D.S., 2011. Heat production and geotherms for the continental lithosphere. Earth and Planetary Science Letters 307, 59–70. Jaupart, C., Mareschal, J.-C., 2010. Heat Generation and Transport in the Earth. Cambridge University Press . (476 pp.). Jiménez-Díaz, A., Ruiz, J., Villaseca, C., Tejero, R., Capote, R., 2012. The thermal state and strength of the lithosphere in the Spanish Central System and Tajo Basin from crustal heat production and thermal isostasy. Journal of Geodynamics 58, 29–37. Karato, S.-I., 2008. Deformation of Earth Materials—An Introduction to the Rheology of the Solid Earth. Cambridge University Press . (474 pp.). Keefner, J.W., Mackwell, S.J., Kohlstedt, D.L., Heidelbach, F., 2011. Dependence of dislocation creep of dunite on oxygen fugacity: implications for viscosity variations in Earth's mantle. Journal of Geophysical Research 116, B05201. http://dx.doi.org/ 10.1029/2010JB007748. Kirby, S.H., Kronenberg, A.K., 1987. Rheology of the lithosphere: selected topics. Reviews of Geophysics 25, 1219–1244. Kohlstedt, D.L., Evans, B., Mackwell, S.J., 1995. Strength of the lithosphere: constraints imposed by laboratory experiments. Journal of Geophysical Research 100, 17587–17602. Korenaga, J., Karato, S.-I., 2008. A new analysis of experimental data on olivine rheology. Journal of Geophysical Research 113, B02403. http://dx.doi.org/10.1029/2007JB005100. Lamontagne, M., Ranalli, G., 1996. Thermal and rheological constraints on the earthquake depth distribution in the Charlevoix, Canada, intraplate seismic zone. Tectonophysics 257, 55–69. Lobkovsky, L.I., Kerchman, V.I., 1991. A two-level concept of plate tectonics: application to geodynamics. Tectonophysics 199, 343–374. Mackwell, S.J., Zimmerman, M.E., Kohlstedt, D.l., 1998. High-temperature deformation of dry diabase with application to tectonics on Venus. Journal of Geophysical Research 103, 975–984. Maggi, A., Jackson, J.A., McKenzie, D., Priestley, K., 2000. Earthquake focal depths, effective elastic thickness, and the strength of the continental lithosphere. Geology 28, 495–498. Mahatsente, R., Ranalli, G., Bolte, D., Götze, H.J., 2012. On the relation between lithospheric strength and ridge push transmission in the Nazca plate. Journal of Geodynamics 53, 18–26. Mei, S., Suzuki, A.M., Kohlstedt, D.L., Dixon, N.A., Durham, W.B., 2010. Experimental constraints on the strength of the lithospheric mantle. Journal of Geophysical Research 115, B08204. http://dx.doi.org/10.1029/2009JB006873.

Mikhailov, V., Stephenson, R., Diament, M., 2010. Modelling of compression and extension of the continental lithosphere: towards rehabilitation of the necking model. Journal of Geodynamics 50, 368–380. O'Reilly, S.Y., Griffin, W.L., 2013. Moho vs. crust-mantle boundary: evolution of an idea. Tectonophysics 609, 535–546 (this volume). Ord, A., Hobbs, B.E., 1989. The strength of the continental crust, detachment zones and the development of plastic instabilities. Tectonophysics 158, 269–289. Pauselli, C., Ranalli, G., Federico, C., 2010. Rheology of the Northern Apennines: lateral variations of lithospheric strength. Tectonophysics 484, 27–35. Ranalli, G., 1995. Rheology of the Earth, 2nd edn. Chapman & Hall, London . (413 pp.). Ranalli, G., 1997. Rheology of the lithosphere in space and time. In: Burg, J.-P., Ford, M. (Eds.), Orogeny through Time: Geol. Soc. Lond. Sp. Publ., 121, pp. 19–37. Ranalli, G., Murphy, D.C., 1987. Rheological stratification of the lithosphere. Tectonophysics 132, 281–295. Regenauer-Lieb, K., Weinberg, R.F., Rosenbaum, G., 2006. The effect of energy feedbacks on continental strength. Nature 442, 67–70. Renshaw, C.E., Schulson, E.M., 2007. Limits on rock strength under high confinement. Earth and Planetary Science Letters 258, 307–314. Rutter, E.H., Brodie, K.H., 1992. Rheology of the lower crust. In: Fountain, D.M., Arculus, R., Kay, R.W. (Eds.), Continental Lower Crust. Elsevier, Amsterdam, pp. 201–267. Schmalholz, S.M., Kaus, B.J.P., Burg, J.-P., 2009. Stress–strength relationship in the lithosphere during continental collision. Geology 37, 775–778. Shelton, G., Tullis, J., 1981. Experimental flow laws for crustal rocks (Abstract). EOS, Transactions of the American Geophysical Union 62, 396. Shimada, M., 1993. Lithosphere strength inferred from fracture strength of rocks at high confining pressure and temperature. Tectonophysics 217, 55–64. Sibson, R.H., 1974. Frictional constraints on thrust, wrench and normal faults. Nature 249, 542–544. Tesauro, M., Kaban, M.K., Cloetingh, S.A.P.L., 2009a. How rigid is Europe's lithosphere? Geophysical Research Letters 36, L16303. http://dx.doi.org/10.1029/2009GL039229. Tesauro, M., Kaban, M.K., Cloetingh, S.A.P.L., 2009b. A new thermal and rheological model of the European lithosphere. Tectonophysics 476, 478–495. Vilà, M., Fernández, M., Jiménez-Munt, I., 2010. Radiogenic heat production variability of some common lithological groups and its significance to lithospheric thermal modelling. Tectonophysics 490, 152–164. Watts, A.B., Burov, E.B., 2003. Lithospheric strength and its relationship to the elastic and seismogenic layer thickness. Earth and Planetary Science Letters 213, 113–131. Wilks, K.R., Carter, N.L., 1990. Rheology of some continental lower crustal rocks. Tectonophysics 182, 57–77. Zang, S.X., Wei, R.Q., Ning, J.Y., 2007. Effect of the brittle fracture on the rheological structure of the lithosphere and its applications in the Ordos. Tectonophysics 429, 267–285.