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Geology; September 2009; v. 37; no. 9; p. 775-778; DOI: 10.1130/G25678A.1
© 2009 Geological Society of America
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Stress-strength relationship in the lithosphere during continental collision

Stefan M. Schmalholz1,*, Boris J.P. Kaus1,2 and Jean-Pierre Burg1

1 Department of Earth Sciences, ETH Zurich, Sonneggstrasse 5, 8092 Zürich, Switzerland
2 Department of Earth Sciences, University of Southern California, Los Angeles, California 90089, USA

Correspondence: *E-mail: schmalholz{at}erdw.ethz.ch.


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 METHOD AND SETUP
 RESULTS
 DISCUSSION
 CONCLUSIONS
 REFERENCES CITED
 
Lithospheric strength profiles generated for a shortening continental lithosphere generally predict excessively high differential stresses in the sub-Moho continental mantle; this seems inconsistent with the relative scarcity of earthquakes at this depth. This inconsistency was put forward as evidence for weak mantle rheology. However, this argument implicitly assumes that strength envelopes are valid in actively deforming regions. We test this assumption on two end-member model lithospheres having identical upper crustal rheologies, but with (1) a weak lower crust and strong mantle, and (2) a strong lower crust and weak mantle. For this purpose, we compare one-dimensional (1-D) with 2-D visco-elastoplastic numerical models of continental shortening. The 2-D models show that strongly heterogeneous deformation typically follows initially homogeneous deformation. Lithospheric-scale buckle folds and shear zones result in strain rate variations of as much as three orders of magnitude. Differential stresses in the upper crust are close to yield, as predicted by 1-D models. Stresses in deeper lithospheric regions, however, are significantly smaller than in 1-D models, especially in actively deforming regions. Systematic numerical simulations as a function of temperature and deformation rate reveal that 1-D models are reliable in hot and/or slowly deforming lithospheres only. The relative scarcity of earthquakes at mantle levels should thus be interpreted as an intrinsic consequence of strong lithospheric deformation rather than as evidence for a weak upper mantle.


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
METHOD AND SETUP
 RESULTS
 DISCUSSION
 CONCLUSIONS
 REFERENCES CITED
 
The deformation of continents departs significantly from the principles of plate tectonics, which state that deformation is localized at plate boundaries. By contrast, continental deformation is often widely distributed and can generate vast areas of high elevation such as the Tibetan Plateau, north of the India-Asia collision zone. The dominating deformation style (e.g., homogeneous shortening and/or thickening or localized faulting) and its spatial and temporal distribution remain matters of dispute (e.g., Tapponnier et al., 2001; Royden et al., 2008). In that respect, the magnitude and spatial distribution of strength in the continental lithosphere are key parameters. Since none of these parameters is directly measurable, there are controversial views on whether the strength of the continental lithosphere resides essentially in the crust or is distributed between the upper crust and upper mantle (Maggi et al., 2000; Jackson, 2002; Watts and Burov, 2003; Burov and Watts, 2006).

Part of the argument in this discussion is based on one-dimensional (1-D) yield strength envelopes (Goetze and Evans, 1979; Kohlstedt et al., 1995). These plots of differential stress versus depth are constructed assuming a specific geotherm, rock composition, and a constant strain rate. Domains with high differential stress are considered stronger than domains with lower differential stress. Hence, the relative scarcity of earthquakes below the Moho seems inconsistent with the belief that differential stress in the mantle lithosphere is close to yielding, and this has therefore been taken as evidence for a weak mantle lithosphere (Jackson, 2002). However, this interpretation implicitly assumes that 1-D strength envelopes are valid in deforming regions, and excludes effects such as lateral and vertical variations in differential stress and strain rate.

Dissipation of energy (i.e., shear heating; Schubert and Yuen, 1978; Brun and Cobbold, 1980) increases temperature and reduces differential stresses in the strongest layers of 1-D strength envelopes (Braeck and Podladchikov, 2007; Hartz and Podladchikov, 2008). Two-dimensional models have shown that energy dissipation might result in lithospheric-scale failure, which produces marked lateral variations of differential stresses (Regenauer-Lieb et al., 2006). There are few investigations, however, about the applicability of 1-D strength envelopes to actively shortening continental lithospheres. We expand the 1-D models to full 2-D models, allowing for vertical and lateral variations in strain rate and comparing 1-D and 2-D numerical simulations to understand the style of lithospheric deformation with respect to its strength distribution. The 2-D results indicate that differential stresses in deep levels of an intensely deforming lithosphere are far below their yield strength. We conclude that the scarcity of earthquakes in these levels indicates intense deformation rather than a weak rheology.


   METHOD AND SETUP
TOP
 ABSTRACT
 INTRODUCTION
 METHOD AND SETUP
RESULTS
 DISCUSSION
 CONCLUSIONS
 REFERENCES CITED
 
In both the 1-D and 2-D simulations, we consider a continental lithosphere compressed under constant strain rate of 3 x 10–15 s–1 (Fig. 1; Table 1), such that 1-D and 2-D results are directly comparable.


Figure 01
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  		Figure 1 A: Model setup for both one-dimensional (1-D) and 2-D models. 1-D model corresponds to vertical line across model lithosphere. See Table 1 for model parameters. In 1-D models, bottom temperature is constant; in 2-D models it is increased in right side of model. In WLC model, basal temperatures are T1 = 1170 °C and T2 = 1200 °C, and in SLC model, T1 = 1200 °C and T2 = 1230 °C. B: Representative distribution of differential stress and temperature for WLC model (without shear heating). C: As in B, but for SLC model. Note that brittle yield stresses for applied Mohr-Coulomb friction angle of 30° are lower than yield stresses predicted by Byerlee law, thus accounting for effects of fluid pressure.

 


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  		TABLE 1 APPLIED ROCK-PHYSICAL PARAMETERS AND RHEOLOGICAL EQUATIONS*

 
The equations describing the visco-elastoplastic rheology (Table 1) and the temperature evolution with shear heating in the 1-D model are 1-D versions of the 2-D equations used in Burg and Schmalholz (2008). The crust has Mohr-Coulomb plasticity and the upper mantle has low-temperature plasticity (Goetze and Evans, 1979). Both crust and lithosphere thicken with progressive shortening. Temperature is continuously calculated and coupled with the effective viscosity and the shear heating term.

The 2-D code is based on the Lagrangian finite element method outlined in Burg and Schmalholz (2008). The difference is the low-temperature plasticity applied here for the mantle (Table 1). Kinematic boundary conditions are applied at both model sides, specifying a horizontal velocity that maintains the constant bulk shortening rate (Fig. 1). Only the far-field bulk shortening rate is constrained; strain rates can evolve freely within the model. Models are sufficiently thick to maintain isostasy. The bottom temperature has a step at the model center (Fig. 1); this step represents any geometrical, thermal, rheological, or compositional heterogeneity in the lithosphere. It is useful to allow for potential deviations from the homogeneous pure shear resulting from the boundary conditions in case of no perturbation (i.e., nonrealistic, perfectly horizontal boundary between crust and mantle). The numerical resolution is 401 (horizontal) x 161 (vertical) nodes. For a bulk shortening of 25% ~1000 time steps are calculated. Resolution tests demonstrated that this is sufficient. For a background strain rate of 3 x 10–15 s–1, a bulk shortening of 25% requires ~3 Ma. The three lithospheric layers (upper crust, lower crust, and mantle) are visco-elastoplastic, yet with different rheological parameters (Table 1). The upper crust parameters are identical for all simulations. Two end members are considered. The weak lower crust model has a weak lower crust and a strong upper mantle, and the strong lower crust model has a strong lower crust and a weak upper mantle (Fig. 1; Table 1). The initial thickness of the lithosphere is 120 km and that of the lower crust 10 km. A slightly thicker initial thickness of upper crust (28 km instead of 25 km) and a 30 °C higher bottom temperature cause the relative weakness of the upper mantle in the strong lower crust model compared to the weak lower crust model (Fig. 1). The initial temperature field is in equilibrium and the initial surface heat flow is ~60 mW/m2 for all simulations.


   RESULTS
TOP
 ABSTRACT
 INTRODUCTION
 METHOD AND SETUP
 RESULTS
DISCUSSION
 CONCLUSIONS
 REFERENCES CITED
 
The second invariant of the strain rate tensor (EII) and the stress tensor (i.e., the differential stress {Delta}{sigma}) are used to visualize strain rate and stress variation. The spatial variations of EII and {Delta}{sigma} increase during shortening of the 2-D models (Fig. 2) because the bottom temperature step triggers processes such as low-amplitude buckling and shear localization. Shear zones develop in both the weak and strong lower crust models; strain localization is more efficient in the strong lower crust model due to activation of Mohr-Coulomb failure in the strong lower crust (Fig. 2D). Shear zones in the mantle of the weak lower crust model are less localized because of the low-temperature plasticity (Fig. 2B). Values of {Delta}{sigma} near the Moho vary laterally and are smallest in regions of largest EII values (Figs. 2C and 2F).


Figure 02
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  		Figure 2 Distribution of second invariant of strain rate tensor (EII) and differential stress ({Delta}{sigma}) at different amounts of bulk shortening (percent value at lower left of each graph with corresponding time of shortening). Solid white line indicates Moho and dashed white line indicates upper crust–lower crust boundary. Color scale for EII is same for graphs A, B, D, and E for {Delta}{sigma} is the same for graphs C and F. A–C show results for model with weak lower crust and strong upper mantle (WLC) and D–F show results for model with strong lower crust and weak upper mantle (SLC). Dotted vertical lines show location where profiles of EII and {Delta}{sigma} have been calculated (Figs. 3 and 4).

 
Profiles of {Delta}{sigma} versus depth in the middle of the 2-D models (vertical dotted line in Figs. 2C and 2F) are notably different from those predicted by 1-D models (Figs. 3A, 3C). Values of {Delta}{sigma} in the lower crust of the strong lower crust model are much reduced (Fig. 3C). The Moho of the 2-D weak lower crust model is ~10 km deeper than that of the 1-D model, whereas the upper crustal thickness of both models is similar, indicating heterogeneous thickening due to folding (Fig. 3A). For comparison, values of {Delta}{sigma} resulting from 1-D models are also shown for cases without shear heating, which yield considerably larger values. Without shear heating the 1-D models predict {Delta}{sigma} values at yield in the lower crust for the strong lower crust model (Fig. 3C) and in the upper mantle for the weak lower crust model (Fig. 3A). Values of EII vary up to three orders of magnitude along a vertical line through the lithosphere in the 2-D model (Figs. 3B and 3D).


Figure 03
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  		Figure 3 A and C: Profiles of {Delta}{sigma} and temperature (T) with depth for one-dimensional (1-D) and 2-D models for both weak (WLC) and strong (SLC) lower crust. 1-D and 2-D results agree well for upper crust. Values of {Delta}{sigma} are significantly smaller in 2-D compared to 1-D models. 1-D results without shear heating predict significantly higher stresses and lower temperatures in upper half of lithosphere (dashed lines). For 2-D SLC model, maximum increase in temperature due to shear heating is ~150 °C compared to 1-D model (at depth of ~60 km in 3C). B and D: Profiles of second invariant of the strain rate tensor (EII) versus depth. Values of EII vary orders of magnitude in 2-D models, indicating formation of detachment levels (high values of EII).

 
The vertically integrated differential stress (Fx) measures lithospheric strength (e.g., Molnar and Lyon-Caen, 1988). Fx was calculated along the middle (vertical dotted line) of both the 2-D weak and strong lower crust models (Figs. 2C and 2F). For all models, values of Fx decrease after increasing (Fig. 4A). The increase is caused by viscoelastic stress buildup, during which 1-D and 2-D models agree well. Decrease in strength is due to combined effects of (1) crustal thickening, resulting in enhanced temperatures in the strong mantle and lower crust, (2) structural softening, i.e., a natural consequence of folding that requires less strain energy than pure shear shortening (Schmalholz et al., 2005), and (3) softening due to shear heating (Regenauer-Lieb et al., 2006; Hartz and Podladchikov, 2008). The decrease of Fx is more pronounced in 2-D than in 1-D models (Fig. 4A). The value of Fx decreases significantly in the middle of the 2-D strong lower crust model after ~20% bulk shortening due to the development of a large-scale shear zone (Fig. 2E).


Figure 04
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  		Figure 4 A: Evolution of vertically integrated differential stress across entire lithosphere (Fx) with bulk shortening for one-dimensional (1-D) models and in middle of 2-D models (see vertically dotted line in Figs. 2C, 2F). B: Ratio of mantle stresses predicted by 1-D models to mantle stresses predicted by 2-D models (R) for different values of initial Moho temperatures (TMOHO) and bulk shortening rates (EB) for weak lower crust (WLC) model. Depending on TMOHO and EB, lithospheric deformation is dominated either by thickening (circles), folding (diamonds), or thrusting (crosses; see text for detailed explanation). Differential stresses in mantle predicted by 1-D models can be as much as five times larger than actual stresses in regions of active deformation (i.e., R = 0.2). SLC—strong lower crust.

 
To further understand the difference between 1-D and 2-D models, we performed systematic numerical simulations for the 2-D weak lower crust model as a function of bottom temperature and shortening rate (Fig. 4B). In these models, mantle strength is quantified by the value of the differential stress integrated vertically across the uppermost 10 km of the mantle (FxM). The ratio of the value of FxM for the 2-D model to the value of FxM for the 1-D model [R = FxM(2-D)/FxM(1-D)] represents the difference of mantle strength predicted by 1-D and 2-D models (R = 1 indicates perfect agreement between 1-D and 2-D models and R = 0.5 indicates that stresses in 2-D models are 50% smaller than in 1-D models). R was calculated after a bulk shortening of 35% or when the 2-D model failed due to the development of a large-scale shear zone like that in Figure 2E. The FxM(2-D) value has been averaged for vertical profiles located between horizontal positions 5 km and 50 km in the 2-D model (i.e., the middle-right region of the 2-D model; Fig. 2B). For hot initial Moho temperatures (TMOHO) and slow shortening rates (EB), lithospheric shortening mainly causes homogeneous thickening, and mantle strengths predicted by 1-D and 2-D models fit well (i.e., R {approx}1; Fig. 4B). For cold TMOHO and fast EB, deformation is characterized by large-scale shear zones (thrusting) and FxM(2-D) values are as much as five times smaller than FxM(1-D) values (0.2 < R < 0.5). Between the two end-member deformation styles, thickening and thrusting, folding dominates lithospheric deformation (Fig. 4B). Folding is here defined by maximum topographic differences >2 km after 35% shortening without generation of large-scale shear zones.


   DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 METHOD AND SETUP
 RESULTS
 DISCUSSION
CONCLUSIONS
 REFERENCES CITED
 
Thin-sheet models applied to the India-Asia collision zone require Fx values between 1.5 x 1013 N m–1 and 2.5 x 1013 N m–1 to fit observed distributions of crustal thickness, stress, and strain rate, and to drive deformation (England and Houseman, 1986). Maximum values obtained here consistently are ~2.2 x 1013 N m–1 (Fig. 4A). Small differential stresses in a strong mantle lithosphere or lower crust can be produced if intense deformation induces local heating. Even if shear heating does not yield localized shear deformation, it has a fundamental impact in reducing differential stresses in actively deforming lithospheric regions (Fig. 3; Hartz and Podladchikov, 2008).

In our 2-D models the continental crust and lithospheric mantle are deformed together, generating relief of as much as 20 km on the Moho depth (Figs. 2C and 2F). This topography agrees with Moho depth variations inferred from seismic images beneath western Tibet (e.g., Wittlinger et al., 2004). In all models, upper crustal stresses are close to the brittle yield stress.

The largest {Delta}{sigma} deviations from 1-D models occur in deep levels of lithospheres whose Moho has marked topography (Figs. 2C and 2F). The 2-D results show that stress reduction below yield and subsequent scarcity of seismogenic fracturing occur naturally in 2-D models of continental collision. This is consistent with reports from the India-Asia collision zone, where earthquakes occur throughout the cratonic Indian lithosphere to the south of the collision zone, while few earthquakes occur in the Indian lower crust and upper mantle underneath Tibet (Priestley et al., 2008). The models presented here are also consistent with earthquake locations in the Swiss Alps (e.g., Deichmann, 1992, 2003). Earthquakes occur at all levels of the European crust to the north of the Alps but are restricted to the upper crust in regions with recognized Moho topography (Waldhauser et al., 1998) and most intense orogenic deformation (Burg and Gerya, 2005). Consequently, the lack of seismicity (assuming slip weakening) should not be taken as evidence for unusually weak mantle lithosphere or lower crust, but rather as an indication of small differential stresses.

During lithospheric shortening, many other weakening and feedback mechanisms might be active (e.g., strain softening; Ellis et al., 2001). These mechanisms have been ignored here, and our results can therefore be considered to be conservative. Nevertheless, our models reproduce the first-order lithospheric deformation modes (1) thickening, (2) buckling, and (3) thrusting on localized shear zones. This reiterates the importance of shear heating and visco-elastoplastic rheologies for lithospheric deformation (Kaus and Podladchikov, 2006; Burg and Schmalholz, 2008).


   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 METHOD AND SETUP
 RESULTS
 DISCUSSION
 CONCLUSIONS
REFERENCES CITED
 
Shortening of the continental lithosphere with a constant bulk shortening rate can generate strain rate variations of as much as three orders of magnitude due to the development of low-amplitude buckle folds and shear zones in a wide variety of lithospheres, from those with strong upper mantle and weak lower crust to those with weak upper mantle and strong lower crust. Crustal-scale folds and shear zones are typical structures of continent collision zones. Models show that intensely deforming regions exhibit the smallest differential stresses, implying that the strength of the continental lithosphere varies drastically with strain. One-dimensional strength envelope models are appropriate to predict differential stresses in the upper crust and in regions with relatively homogeneous deformation, but tend to strongly overestimate the strength of both the upper mantle and lower crust in actively deforming areas. The main reasons for this are that most 1-D strength envelope models neglect homogeneous shear heating and do not include 2-D effects such as localized shear heating, strain localization, and structural softening. The 1-D and 2-D models agree well only in cases of very hot lithospheres and in cases with slow background deformation rates, which both yield deformation patterns dominated by pure shear thickening rather than by laterally heterogeneous deformation.

We conclude that for realistic parameters, differential stresses in the upper mantle lithosphere or in a strong lower crust (1) vary significantly and (2) are most likely far below their yield strength in an actively deforming continental lithosphere. Therefore, the scarcity of deep earthquakes in areas of intense lithospheric deformation is likely the consequence of a differential stress reduction (far below yield) caused by both shear heating and structural softening in a basically strong mantle, where differential stresses could be as high as ~1000 MPa. If so, the scarcity of deep earthquakes is not indicative of a weak mantle lithosphere.


   ACKNOWLEDGMENTS
 
We thank Susan Ellis, Ebbe Hartz, and an anonymous reviewer for helpful and constructive reviews. This work has been supported by the ETH Zurich.


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 METHOD AND SETUP
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 DISCUSSION
 CONCLUSIONS
 REFERENCES CITED
 

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Received for publication 30 November 2008

Revised manuscript received 19 March 2009

Manuscript accepted 9 April 2009





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